text: how do you type a division sign on a chromebook. You can currently format numbers 0-9, basic mathematical symbols (=, +, – ), and alphabets using this function. Excel offers the IMSQRT () and IMPOWER () functions to return roots and. To do that, we'll switch the numerator and denominator. XtraMath needs your help! XtraMath ® is a 501(c)(3) non-profit dedicated to math achievement for all. How to type a fraction on a chromebook. - You can rearrange your server list by dragging server icons up and down. Long Division Symbol -- from Wolfram MathWorld Search. Insert the numbers into the fraction boxes. About Division Google Docs Long Symbol In. > From the Region and Language box, go to the Keyboards and Languages tab. how to type radical sign on google docshow to type radical sign on google docs. Shift + click a button to insert its upper-case form. To type the peso sign (₱) using your computer keyboard, hold down the Alt key while pressing 8369 on the number pad. Both in Office for Windows and Office for Mac. In case you are not sure what to do or don't want to dive into the BIOS configuration stuff, you can check if the keyboard layout and input language is set to English US. Create a document in Word - support. Or, for practice using Word features, try a learning guide like Welcome to Word, …. toucan language login » heat intolerance hyperthyroidism » what is kernel-level access. Just obey the following instructions: Place the insertion pointer where you need to type the division symbol. Whilst holding on to the Alt key, press the symbol’s Alt Code ( 0247) on the numeric keypad. If you want fancier lines, try these tricks: 1. ” x divided by 3 ” you would type x/3. How to type exponents on Mac — in native mac programs like. Press the 2 key on your Number pad. It’s WAY quicker than clicking and typing in the dollar ($) signs to change a reference into an absolute reference. Microsoft uses a variant of ↵, with long horizontal line. Each signed-in user can set their own display language. You can type the degree ° symbol on a PC keyboard you can try one of these methods: Hold down the ALT key and type 0176 or 248 on the numeric keypad Find the symbol you want to type in the keyboard. Just type Ctrl+Shift+u at the same time, release them, type in the 4-character unicode, then press the space bar. Using this code, you can add any symbol to your document if you know the. About Long In Google Symbol Division Docs. In the ribbon at the top click on “Insert’. To type a division symbol using the keyboard, press Alt-0247 on the numeric keypad with Num Lock turned on; Where is the divide button on a Chromebook? How do you type a division sign on your keyboard – Chromebook Community. To do so, it will depend on the power used: 1. How do you write 2 divided by 3? Answer: 2 divided by 3 as a fraction is 2/3. If you … ALT Codes for Math Symbols: Multiplication & Division Sign Operators Read More ». In the Device section, select Keyboard. There is no easy way to type in the division symbol, but you can copy paste this one: ÷ Typically when you write mathematical forumulas on a computer, you simply use the / forward slash to denote a devision between two numbers and round brackets to …. edu and click on the G Suite link Enter your student ID number and click Next Example: [email protected] Out of all these Chromebook keyboard. Insert the degree symbol by using the ribbon. Get products · Sign up for updates . Consumers should immediately stop using the recalled charger for the HP Chromebook 11 and contact Google for a free replacement charger. Undoubtedly one of the most useful Google Sheets formula shortcuts to learn. Answer (1 of 2): How can I type \text{divisor}\,)\overline{\,\text{dividend}\,\,} (the long division gazinta symbol) on a computer? I’ve often lamented the lack of. Note: If you do not see a secondary menu, click the option for Language Bar. To get division symbol (÷) in Ms Word, use Alt + 246 or Math Autocorrect shortcut "\div". To type just the modifier key by itself, type the following the modifier key. If you are looking for how you can type this …. Accuro’s transcription service is provided in any style or format you want. How to type infinity symbol on chromebook. Student 1 writes the Contract and Purpose for Problem A, Student 2 writes the Contract and Purpose for Problem B, and so on. new" into a new tab's address bar. We’re wanting the current balance to show up below the existing one of$10,000. How to type ¢ Cent Sign symbol from keyboard?. I type \delta and it immediately turns into: δ. ” This tool does not provide medical advice. MIKE'S TIP: While you might not personally have to answer all of these Google interview questions, it doesn’t hurt to spend a moment consider each one (barring the technical ones if you’re trying to land a non-tech role). How do you insert a division symbol in Word on a Mac? On Mac, press “Option + /” keys to type division sign ÷. It may be crude, but it is a way to do what is being asked. Press and hold the ALT key and type 0176 on your keyboard. Microsoft Word Only Keyboard Shortcut In Microsoft Word only there is an additional shortcut that will let you type the does not equal sign – just type 2260 and then press Alt+x and the ≠ sign will substitute for the numbers. Go to System Preferences > Keyboard > Text. At that point, enter the Unicode for the symbol you want . The on-screen keyboard aka virtual or touchscreen keyboard, as the name suggests, appears on your Chromebook's screen. A much faster way to do this is to select the text and then …. Alt + click a button to copy a single character to the clipboard. interpunct) is often used as a list separator, or in e. In the Search screen, in the Search field …. Drag your touch to select the desired unique character. For example, to determine if the value in cell B2 is greater than five, you could use the formula =B2>5. The Insert, Symbol command gives you access to a wide variety of superscripts. The option to print a tax return becomes visible only after you pay for that specific return. Scroll through the symbols, and click the degree sign. The numerator is the top number in the fraction. For those of you who've skimmed all this and are just looking for the solution, here it is:How to Type Special Characters with a Chromebook: Unicode InputOn a Chromebook, when you press CTRL+SHIFT+U in an input field, you'll get a little underlined "u" on your screen. Division sign on keyboard isn't something like you studied in school. The same key used to type an apostrophe (the single straight tick mark) is used for a prime. In every group, each student is given their own word problem. Fn + Delete (ideapad 100-14IBY): Fn + Insert (ideapad 100-15IBY): Enables/disables the scroll lock. Press ctrl+shift+u and an underlined u will appear. Find the Right Product and Accessories To Fulfill Your Requirement. Then, while continuing to hold down on the touchpad, drag your finger in the direction you want to highlight text. About A How Type A Chromebook Division Sign Do On You. Type in the number you want to format as an exponent. In the "Symbols" section, choose "Equation". Save anything on the web to OneNote. m + [space key] for ם Type an apostrophe ['] to add the daguesh: ex. To open an existing file or to create a new file, type nano followed by the file name: nano filename. Make sure you switch on the Num Lock from the keyboard and you type the number from the Numpad and not from the top row of the keyboard. The “pitch,” or the distance center to center between two …. dictionaries for separating parts of a word (though in Unicode that is technically a different, but most often identical looking character: ‧ ). You can copy and paste this sign anywhere you need to. How do you type a division sign on your keyboard best support. Converting decimal numbers to a fraction: Yes, we can use excel to directly convert decimal points into fractions. Here’s how to type exponents on a Chromebook by using Google Docs: Open your file in Google Docs. How students can join a Zoom meeting within Schoology Step 1 - Sign in to Chrome with your fcpsschools. Search: Chromebook Keyboard Symbols. All formulas in Excel begin with an equal sign, so the first symbol there is introducing the formula, not specifying anything about equality. Create your Star Stable Account. To type the degree symbol on a windows PC using the shortcut, obey the following instructions: Enable Num Lock by pressing the Num Lock key. Even if you want the former one Type Alt-246. All you will have to do is make sure the exponent you type in is represented as an exponent using special symbols. keyboard On English PC and Mac keyboards, the less than is on the same key as the comma. If you do it correctly, you will find that everything on the page is highlighted. For the division symbol, the Unicode is "00f7". You should now see the Division symbol on the character map dialog. Anytime, when you type the text in the Word document, you can switch to the Symbol font and use the corresponding Latin letters to enter Greek letters:. How to Type the Peso Sign on a Windows Computer. Aadhya Aaliyah Abbie Ada Adalyn Adelaide Adele Adilah Adriana Adrienne Agafya Agata Agathe Aglaya Agnes Agrippa Agustina Aisha Aitana Aixa Akiko Alanis Alba Alda. In the case of Chromebooks, keyboard shortcuts can help new users cope with some of the differences between the design of the Chromebook and other systems, such as Macs and Windows PCs. To insert math symbol type Ctrl+M (Command M on Mac OS) and then type symbol mentioned in the table below. Click on the Start menu and type "partitions. Our goal is to help you work faster in Excel. The 'u00f7' that you just entered will magically turn into ÷. For instance, if you want to type a degree symbol by using the ALT Code on the keyboard. ; If using Microsoft Word, in the Symbols pop-down menu, click the More Symbols option. EquatIO does not have symbol buttons you press like CodeCOGS. List of Division Sign signs, make over 8 division sign symbols text character. how to type an exponent on google chrome. You can type a lot of characters that may not have a corresponding key on your keyboard – such as European language alphabetic characters, ASCII symbols, and even Chinese characters (also known as Hanzi, Kanji, or Hanja). Use an asterisk (*) if you need to match the UI. RewardOps is an engagement commerce platform creating positive disruption in the rewards space. 325 +1 and share a web page, anywhere you go on the web. You'll see an em dash (Figure D). This is a mathematical symbol used to denote a division into parts — most notably feet or minutes. However, if you just want to type this symbol on your keyboard, the actionable steps below will show you how. Click on the cell where you want to type the exponent. County Public Schools Citizenship Agreement - Form for parents and students to sign . Microsoft Office programs, for instance, add exponents through the symbols section of the ribbon's Insert tab. The fingerprint reader module is located within the keyboard area on HP ENVY and Spectre notebook PC models and will integrate the following new functions in the keyboard. Mathematical symbols, equations in Google Docs. Is it a USA keyboard? Is there an @symbol above the number "2" key? As a temporary workaround you can use the "onscreen keyboard" just type onscreen in the text box at the left of the task bar, bring up the onscrren keyboard and try shift + 2. In the list which follows, all the keys which differ from the standard keyboard are described, and the key action which is generated is indicated when you press the key alone (by itself with no modifier), Shifted. To get the "exponent", hold down the following keys - shift, control, command, then click the + sign key, then release them. The caret (^) key (Shift-6) plus a letter will make a circonflexe : û, ê. The second section is the modifier keys (in light yellow). How do you type special characters?. The Clicker Communicator keyboard has been specifically designed for children, with a simplified layout and a child-friendly, lower case font. This tool is very convenient to help you preview the symbol, including viewing the details of the symbol display and the effect displayed on the web page. The Chromebooks have a lot more in common with tablets than other laptops that run Windows. You can also type 2 first and then select or highlight it before clicking the x² button. You must use the numeric keypad for the ASCII codes, not the top row numbers. To type the ¢ using the keyboard you can the Alt code from the shortcode section. Related advices for Is There A Fraction Button On The TI-84 Plus? How do you do fractions on a scientific. You can press the Alt key in combination with numbers on the numeric keypad to insert the degree symbol. For the the complete list of the ASCII based Windows ALT Codes, refer to Windows ALT Codes for Special Characters & Symbols. How do you make a square root symbol on a Chromebook? Hold down the "Alt" key and, at the same time, type in …. The divisor is represented by the denominator, i. To set up your Mac so that you can type macrons: Click on the Apple logo in the top left and choose System Preferences; Click Language & Region; Click Keyboard Preferences; Click the + icon and find Maori in the list. Select Insert and symbol on the right of the ribbon menu and use the same method as in Word to select it. Learn about Chromebooks, get started with Google Workspace, or apply for Google Cloud credits now. For example, type Alt + E to type € on a French or German keyboard. To add an exponent symbol after a number or letter when using a Windows operating system, the alt key should be held down and the code 0176, 0185, 0178 or 0179 should be pressed. How Can You Write a Division Symbol Using a Keyboard? Assuming use of a standard US keyboard, a Windows PC, and a desired symbol of “÷”; hold the “Alt” key and press “246” on the number pad. Make the symbol "degree" on Mac / MacBook : The technique : You keep the Option ⌥ and Shift ⇧ key. The multiplication sign is the " * " symbol, and the division sign is the " / " symbol. ; Natural Logarithm: Denoted by LN, this key displays log e of the number you enter. A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae. how to type radical sign on google docs; 17 Feb February 17, 2022. How do I sign up for a Canvas account as a parent?. Ohm (or Omega) symbol Quick Help. Click on the option to either create a new or a random emoji. As you can see, there’s more than one way to type exponents or powers on a Chromebook. You should see a little balloon change from US to Intl. The Input menu now appears in the menu bar as shown (the 5th icon on the right): 3. In common text messaging and email marketing it is also used to separate some pieces of information in titles. This will open the annotation tools. If you want to generate a capitalized letter with an accent, make sure to enter a capitalized version of the letter (A, E, I, O, U). Step 3: Open the Tools menu, and then click Preferences. On a Mac you can hold down the Option key followed by / On an iWhatever with a virtual keyboard you can use an emoji ( , not really very elegant), search for a …. (Note, if you haven’t installed the readr package, you’ll have to do that first. Be sure, that after Current assign to: there are no active Word functions. The bullet symbol may take any of a variety of shapes, such as a middle dot symbol •, square, diamond, arrow , bullet symbol ⁍, etc. To get division symbol (÷) in Ms Word, use Alt + 246 or Math Autocorrect shortcut “\div”. About A How Sign You Chromebook Type Do A On Division. Without any wasting much time, let's get started. typicality social studies taylor's clinical nursing skills quizlet unicode division symbol. Hi, I am trying to add the vital superscript trademark (TM) on our product names. Subtract the product from the divisor. Tap the pencil icon again to close the annotation tools. To type the root symbol in Mac OS X, all you would like to try to to is clicking ALT+V. For the @ symbol, type 64 or 064. This is a very thorough guide with several methods you can use to insert or type the Plus or Minus Symbol or Sign (±) anywhere on your PC, like Microsoft Word/Excel/PowerPoint, whether you are using the Windows or on the Mac keyboard. Alternatively, you can draw the symbol in the square, which will then open all the possibilities that are similar to your drawing. You can also press "Alt+=" on your keyboard. The easiest way to use the congruent symbol is to copy it from another source that uses the symbol and paste it where you need it. When you release Alt, the musical symbol will appear. To use this method to type an exponent on a computer, you need to: Move your mouse pointer to wherever on your screen you want to type the exponent. Watch the video below to see it in action: Here is a. How do I type a dash on my Asus laptop? Use built-in keyboard shortcuts. A brief description of the tax. Make a click to highlight the symbol sign then click the Insert button. Alt-Codes can be typed on Microsoft Operating Systems: First make sure that numlock is on, Then press and hold the ALT key, While keeping ALT key pressed type the code for the symbol that you want and release the ALT key. How do you make a division sign on an iPad? I provided you here 2 options. The division sign's Unicode value is 0247 in decimal notation, and you must type the leading zero for the shortcut to work. Writing basic equations in LaTeX is straightforward, for example: \documentclass{ article } \begin{ document } The well known Pythagorean theorem \ (x^2 + y^2 = z^2\) was proved to be invalid for other exponents. Backblaze is the solution I use. The copyright symbol (©) is a special character commonly used by photographers and other content creators. If you do not want to use these, click the small trash can icon to the right of each shortcut to remove them. ALT Codes for Mathematical Symbols Alt Code Symbol Description Alt 48 - 57 0 - 9 zero to nine Alt Codes for Basic Operators ; Alt Code Symbol Description Alt 43 + Plus Sign Alt 45 - Minus Sign Alt 0215 × Multiplication Sign Alt 0247 ÷ Obelus / Division ign Alt Codes for Pers. how do you type a division sign on a chromebook. How Do I Type a Degree Symbol? Degree Symbol on Microsoft Windows PC. After you select the symbol, it’ll appear in the Characters to copy: field, hit on the Copy button to copy. To type commonly used symbols on Linux and Chromebook, press Ctrl +. In statistics you’ll often see results given as …. Support is also available on your mobile device through the Samsung Members App. Option 1 – Copy and Paste: Copy (Ctrl + C or right-click > Copy) the peso sign located at the end of this article or Character Map (see section B above) and paste (Ctrl + V or right-click > Paste) it on the file. What Is The Divide Symbol On A Keyboard? In Microsoft Windows, this division sign is produced with Alt+0247 (or 246 with no zero) on the number pad, or by. To Chromebook How On Type Square Root. When composing an email, type #Encrypt# into the subject line before you type the subject of the email. Unbeatable deals on Laptops at Harvey Norman. Please make sure the NumLock is on and type 0176 with the leading zero. Note: TeXr might not be compatible with some older versions of Chromebooks. What is the division sign on a laptop keyboard? You will not find the traditional division sign (Ã·) on a computer keyboard. Where is the divide button on a Chromebook? How do you type a division sign on your keyboard – Chromebook Community. Using the Symbol font: This method is very useful when you need to insert symbols rarely and it works only for Latin or Greek letters. You can also switch the language input to Unicode Hex Input and type Option + 00F7 to type ÷ sign. How do you do exponents on a Chromebook? Here's how to do it: Press "CTRL + /" to access the list of features. How do you type math symbols on a Chromebook? I’ve always typed em dashes normally to press Alt and type 0151 on the numerical keypad. Step 4: If you'd like the Clever Portal to open on startup, be sure to enter you Portal URL into the setting "Pages to load on startup". You can hold [Shift] for the upper case Greek characters. Now, type in the Unicode entry for the special. under secretary of defense for personnel and readiness covid. The United Kingdom and Ireland use …. This will make the left column one big cell for your whole number. Press F2 to get into the edit mode. Occasionally you may find yourself wanting to type superscript (see the adjacent graphic) or subscript (10 2). Obey the following instructions to type this symbol on Mac using the keyboard shortcut stated above: With this keyboard shortcut, you can type the Division symbol anywhere on your Mac computer. In your document, open the “Insert” menu and then click the “Special Characters” command. Press Shift-Ctrl-U, type the numbers 2014 (one by one), and press spacebar. ) Inside the parentheses of the function, write the name of the file in quotes, including the file extension (. Select Latin-1 Supplement from the Subset drop-down menu. How do you type an N with a tilde on a Chromebook? Now that you see INTL over on the underside proper of yourscreen you can type your accents. Tap the type of content that you want to share. Hi Kurious, You can enter ÷ by pressing and releasing Ctrl + Shift + U. On Mac OS: Hold down the Option key and press / (slash) In HTML: ÷ or ÷. Highlight that number by clicking and dragging over it with your cursor. While holding down the Alt key, type Alt code 0247 on the numeric keypad (Note: on the right side). We know that every time we make it easier for you to support your students and teachers, magic happens. Typing Division Sign in HTML & CSS. Analyst fires back after Elon Musk shut him down in Tesla earnings call. Select a cell you will insert division symbol, hold the Alt key, type 0247 and then release the Alt key. As I mentioned earlier, you can use Alt codes to type characters you could otherwise type on your keyboard. py" (you can save it in any folder you want). Bullet points also called as Bullet List in HTML or Word terminology. How Do You Type A Division Sign On A Keyboard. Choose Use Math AutoCorrect outside of Math regions. Change the division sign into a multiplication sign and multiply. Learn how to use degree sign alt code on Word, Excel, Mac, LaTeX and more. : Press and hold AltGr and while holding it press numerical 2. For typing division sign use "÷" instead of "/". The Division symbol is used to represent the mathematical operator division. About Google Docs Long Symbol Division In. About Google Symbol Docs Division Long In. Next click on 'Add a language' and choose 'English'. To get the theta sign, your TI-84 calculator needs to be in Polar mode. In any case, you can reassign any existing function in Word to a …. Examples 48 × 48 pixels The application icon should be 62 × 62 pixels and PNG format. marzo 26, 2022 Posted by seaside inn san francisco bed bugs. Where is the divide symbol on a Chromebook? How do you type a division sign on your keyboard - Chromebook Community. If you are referring to , for instance the number 1 next to the letter J, I have them; they are blue on my keyboard. Your data will now be imported into R environment. To paste the peso symbol, go to your document or application and then right-click and select Paste. When you press Ctrl+Shift+U in an input field on a Chromebook, you'll get a little underlined "u" on your screen. About How A Type Do You Chromebook Sign A Division On Select Insert and symbol on the right of the ribbon menu and use the same method as in Word to select it. You can find the formulas for these characters via spreadsheetpoint here. If you are comfortable with creating fields “by hand,” press Ctrl+F9 to insert the field delimiters (the characters that look like braces but cannot be typed from the keyboard) and type the required text between them. After paying, the FILE menu lists the available print options. To type the Letter O with an accent mark, first get to know the alt code of the character you want to type (for example, the O with acute Ó alt code is 0211), then press down the alt key and type the alt code (e. If you have selected a different keyboard language from what the Chromebook was designed for, it gets complicated. Now it’s time to look for the symbol you want to insert. So if you type “=CHAR(67)” in a cell and press Enter, you’ll get C. Different keyboard layouts have different alternate characters. You may send a letter instead of using Form 3949-A. Type \div and press space, so that Word will automatically convert it into ÷ sign. The Division symbol is used to represent the mathematical operator division The division sign's Unicode value is 0247 in decimal notation, and you must type the leading zero for the shortcut to work. Only use a clean, soft cloth to clean the screen; no cleansers of any type. ; Angle Functions: Scientific calculators …. Emoticons(Emojis) on a Chromebook. 9 in number row key: Alone 9 In other game sessions the key produces the expected characters. For example, 3 times 4 is "3*4. … You can use the 'shift' key to type any symbol at the top of a key. The switcher will be relocated to the top of the page. In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the …. Then look again at the top right area of the dialog and select Latin-1 Supplement in the Subset: drop …. com/ehowtechTyping the divide symbol is something tha. Inserting a Square Root using the Alt code. I can do the symbols in the lower number range. These two symbols are immediately above the " - " sign. Type the ^ symbol, located on the 6 key on a standard keyboard. At the bottom right, select the time. Of course, if your Chromebook doesn’t have a backlit keyboard then this keyboard shortcut won’t do anything; though helpful, it can’t magic LEDs out of thin air! But for an easy way to turn on the backlit keyboard or even turn it off, it’s among the handy Chromebook keyboard shortcuts you should know. If you want a computer that is smart, secure and ready when you are, it’s time for a Chromebook. Once you have selected the equation tools you’ll see a new tool bar has opened up underneath the existing one. Click the arrow next to the name of the symbol set, and then select the symbol set that you want to display. Use the Enter key when finished to continue typing the rest of your equation. The division sign should appear after you type the last number in the sequence. How to move a semi circle in desmos — semi. On Windows: Hold down the ALT key and type 0247 on the keypad. It is located between the Control key and Command key on a typical Mac keyboard. Note: You MUST use the numeric keypad because the number keys at the top of the keyboard won’t work. You can copy Division sign by pressing the copy button below. Fn + Delete (ideapad 100-15IBY): Enables/disables the numeric keypad. Obey the following instructions for a step-by-step instructions to Type Exponents in Word using the keyboard shortcut. Whilst holding on to the Alt key, press the symbol’s Alt Code (0247) on the numeric keypad. How to type division symbol on chromebook - Top vector Use Fn key + NumLk keys to turn on Num Lock on laptops without a numeric keypad. Locate the Symbol; To easily find the Divide By symbol in the dialog box, look at the bottom right area of the Symbol dialog box and select Unicode (hex) in the from: drop-down. What division sentence means the same as 12/4 8?. The Division symbol [÷] Alt Code for Windows. First of all, make sure you turn on the NumLock. If you do not have Alt GR in your keyboard, simply press and hold left ALT then type 92 ; To get the letter, character, sign or symbol \: ( Backslash, reverse slash ) on computers with Windows operating system: 1) Press the Alt key on your keyboard, and do not let go. Look:Now, type in the Unicode entry for the special character you want. Using “/” to do division this way is deprecated; if you want floor …. To use the Divide By symbol (÷) shortcut on a keyboard, press down the Alt key. Hold the ALT key and type 236 on the num-lock keypad. Below is the Insert Symbol dialog box: 2. In the horizontal toolbar at the top of the screen. On a Mac you can hold down the Option key followed by / On an iWhatever with a virtual keyboard you can use an emoji ( , not really very elegant), search for a site that shows the symbo. On an international Spanish keyboard, press Alt Gr + 2. You can use it for division purpose. For Chromebooks that have no mouse buttons, you can copy and paste text by following the steps below. University of Vermont: “Alexander’s and Cushman’s Study Finds Blood Type and Memory Loss Link. com/subscription_center?add_user=ehowtechWatch More:http://www. The EasyBib Writing Center Use our resources and guides to write the perfect paper!. java type the boldfaced text below at …. You can use the built-in text formatting features to make any character appear in exponent form. Select the accent you need or press the corresponding number. Optimum staffing size = square root of work months. A list of accented letters will pop up for you to choose from. Google Bookmarks extension for chrome. Answer (1 of 25): It depends on your OS. I am simply trying to find out if there is a function on this keyboard that will give my the division sign. Although Math Autocorrect shortcut works only inside equation editor, you enable it to use outside equation editor. How to create the pi symbol on Chromebook? Method 2 Type Pi symbol on PC. How do I enter a division sign on a keyboard? You didn't learn keyboard partitions in school. When typing math text on the computer, people typically use the slash mark, the character sharing . they discussed the material for a few minutes, then typed in their conclusions. Here's how you can enable it: Go to the 'Settings' app on our Device. To create a keyboard shortcut for the square root symbol: Position the cursor in a Word document where you want to insert the symbol. On the right-hand side, you’ll see the ”CTRL +. How to type special characters on a Windows PC. The 'shift' keys are on the left and right of the keyboard, with the arrow pointing upwards. How to type the registered trademark symbol on a Chromebook Make the arrows with keyboard shortcuts under Windows. You’ll learn all the easy methods including the Plus or Minus Alt code shortcut. Is there a divide symbol on a keyboard? Typing Division Sign in Windows Now, hold one of the alt keys and type 0247 to make ÷ sign. About On Square How Chromebook Type To Root. How do you type a multiply symbol on the keyboard Mac? Just press option-shift-9 for a dot multiplication symbol. Philosophy Of Education By Nyirenda And Ishumi Pdf, Escape From Tomorrow, Feel More / With Less, The 42nd Parallel Analysis, Manservant And Maidservant, Moist Diane Hk, The Big Fix, Will Kane Allergan, Super Mario Sunshine, All Projects. Use an en dash for a minus sign. Bullet point symbol and • middle dot point symbol are icons used to introduce items in a list. How do you type the division symbol on a laptop without a number pad?. The character / Division slash and the : colon. · After that, it is time to type in the code that . On Microsoft Word documents, you can type 00F7 and press alt + x keys together to make a division sign. You're now using the US International layout. On the right-hand side, you'll see the "CTRL +. Note that division sign is also referred as obelus sign in typography If you haven't already, sign in to your Chromebook. 2) HRC Podcast: Your Personnel File. Method #2… If your keyboard doesn’t have a numeric keypad, you can do it this way: 1 – Select the Insert tab in MS Word. How to type multiplication & division symbol in Word. Press the Mode key (near the top of the keyboard). If you don't see the symbol you want to type, …. So for example if we want to write the square root of 16, it would look like this: Additional Commands \div = division sign \times = multiplication sign \cdot = dot multiplication sign. Find the "Text Formatting" section. Hold one of the alt or option key. If you’re working with Microsoft Word, WordPad, Outlook, or another Microsoft app, you can type in the Unicode value and then type Alt-X; …. How do I type a division sign on a Chromebook? Type Alt-246. Even if you’ve tried to fix your keyboard and failed, don’t give up hope. Click ‘Create New’ to start new document or open any existing document file. So I found the combining over line in special characters. Select ‘symbol’ from the left drop-down menu and ‘Maths’ from the right drop-down. If you made the correct choice, the curve should match up well with the points. This opens a new editor window, and you can start editing the file. Press the F4 key to toggle between relative and absolute references in ranges in your Google Sheets formulas. Now all you have to do is tap ds and the space bar for your division sign Will appear. Chrome is slowly implementing a way to type emojis in Chrome, but this trick gives you the functionality right now (rather than needing to use one of your Chromebook's development channels) and lets you use. Methods of Absolute Functions in Excel. 5 5^2 (9 raise power 3) sqrt(55) (Square root of 15). Tip: To start from scratch, select Blank document. The data file type you select will be exported to your PC’s, Mac’s, tablet’s, or smart phone’s download folder (the location of which varies by web browser and device type). com and login using Google account. This is a mathematical extension that will allow you to type in Subscript characters in all Google Forms and Google Sheets. *For the detailed usage method, refer to the user manual. What is the division sign on a Mac keyboard? – AnswersToAll. Alternatively, you can also press Shift + 8 twice to type in two asterisks (*). How do you make a division symbol on a laptop? Press and hold the Alt key and type 0247 on your keypad. Cube Root (Symbol/sign/mark) Preview and HTML-code. Only thing is that Chromebooks don’t have numerical keypads. In the Press new shortcut key field, type the key combination that you want to use for the symbol or character (e. This expert-written guide covers small circle used with temperatures, geo coordinates, and angles of a circle. Whilst holding on to the Alt key, press the symbol's Alt Code (0247) on the numeric keypad. How to get the @ symbol on a Windows laptop? On a laptop with a numeric keypad, press Ctrl + Alt + 2, or Alt + 64. Learn how to open special font characters in Cricut Design Space, also known as glyphs, dingbats, ornaments, swirls, curls and font extras. how to make the infinity symbol on chromebook. The Department of Technology (DOT) requires several security improvements across the division which changes the students' Chromebook login experience. How do you insert a divide sign in Word? Open the Insert tab, click Symbol and pick the ÷ division symbol to insert it in your document. Whether it's a gaming laptop or a Chromebook, a good first step is investing in a laptop stand, like our favorite Rain Design mStand (available at …. Choose which style fraction you want. About Division Symbol Long Google Docs In. Use the Text tool to place an insertion point in the file. How can I type @ on my Acer Laptop. About Keyboard Chromebook Symbols. Point to (but do not click) the lower-right or top-right corner of the screen, and then click to select the Search charm. You can use a keyboard shortcut to make a ÷. Go to settings> general> keyboard> shortcuts> tap the plus sign> paste the division sign on the top line, ds on the bottom lne. The Gentium Plus font family supports Yorùbá and is installed with the keyboard. Method 3: Using the indexing method, you can type in many different indexes, not just m2, m3. You’ll be able to choose the option you need. To input this symbol, type in "00f7" beside the Unicode "u. To insert a division symbol, hold down the ALT key and type 0247. This represented an insurance against the items being lower than the statutory weight, or of lower than. But Windows also supports shortcuts for inserting exponents via your. How To Type A Division Symbol In Microsoft Word Word Symbols Words Microsoft Word. Annotation tools for shared screen or whiteboard. The EMU Wireless Login page: If your department would like to set up a wireless network access point, contact the Help Desk to obtain the specific equipment security and feature requirements and/or consultation for a wireless environment plan. Select everything on that page by typing "Ctrl+A" on your keyboard. Division sign on keyboard chromebook Jan 04 2010 A keyboard is saying input device that uses. Sign back into your own account and then select Next and enter in your info. Understand and utilize EDGAR applicant types. #1 in 22 categories, including Customer Satisfaction as measured by American Customer Satisfaction Index (ACSI) in 2021. Now type in the numbers you want for the numerator, denominator, and whole number (if needed). When should you use the slash symbol? IT. divide symbol on chromebook. In addition, there is also a divide emoji that you can use similar to division sign. Users can search and access all recommended login pages for free. Symbols with codes between 9000 and 9999 rely on certain fonts, and may not work everywhere. Place the cursor where you want the degree symbol to go in your text. If you like this Page, please click that +1 button, too. Microsoft English US-International Keyboard. Type both the base number and the exponent or power. These are general questions about Penn Foster, and are not program-specific. The Division (÷) symbol should appear in …. Don’t type the quotation marks “”, just the text inside them! Works for Window 7 desktop and laptop, chromebook, and tablets. Baker’s Dozen If you go to the search bar and look for “baker’s dozen”, you will see the number 13 on the calculator. Shift + 2, just like any other keyboard. The individual’s Social Security number or the business’ Employer Identification Number. keyboard function for division sign. Press and hold Alt and type 0247. Navigate to the file where you would like the root symbol inserted in. You can type the hexadecimal numbers then press alt and x keys together. Plus or Minus Symbol [±] Quick GuideTo type the Plus or Minus Symbol on Mac, press Option + Shift + = shortcut on your keyboard. How do you work out 63 divided by 7? Using a calculator, if you typed in 63 divided by 7, you'd get 9. How to type a subscript on a chromebook keyboard. There is not a numeric key pad, therefore no 'number lock' key, nor a 'skroll lock' key. Below are the steps to type the Delta symbol on your keyboard (using the unicode): Place the insertion pointer at where you want to insert the symbol. Type over text in Word for Windows. Type the numerator into your calculator, then push the divide button. Voila! You have your own personal symbol to express yourself. Create a free Grammarly account and start eliminating writing mistakes in seconds. The form of the obelus as a horizontal line with a dot above and a dot below, ÷ , was first used as a symbol for …. If you find yourself in Silicon Valley and you need a laptop, try the library. You won't see any text appear even though you just pressed some keys. Add numbers or substitute variables in the box. Place the cursor at where you want to type the exponent. View notifications of Google+ activity when signed in, anywhere you go. Now save the program: select File from the menu, then Save. Pressing and holding down the Shift while pressing the < creates and inserts the less than symbol. The Division sign shows a dash, similar to an ordinary hyphen. word 2010 insert division symbol. Microsoft word shortcut for inserting division sign using Math Autocorrect. How to Insert or Type the Degree Symbol in Word (5 Ways to. By default, the TI-84 is in Normal mode. Watch the video below to learn how to use the F4 shortcut. Type Special Characters On a Chromebook (Accents, Symbols, Em Dashes ; d is a great tool for Spanish-speaking countries. class extension (the computer-language version). Once you do that, while your cursor is in any text input field, press Ctrl-space. Type the Alt Code value of the degrees symbol 0176 on the Numeric Pad. Once you have signed in, click in the bottom right corner, . If you are looking for how you can type this symbol on the keyboard, then you’ve found it. How to highlight text on a Chromebook. Use t-his to open Google Assistant on your Chromebook. Now I want the greater than sign that's at the top right hand coner and I can't access it. Spreadsheet math using cell references. The greater than or equal to symbol is used in math to express the relationship between two math expressions. Hold down the ALT key and press 8800. Fn + Pause: Activates the break function. Shop Chromebooks Working from home Studying at home Acer Chromebook Spin 514. Repeat the same step for each symbol you need, or paste the first division symbol. This sign informs Excel that you are entering a formula. It looks like this: * The one that is made by holding Shift and …. The second one looks like a semi-circle to me. org" your system routes the request to domain name system servers, which may be chosen by Advanced Chromebook users could configure either a virtual private network or Tor network connection. Pre Algebra Worksheets Algebraic Expressions Worksheets Algebra Worksheets Pre Algebra Worksheets Math Vocabulary. Press Shift + 6 to type in the caret symbol (^). There's a whole lot more, however, lurking just beneath the surface, including accented (diacritic) characters. Clever is your partner in providing secure and reliable data management, simple and secure implementation, and classroom …. Open the Control Panel - Language. A Chromebook costs less than an iPad and has a keyboard. Select the kind you want and this will appear beneath the one. Press Alt with the appropriate letter. About Symbols Keyboard Chromebook. Here's a handy reference to show you how. About In Google Docs Symbol Division Long. On the face of it one can suggest to use the SHIFT key to do this task but it's the bracket sign that comes up. Click on user icon [bottom right corner of desktop] -> Settings -> scroll down to Show Advanced Settings-> click on Advanced Settings. You've somehow managed to have a US desktop/laptop keyboard with a UK settings OR vice versa, but what's the difference in keyboard layouts? The major difference you'll probably notice first when typing are the @ and " keys beings swapped, and the US keyboard layout has a second Alt key instead of the UK AltGr key 1 ". Depending on the font you are uesing, there may be exponent characters. If you go to type on your Chromebook and the wrong letter is showing up it can be extremely frustrating! How to type the squared symbol in Chromebook. For typing division sign use “÷” instead of “/”. Is it possible to type pi on a Chromebook?. Typing special characters with a Chromebook can be done using unicode. Use Auto Format to Insert a Line in Word. A On You How A Sign Division Do Chromebook Type. Next, type the bottom number into the calculator, which is the denominator. Calculate the roots of imaginary numbers. The algebra section allows you to expand, factor or simplify virtually any expression you choose. For example, to type ⊂, ⊆ or ⊄, hold Alt and press C one, two or three times. On a Spanish keyboard for Latin America, press Alt Gr + Q. Find out which Pixel is right for you. Please, read a guide if you're running a laptop. Keyboard shortcuts: Enter math expressions Updated 01 Aug 2021. For example, three-fourths are. With this tool, you can adjust the size, color, italic, and bold of Long Division(symbol), then you can copy and paste it. How To Make Infinity Sign On Mac? Hold the Option key and press 5. Choose From a Huge Collection Of Chromebooks Available At B&H. The division sign or is written as a horizontal line with dot above and dot below (obelus), or a slash or horizontal line: ÷ / — The division sign indicates division operation of 2 numbers or expressions. Or on Android, iOS, desktop, and through Google Assistant or Chrome. May 07, 2021 · To use the Divide By symbol (÷) shortcut on a keyboard, press down the Alt key. Harassment is any behavior intended to disturb or upset a person or group of people. The division sign ÷ is a symbol consisting of a short horizontal line with a dot above and another dot below, used to indicate mathematical division. How do you get French accents on a Chromebook? The apostrophe key (left of the enter key) followed by the letter will make an acute accent (aigu) , or a cédille if you type a c: á, é, ç. how to type radical sign on google docs. " For instance, the code for lower-case á is ALT+0225, but capital Á is ALT+0193. It is different from the subtraction key. How do you type the division sign on a keyboard? How do you type a division sign? Whilst holding down the [Alt] key, type [0247] on the numeric keypad (on the right). Check Alt Codes and learn how to make specific symbols on the keyboard. See also Keys and keyboard shortcuts term collection. How do you do the copyright symbol on a Chromebook? The normal way I’ve always typed em dashes is to press Alt and type 0151 on the numerical keypad. There is a character code for all characters and symbols in Microsoft Word. The = sign is always necessary to start off any formula in a spreadsheet. 2019 by Katy Challis | Updated 10. There should be two letters at the end of the model number. Note: The number 0247 you typed must in the numeric keypad. Typing Division Sign in Windows Now, hold one of the alt keys and type 0247 to make ÷ sign. If you want a computer that is smart, secure and ready when you are, it's time for a Chromebook. The least lucrative option is microtasks, where the average pay is just \$2 to 3 an. To type the alternate character on a key, press the right Alt key and the desired key. Go to the menu bar above the file. How To Type the Peso Sign on Your Computer or Smartphone. Where are the function keys on a Chromebook? Turning Chromebook’s Shortcut Keys into Function KeysClick the Settings panel in the bottom right corner of your Chromebook screen. For Windows users, simply press down the Alt key and type 0177 using the numeric keypad, then let go of the Alt key. Explanation: The best way to make one is to search for one then copy and paste. The rest is pretty straightforward: Take the current balance (C2) minus the …. Now if you wish to take it one step further and not only write Python programs but also build games using Python, then you can install PyGame on your Chromebook. 2020 by Katy Challis See also: Signing in to a Chromebook Removing a user from a Chromebook Connecting a Chromebook to the internet (at school) Basic Sign-in In “basic” sign in view, students will type their ent. For subscript, press CTRL + = (press and hold Ctrl, then press =). RewardOps' cloud-hosted infrastructure, SaaS approach and agile methodology sets it apart from other companies in. Understand and utilize EDGAR CIKs, passphrases. For US version, they use the word “return”. Search: Long Division Symbol In Google Docs. How to Open Special Characters and Font Glyphs in Cricut. TypingClub is the most accessible typing program available. Applicants with a CIK but no access codes (formerly “Convert Paper Only Filer to Electronic Filer”) Create and obtain EDGAR access for asset-backed securities (ABS) issuing entities. Go into Settings and check the keyboard "Input method" - what is selected?. Click Set up and then click Add Fingerprint. From the iOS keyboard on your iPhone or iPad: First hit the “123” button to see the numerical keypad. Find the symbol and tap it to insert it in the document. For example, if you wanted to do a simple math problem, using Google Sheets like a calculator, you could type something like the following into a cell: =27/3. Sign Up to receive special offers and get the latest news from P. : Please check to the near the space button. Hold the Alt key and enter the number below using the numeric pad on your keyboard. If your keyboard has a number pad on the right hand side (or your laptop keyboard has a number lock) you can use Alt codes to get characters you wouldn't find normally. If you can't remember the name, open the virtual keyboard, and then in the bottom left of the keyboard press the gear icon. How do you make division sign using keyboard? How do you type a division sign? Whilst holding down the [Alt] key, type [0247] on the numeric keypad (on the right). Contour Software Karachi Division, Pakistan4 weeks agoBe among the first 25 applicantsSee who Contour Software has hired for this role. Site officiel de l’office de Tourisme d’Argentan Intercom – volatile crossword clue 5 8 – when is the nba skills challenge 2022. Hold down Ctrl and Shift and hit the Q key twice to log out of your Chromebook. 12 divided by 3 is 4, and 4 times 4 is 16. Interview questions are a reflection of a company’s priorities, so it doesn’t hurt to take advantage of the opportunity and gain some valuable insights that …. Division sign, Backslash / How do you type a dash on a Chromebook? Chrome OS: Type. Hold down the Alt key, and on the numeric keypad on the right of the keyboard, type 0176 or Alt+ 248. The general equation for a circle is x^2 + y^2 = r^2 where r is the radius of the circle you're making. User-added image The majority of Chromebooks today have two types of USB ports: 2. Use square root and cube root symbols to …. While you can use the above keyboard shortcut to open the Crosh, you can click on the following link to see what Chrosh commands you can run to customize your Chrome OS experience. Under Editing options, do one of the following: To use Insert key to control Overtype mode, select the Use Insert key to control overtype check box. Open the Microsoft Word, Excel, or PowerPoint application. This table is organized by Accent type. In order to easily type in accent marks ( é è à ê ç ï ù ë), it is a simple matter to install a keyboard that will allow you to type in accents conveniently. To calculate the cube root of a number in Excel, use the caret operator (^) with 1/3 as the exponent in a simple formula. Then you can see the ÷ symbol is showing in the selected cell. " When you're done, press the "Enter" key. Where is the division sign on a chromebook in word? How do you type a division sign on your keyboard – Chromebook Community. 6̅6̅ Eureka! Type a single 6 and then add the special character by double clicking on it. Click Insert>Special Characters from the Google Docs menu. Note: You must use the numeric keypad, not the numbers on the top row of your keyboard. Typing Symbols in Unicode ; 00F7, ÷, Division sign ; 00BA, º, Masculine ordinal indicator ; 00BB, », Right-pointing double angle quotation mark. The following general math notation can be entered in Show My Work boxes. By combining them with the FN key, you'll get access to F1-F12 keys, which can be associated with particular application features (useful in Photoshop for example). If you're an average internet user, you might be wondering how to type letter 'enye' or letter ñ in your keyboard. Here is how to easily make your own emoji with emojily: Open the app. So, go ahead, type the symbols using Alt key in your word document and have fun creating an interesting document.


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Scale and Indirect Measurement | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Go to the latest version.

# 3.6: Scale and Indirect Measurement

Created by: CK-12

## Learning Objectives

• Use scale on a map.
• Solve problems using scale drawings.
• Use similar figures to measure indirectly.

## Introduction

We are occasionally faced with having to make measurements of things that would be difficult to measure directly: the height of a tall tree, the width of a wide river, height of moon’s craters, even the distance between two cities separated by mountainous terrain. In such circumstances, measurements can be made indirectly, using proportions and similar triangles. Such indirect methods link measurement with geometry and numbers. In this lesson, we will examine some of the methods for making indirect measurements.

## Use Scale on a Map

A map is a two-dimensional, geometrically accurate representation of a section of the Earth’s surface. Maps are used to show, pictorially, how various geographical features are arranged in a particular area. The scale of the map describes the relationship between distances on a map and the corresponding distances on the earth's surface. These measurements are expressed as a fraction or a ratio.

So far we have only written ratios as fractions, but outside of mathematics books, ratios are often written as two numbers separated by a colon (:). Here is a table that compares ratios written in two different ways.

Ratio Is Read As Equivalent To
1:20 one to twenty $\left (\frac{1} {20}\right )$
2:3 two to three $\left (\frac{2} {3}\right )$
1:1000 one to one-thousand $\left (\frac{1} {1000}\right )$

Look at the last row. In a map with a scale of 1:1000 (“one to one-thousand”) one unit of measurement on the map (1 inch or 1 centimeter for example) would represent 1000 of the same units on the ground. A 1:1 (one to one) map would be a map as large as the area it shows!

Example 1

Anne is visiting a friend in London, and is using the map below to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end?

The scale is the ratio of distance on the map to the corresponding distance in real life.

$\frac{\text{dist}.\text{on map}} {\text{real dist}.} = \frac{1} {100,000}$

We can substitute the information we have to solve for the unknown.

$\frac{8.8\ cm} {\text{real dist}. (x)} & = \frac{1} {100,000} & & \text{Cross multiply}.\\880000 \ cm & = x 100 & & cm = 1\ m.\\x & = 8800 \ m & & 1000 \ m = 1\ km.$

Solution

The distance from Fleet Street to Borough Road is 8.8 km.

We could, in this case, use our intuition: the $1 \ cm = 1 \ km$ scale indicates that we could simply use our reading in centimeters to give us our reading in km. Not all maps have a scale this simple. In general, you will need to refer to the map scale to convert between measurements on the map and distances in real life!

Example 2

Antonio is drawing a map of his school for a project in math. He has drawn out the following map of the school buildings and the surrounding area.

He is trying to determine the scale of his figure. He knows that the distance from the point marked $A$ on the baseball diamond to the point marked $B$ on the athletics track is 183 meters. Use the dimensions marked on the drawing to determine the scale of his map.

We know that the real-life distance is 183 m. To determine the scale we use the ratio:

$\text{Scale} = \frac{\text{distance on map}} {\text{distance in real life}}$

To find the distance on the map, we will use Pythagoras’ Theorem $a^2+b^2=c^2$.

$\text{(Distance)}^2 & = 8^2+14^2\\ \text{(Distance)}^2& = 64+196\\ \text{(Distance)}^2& = 260\\\text{Distance} &= \sqrt{260}=16.12 \ cm$

$\text{Scale} & =\frac{16.12\ cm}{\text{real dist}.}\\\text{Scale} & = \frac{16.12\ cm} {183\ m} & & 1 \ m = 100 \ cm.\\\text{Scale} & = \frac{16.12\ cm} {18300\ cm} & & \text{Divide top and bottom by} \ 16.12.\\\text{Scale} & \approx \frac{1} {1135.23}& & \text{Round to two significant figures}:$

Solution

The scale of Antonio’s map is approximately 1:1100.

## Solve Problems Using Scale Drawings

Another visual use of ratio and proportion is in scale drawings. Scale drawings are used extensively by architects (and often called plans). They are used to represent real objects and are drawn to a specific ratio. The equations governing scale are the same as for maps. We will restate the equations in forms where we can solve for scale, real distance, or scaled distance.

$\text{Scale} = \frac{\text{distance on diagram}} {\text{distance in real life}}$

Rearrange to find the distance on the diagram and the distance in real life.

$(\text{distance on diagram}) & = (\text{distance in real life}) \times (\text{scale})\\(\text{Distance in real life}) & = \frac{\text{distance on diagram}} {\text{scale}} = (\text{distance on diagram}) \cdot \left (\frac{1} {\text{scale}}\right )$

Example 3

Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 feet long. He would like his drawing to be 1:500 scale. How long, in inches, must the paper that he uses be?

We can reason intuitively that since the scale is 1:500 that the paper must be $\frac{883}{500}=1.766 \ feet$ long.

Converting to inches gives the length at $12(1.766) \ in=21.192 \ in$.

Solution

Oscar’s paper should be at least 22 inches long.

Example 4

The Rose Bowl stadium in Pasadena California measures 880 feet from north to south and 695 feet from east to west. A scale diagram of the stadium is to be made. If 1 inch represents 100 feet, what would be the dimensions of the stadium drawn on a sheet of paper? Will if fit on a standard (U.S.) sheet of paper $(8.5 \ in \times 11 \ in)$?

We will use the following relationship.

$(\text{distance on diagram}) & = (\text{distance in real life}) \times (\text{scale}) \\\text{Scale} & = 1\ inch \ \text{to}\ 100\ feet = \left (\frac{1\ inch} {100\ feet}\right )\\\text{Width on paper} & = 880\ feet \times \left (\frac{1\ inch} {100 \ feet}\right ) = 8.8\ inches\\\text{Height on paper} & = 695\ feet \times \left (\frac{1\ inch} {100\ feet}\right ) = 6.95\ inches\\$

Solution

The dimensions of the scale diagram would be $8.8 \ in \times 6.95 \ in$. Yes, this will fit on a $8.5 \ in \times 11 \ in$ sheet of paper.

Example 5

The scale drawing below is sent to kids who attend Summer Camp. Use the scale to estimate the following:

a) The distance from the mess hall to the swimming pool via the path shown.

b) The distance from the lodge to the swimming pool via the horse corral.

c) The direct distance from the mess hall to the lodge

To proceed with this problem, we need a ruler. It does not matter whether we use a ruler marked in inches or centimeters, but a centimeter scale is easier, as it is marked in tenths. For this example, the ruler used will be a centimeter ruler.

We first need to convert the scale on the diagram into something we can use. Often the scale will be stated on the diagram but it is always worth checking, as the diagram may have been enlarged or reduced from its original size. Here we see that 500 feet on the diagram is equivalent to 3.0 cm on the ruler. The scale we will use is therefore $3 \ cm = 500 \ ft$. We can write this as a ratio.

$\text{Scale} = \left (\frac{3\ cm} {500 \ ft}\right )& & \text{Do not worry about canceling units this time}!$

a) We are now ready to move to the next step. Measuring distances on the diagram. First, we need to know the distance from the mess hall to the swimming pool. We measure the distance with our ruler. We find that the distance is 5.6 cm. We divide this by the scale to find the real distance.

$\frac{\text{disance on diagram}} {\text{scale}} = \frac{5.6\ cm} {\left (\frac{3\ cm} {500 \ ft}\right )} = 5.6\ cm \cdot \left ( \frac{500\ ft} {3 \ cm} \right )$

Multiply this out. Note that the centimeter units will cancel leaving the answer in feet.

Solution

The distance from the mess hall to the swimming pool is approximately 930 feet (rounded to the nearest 10 feet).

b) To find the distance from the lodge to the swimming pool, we have to measure two paths. The first is the distance from the lodge to the horse corral. This is found to be 3.4 cm.

The distance from the corral to the swimming pool is 5.5 cm.

The total distance on the diagram is $(3.4 + 5.5) = 8.9 \ cm$.

$\text{Distance in real life} = \frac{\text{disance on diagram}} {\text{scale}} \approx 8.9\ cm \left (\frac{500\ ft} {3\ cm}\right )$

Solution

The distance from the lodge to the pool is approximately 1480 feet.

c) To find the direct distance from the lodge to the mess hall, we simply use the ruler to measure the distance from one point to the other. We do not have to go round the paths in this case.

$\text{Distance on diagram} & = 6.2 \ cm\\\text{Distance in real life} & = \frac{\text{distance on diagram}} {\text{scale}} \approx 6.2\ cm \cdot \left (\frac{500\ ft} {3\ cm} \right )$

Solution

The distance from the lodge to the mess hall is approximately 1030 feet.

## Use Similar Figures to Measure Indirectly

Similar figures are often used to make indirect measurements. Two shapes are said to be similar if they are the same shape but one is an enlarged (or reduced) version of the other. Similar triangles have the same angles, and are said to be “in proportion.” The ratio of every measurable length in one figure to the corresponding length in the other is the same. Similar triangles crop up often in indirect measurement.

Example 6

Anatole is visiting Paris, and wants to know the height of the Eiffel Tower. Unable to speak French, he decides to measure it in three ways.

1. He measures out a point 500 meters from the base of the tower, and places a small mirror flat on the ground.
2. He stands behind the mirror in such a spot that standing upright he sees the top of the tower reflected in the mirror.
3. He measures both the distance from the spot where he stands to the mirror (2.75 meters) and the height of his eyes from the ground (1.8 meters).

Explain how he is able to determine the height of the Eiffel Tower from these numbers and determine what that height is.

First, we will draw and label a scale diagram of the situation.

A fact about refection is that the angle that the light reflects off the mirror is the same as the angle that it hits the mirror.

Both triangles are right triangles, and both have one other angle in common. That means that all three angles in the large triangle match the angles in the smaller triangle. We say the triangles are similar: exactly the same shape, but enlarged or reduced.

• This means that the ratio of the long leg in the large triangle to the length of the long leg in the small triangle is the same ratio as the length of the short leg in the large triangle to the length of the short leg in the small triangle.

$\frac{500m}{2.75m} & = \frac{x}{1.8m}\\ 1.8 \cdot \frac{500}{2.75}& = \frac{x}{1.8}\cdot 1.8\\ 327.3& = x$

Solution

The Eiffel Tower, according to this calculation, is approximately 327.3 meters high.

Example 7

Bernard is looking at a lighthouse and wondering how high it is. He notices that it casts a long shadow, which he measures at 200 meters long. At the same time he measures his own shadow at 3.1 meters long. Bernard is 1.9 meters tall. How tall is the lighthouse?

We will again draw a scale diagram:

Again, we see that we have two right triangles. The angle that the sun causes the shadow from the lighthouse to fall is the same angle that Bernard shadow falls. We have two similar triangles, so we can again say that the ratio of the corresponding sides is the same.

$\frac{200m}{3.1m} & = \frac{x}{1.9m}\\ 1.9 \cdot \frac{200}{3.1}& = \frac{x}{1.9}\cdot 1.9\\ 122.6& = x$

Solution

The lighthouse is 122.6 meters tall.

## Lesson Summary

• Scale is a proportion that relates map distance to real life distance. $\text{scale} = \frac{\text{distance on map}}{\text{distance in real life}}$
• Two shapes, like triangles, are said to be similar if they have the same angles. The sides of similar triangles are in proportion. The ratio of every measurable length in one triangle to the corresponding length in the other is the same.

## Review Questions

1. Use the map in Example One. Using the scale printed on the map, determine the distances (rounded to the nearest half km) between:
1. Points 1 and 4
2. Points 22 and 25
3. Points 18 and 13
4. Tower Bridge and London Bridge
2. The scale diagram in Example Five does not show the buildings themselves in correct proportion. Use the scale to estimate:
1. The real length the indicated pool would be if it was drawn in proportion.
2. The real height the lodge would be if it was drawn in proportion.
3. The length a 50 ft pool on the diagram.
4. The height a 20 ft high tree would be on the diagram.

3. Use the scale diagram to the right to determine:
1. The length of the helicopter (cabin to tail)
2. The height of the helicopter (floor to rotors)
3. The length of one main rotor
4. The width of the cabin
5. the diameter of the rear rotor system
4. On a sunny morning, the shadow of the Empire State Building is 600 feet long. At the same time, the shadow of a yardstick (3 feet long) is 1 foot, $5 \frac{1}{4} \ inches$. How high is the Empire State building?

1. 3 km
2. 7 km
3. $12 \frac{1}{2} \ km$
4. $4 \frac{1}{2} \ km$
1. $\text{Pool} = 600 \ feet$ long
2. $\text{Lodge} = 250 \ feet$ high
3. Pool should be 0.3 cm
4. Tree should be 0.12 cm
1. $\text{length} = 21 \ ft$
2. $\text{height} = 10 \ ft$
3. $\text{Main rotor} = 12 \ ft$
4. $\text{cabin width} = 5 \frac{1}{2} \ ft$
5. $\text{rotor diameter} = 4 \ ft$
1. 1250 ft

## Date Created:

Feb 22, 2012

Sep 17, 2014
Files can only be attached to the latest version of None

format: plain


text:
# Metalearning: Learning to learn by gradient descent by gradient descent

So I read the Learning to learn paper a while ago, and I was surprised that the Decoupled Neural Interfaces paper didn’t cite them. For me the ideas are pretty close, where you try to predict the gradient used in each step of gradient descent, instead of computing it by backpropagation. Taking into account that they are all from DeepMind, won’t it be nice to cite each other and increase the impact factors for both of them?

Nevertheless, I enjoyed the paper. The key idea is instead of doing a normal update $\theta_{t+1} = \theta_{t} - \alpha_t \nabla f\left(\theta_t\right)$, we do it as $\theta_{t+1} = \theta_{t} + g_t\left(\nabla f\left(\theta_t\right), \phi\right)$ where $g_t$ is some function parameterized by $\phi$.

Now one can use any function approximator for $g_t$ (called optimizer, to differentiate with $f\left(\theta\right)$ – the optimizee), but using RNNs has a particular interesting intuition as we hope that the RNNs can remember the gradient history and mimic the behavior of, for instance, momentum.

The convenient thing about this framework is that the objective function for training the optimizer is the expected weighted sum of the output of the optimizee $f\left(\theta\right)$. Apart from this main idea, everything else is nuts and bolts, which of course are equivalently important.

The first obstacle that they had to solve is how to deal with big models of perhaps millions of parameters. In such cases, $g_t$ has to input and output vector of millions of dimensions. Instead, the authors solved this problem very nicely by only working with one parameter at a time, i.e. the optimizer only takes as input one element of the gradient vector and output the update for that element. However, since the optimizer is a LSTM, the state of the gradient coordinates are maintained separately. This also has a nice side effect that  it reduces the size of the optimizer, and you can potentially re-use the optimizer for different optimizees.

The next two nuts and bolts are not so obvious. To mimic the L2 gradient clipping trick, they used the so-called global averaging cell (GAC), where the outgoing activations of LSTM cells are averaged at each step across all coordinates. To mimic Hessian-based optimization algorithms, they wire the LSTM optimizer with an external memory unit, hoping that the optimizer will learn to store the second-order derivatives in the memory.

Although the experimental results look pretty promising, many people pose some doubts about the whole idea of learning to learn. I was in the panel discussion of Learning to learn at NIPS, and it wasn’t particularly fruitful (people were drinking sangria all the time). It will be interesting to see the follow-ups on this line of work, if there is any.

Advertisements


text:
# Conjugate symmetry to prove inner product

We have to show that $$\langle p,q\rangle=\int_a^b \overline{p(t)}q(t)$$ is an inner Product.

I (think) I know what to do: I have to prove linearity, conjugate symmetry and positive definiteness. I have proven linearity and positive difiniteness already but I have difficulties with proving conjugate symmetry. I've come so far (which is not very far): $$\overline{\langle q,p \rangle } = \overline{\int_a^b \overline{p(t)}q(t)}$$

I know that it continues with $\overline{\int_a^b \overline{p(t)}q(t)}=\int_a^b \overline{\overline{p(t)}q(t)}= \int_a^b p(t)\overline{q(t)}=\langle p,q \rangle$ but I don't fully understand why $$\overline{\int_a^b \overline{p(t)}q(t)}=\int_a^b \overline{\overline{p(t)}q(t)}$$ that is the case.

It would be great if someone could explain! :)

By definition, for a complex function $f(t) = u(t) + iv(t)$ ($u$ and $v$ real) we have $$\int_a^b f(t) := \int_a^b u(t) + i \int_a^b v(t).$$
Thus $$\overline{\int_a^b f(t)} = \overline{\int_a^b u(t) + i\int_a^b v(t)} = \int_a^b u(t) - i\int_a^b v(t) = \int_a^b \overline{f(t)}.$$
For a complex-valued function $f$, we may always decompose it into its real and imaginary parts $$f = u + iv$$ where $u$ is the real part of $f$, $v$ is the imaginary part of $f$, and $u$ and $v$ are both real-valued functions. Then $$\int f := \int u + i \int v \implies \overline{\int f} = \overline{\int u + i \int v} = \int u - i \int v = \int \overline{f},$$ where $\overline{f} = u - i v$.


text:
# get, set

Read or set the value of a variable or a kdb+ data file

## get

Read or memory-map a variable or kdb+ data file

Syntax: get x, get[x]

Reads or memory maps kdb+ data file x. A type error is signalled if the file is not a kdb+ data file.

Used to map columns of databases in and out of memory when querying splayed databases, and can be used to read q log files, etc.

q)a:42
q)get a
42

:NewTrade
q)t:get:NewTrade                       / t is a copy of the table
q):SNewTrade/ set .Q.en[:.;trade]     / save splayed table
:SNewTrade/
q)s:get:SNewTrade/                     / s has columns mapped on demand

get and value

get has several other uses. The function value is a synonym for get. By convention, it is used for other purposes. But the two are completely interchangeable.

q)value "2+3"
5
q)get "2+3"
5

## set

Assign a value to a variable or file

Syntax: x set y, set[x;y]

Where x is

assigns the value of y to variable name or filename x.

q)a set 1 2 3            / set name a
a
q)a
1 2 3

q)a:t
q)a set 1 2 3             / set name t (indirect assignment)
t
q)t
1 2 3

q)a:"t"
q)a set 1 2 3             / fails, as name must be a symbol
:["type"]

If x is a file symbol, the values are written to file.

q):work.dat set 1 2 3    / write values to file
:work.dat
q)get :work.dat
1 2 3

Write a table to a single file:

q)\l sp.q
q):mytable.dat set sp
:mytable.dat
q)get :mytable.dat
s  p  qty
---------
s1 p1 300
s1 p2 200
s1 p3 400
..

To save a table splayed across a directory, x must be a path (i.e. ends with a /), and the table must be fully enumerated, with no primary keys:

q):mydata/ set sp
:mydata/
q)\ls mydata
,"p"
"qty"
,"s"
q)get :mydata
s  p  qty
---------
s1 p1 300
s1 p2 200
s1 p3 400
..

set saves the data in a binary format akin to tag+value, retaining the structure of the data in addition to its value.

q):data/foo set 10 20 30
:data/foo

These are .h, .j, .Q, .q, .z, and any other namespaces with single-character names.


text:
European vanilla option pricing with C++ via Monte Carlo methods

European vanilla option pricing with C++ via Monte Carlo methods

In the previous article on using C++ to price a European option with analytic solutions we were able to take the closed-form solution of the Black-Scholes equation for a European vanilla call or put and provide a price.

This is possible because the boundary conditions generated by the pay-off function of the European vanilla option allow us to easily calculate a closed-form solution. Many option pay-off functions lead to boundary conditions which are much harder to solve analytically and some are impossible to solve this way. Thus in these situations we need to rely on numerical approximation.

In this article we will price the same European vanilla option with a very basic Monte Carlo solver in C++ and then compare our numerical values to the analytical case. We won't be concentrating on an extremely efficient or optimised implementation at this stage. Right now I just want to show you how to get up and running to give you an understanding of how risk neutral pricing works numerically. We will also see how our values differ from those generated analytically.

## Risk Neutral Pricing of a European Vanilla Option

The first stage in implementation is to briefly discuss how risk neutral pricing works for a vanilla call or put option. We haven't yet discussed on QuantStart about risk neutral pricing in depth, so in the meantime, it might help to have a look at the article on Geometric Brownian Motion, which is the underlying model of stock price evolution that we will be using. It is given by the following stochastic differential equation:

\begin{eqnarray*} dS(t) = \mu S(t) dt + \sigma S(t) dB(t) \end{eqnarray*}

where $S$ is the asset price, $\mu$ is the drift of the stock, $\sigma$ is the volatility of the stock and $B$ is a Brownian motion (or Wiener process).

You can think of $dB$ as being a normally distributed random variable with zero mean and variance $dt$. We are going to use the fact that the price of a European vanilla option is given as the discounted expectation of the option pay-off of the final spot price (at maturity $T$):

\begin{eqnarray*} e^{-rT} \mathbb{E}(f(S(T))) \end{eqnarray*}

This expectation is taken under the appropriate risk-neutral measure, which sets the drift $\mu$ equal to the risk-free rate $r$:

\begin{eqnarray*} dS(t) = r S(t) dt + \sigma S(t) dB(t) \end{eqnarray*}

We can make use of Ito's Lemma to give us:

\begin{eqnarray*} d \log S(t) = \left(r - \frac{1}{2} \sigma^2 \right) dt + \sigma dB(t) \end{eqnarray*}

This is a constant coefficient SDE and therefore the solution is given by:

\begin{eqnarray*} \log S(t) = \log S(0) + \left(r - \frac{1}{2} \sigma^2 \right) T + \sigma \sqrt{T} N(0,1) \end{eqnarray*}

Here I've used the fact that since $B(t)$ is a Brownian motion, it has the distribution as a normal distribution with variance $T$ and mean zero so it can be written as $B(t) = \sqrt{T} N(0,1)$.

Rewriting the above equation in terms of $S(t)$ by taking the exponential gives:

\begin{eqnarray*} S(t) = S(0) e^{ (r - \frac{1}{2} \sigma^2) T + \sigma \sqrt{T} N(0,1) } \end{eqnarray*}

Using the risk-neutral pricing method above leads to an expression for the option price as follows:

\begin{eqnarray*} e^{-rT} \mathbb{E} (f(S(0) e^{ (r - \frac{1}{2} \sigma^2) T + \sigma \sqrt{T} N(0,1)} )) \end{eqnarray*}

The key to the Monte Carlo method is to make use of the law of large numbers in order to approximate the expectation. Thus the essence of the method is to compute many draws from the normal distribution $N(0,1)$, as the variable $x$, and then compute the pay-off via the following formula:

\begin{eqnarray*} f(S(0) e^{ (r - \frac{1}{2} \sigma^2) T + \sigma \sqrt{T} x} ) \end{eqnarray*}

In this case the value of $f$ is the pay-off for a call or put. By averaging the sum of these pay-offs and then taking the risk-free discount we obtain the approximate price for the option.

## C++ Implementation

Let's now fire up your favourite C++ development environment and get started on the implementation. As with the previous article, we won't be utilising any object oriented or generic programming techniques right now. At this stage the goal is to generate understanding of a basic C++ Monte Carlo implementation. I've written out the program in full and then below I'll explain how each component works subsequently:

#include <algorithm>    // Needed for the "max" function
#include <cmath>
#include <iostream>

// A simple implementation of the Box-Muller algorithm, used to generate
// gaussian random numbers - necessary for the Monte Carlo method below
// Note that C++11 actually provides std::normal_distribution<> in
// the <random> library, which can be used instead of this function
double gaussian_box_muller() {
double x = 0.0;
double y = 0.0;
double euclid_sq = 0.0;

// Continue generating two uniform random variables
// until the square of their "euclidean distance"
// is less than unity
do {
x = 2.0 * rand() / static_cast<double>(RAND_MAX)-1;
y = 2.0 * rand() / static_cast<double>(RAND_MAX)-1;
euclid_sq = x*x + y*y;
} while (euclid_sq >= 1.0);

return x*sqrt(-2*log(euclid_sq)/euclid_sq);
}

// Pricing a European vanilla call option with a Monte Carlo method
double monte_carlo_call_price(const int& num_sims, const double& S, const double& K, const double& r, const double& v, const double& T) {
double S_adjust = S * exp(T*(r-0.5*v*v));
double S_cur = 0.0;
double payoff_sum = 0.0;

for (int i=0; i<num_sims; i++) {
double gauss_bm = gaussian_box_muller();
S_cur = S_adjust * exp(sqrt(v*v*T)*gauss_bm);
payoff_sum += std::max(S_cur - K, 0.0);
}

return (payoff_sum / static_cast<double>(num_sims)) * exp(-r*T);
}

// Pricing a European vanilla put option with a Monte Carlo method
double monte_carlo_put_price(const int& num_sims, const double& S, const double& K, const double& r, const double& v, const double& T) {
double S_adjust = S * exp(T*(r-0.5*v*v));
double S_cur = 0.0;
double payoff_sum = 0.0;

for (int i=0; i<num_sims; i++) {
double gauss_bm = gaussian_box_muller();
S_cur = S_adjust * exp(sqrt(v*v*T)*gauss_bm);
payoff_sum += std::max(K - S_cur, 0.0);
}

return (payoff_sum / static_cast<double>(num_sims)) * exp(-r*T);
}

int main(int argc, char **argv) {
// First we create the parameter list
int num_sims = 10000000;   // Number of simulated asset paths
double S = 100.0;  // Option price
double K = 100.0;  // Strike price
double r = 0.05;   // Risk-free rate (5%)
double v = 0.2;    // Volatility of the underlying (20%)
double T = 1.0;    // One year until expiry

// Then we calculate the call/put values via Monte Carlo
double call = monte_carlo_call_price(num_sims, S, K, r, v, T);
double put = monte_carlo_put_price(num_sims, S, K, r, v, T);

// Finally we output the parameters and prices
std::cout << "Number of Paths: " << num_sims << std::endl;
std::cout << "Underlying:      " << S << std::endl;
std::cout << "Strike:          " << K << std::endl;
std::cout << "Risk-Free Rate:  " << r << std::endl;
std::cout << "Volatility:      " << v << std::endl;
std::cout << "Maturity:        " << T << std::endl;

std::cout << "Call Price:      " << call << std::endl;
std::cout << "Put Price:       " << put << std::endl;

return 0;
}

Let's now look at the program in a step-by-step fashion.

The first section imports the algorithm, iostream and cmath libraries. The algorithm library provides access to the max comparison function. This allows us to use the std::cout command to output code. In addition we now have access to the C mathematical functions such as exp, pow, log and sqrt:

#include <algorithm>
#include <cmath>
#include <iostream>

The first function to implement is the Box-Muller algorithm. As stated above, it is designed to convert two uniform random variables into a standard Gaussian random variable. Box-Muller is a good choice for a random number generator if your compiler doesn't support the C++11 standard. However, if you do have a compiler that supports it, you can make use of the std::normal_distribution<> template class found in the <random> library. In later articles, we will introduce this as a means of avoiding the need to "roll our own code":

// A simple implementation of the Box-Muller algorithm, used to generate
// gaussian random numbers - necessary for the Monte Carlo method below
// Note that C++11 actually provides std::normal_distribution<> in
// the <random> library, which can be used instead of this function
double gaussian_box_muller() {
double x = 0.0;
double y = 0.0;
double euclid_sq = 0.0;

// Continue generating two uniform random variables
// until the square of their "euclidean distance"
// is less than unity
do {
x = 2.0 * rand() / static_cast<double>(RAND_MAX)-1;
y = 2.0 * rand() / static_cast<double>(RAND_MAX)-1;
euclid_sq = x*x + y*y;
} while (euclid_sq >= 1.0);

return x*sqrt(-2*log(euclid_sq)/euclid_sq);
}

The next two functions are used to price a European vanilla call or put via the Monte Carlo method. We've outlined above how the theory works, so let's study the implementation itself. The comments will give you the best overview. I've neglected to include the function for the put, as it is almost identical to the call and only differs in the nature of the pay-off. In fact, this is a hint telling us that our code is possibly repeating itself (a violation of the Do-Not-Repeat-Yourself, DRY, principle):

// Pricing a European vanilla call option with a Monte Carlo method
double monte_carlo_call_price(const int& num_sims, const double& S, const double& K, const double& r, const double& v, const double& T) {
double S_adjust = S * exp(T*(r-0.5*v*v));   // The adjustment to the spot price
double S_cur = 0.0;   // Our current asset price ("spot")
double payoff_sum = 0.0;  // Holds the sum of all of the final option pay-offs

for (int i=0; i<num_sims; i++) {
double gauss_bm = gaussian_box_muller();   // Generate a Gaussian random number via Box-Muller
S_cur = S_adjust * exp(sqrt(v*v*T)*gauss_bm);   // Adjust the spot price via the Brownian motion final distribution
payoff_sum += std::max(S_cur - K, 0.0);   // Take the option pay-off, then add it to the rest of the pay-off values
}

// Average the pay-off sum via the number of paths and then
// discount the risk-free rate from the price
return (payoff_sum / static_cast<double>(num_sims)) * exp(-r*T);
}

The main function is extremely similar to that found in European vanilla option pricing with C++ and analytic formulae. The exception is the addition of the num_sims variable which stores the number of calculated asset paths. The larger this value, the more accurate the option price will be. Hence an inevitable tradeoff between execution time and price accuracy exists. Unfortunately, this is not a problem limited to this article! Another modification is that the call and put values are now derived from the Monte Carlo function calls:

int main(int argc, char **argv) {
// First we create the parameter list
int num_sims = 10000000;   // Number of simulated asset paths
double S = 100.0;  // Option price
double K = 100.0;  // Strike price
double r = 0.05;   // Risk-free rate (5%)
double v = 0.2;    // Volatility of the underlying (20%)
double T = 1.0;    // One year until expiry

// Then we calculate the call/put values via Monte Carlo
double call = monte_carlo_call_price(num_sims, S, K, r, v, T);
double put = monte_carlo_put_price(num_sims, S, K, r, v, T);

// Finally we output the parameters and prices
std::cout << "Number of Paths: " << num_sims << std::endl;
std::cout << "Underlying:      " << S << std::endl;
std::cout << "Strike:          " << K << std::endl;
std::cout << "Risk-Free Rate:  " << r << std::endl;
std::cout << "Volatility:      " << v << std::endl;
std::cout << "Maturity:        " << T << std::endl;

std::cout << "Call Price:      " << call << std::endl;
std::cout << "Put Price:       " << put << std::endl;

return 0;
}

The output of the program is given as follows:

Number of Paths: 10000000
Underlying:      100
Strike:          100
Risk-Free Rate:  0.05
Volatility:      0.2
Maturity:        1
Call Price:      10.4553
Put Price:       5.57388

We can compare this with the output from the analytical formulae generated in European vanilla option pricing with C++ and analytic formulae, which are given below for convenience. This serves two purposes. Firstly, we can see that the values are almost identical, which provides an extra level of validation that the code in both instances is pricing correctly. Secondly, we can compare accuracy:

Call Price:      10.4506
Put Price:       5.57352

As can be seen the prices are relatively accurate for $10^7$ simulation paths. On my MacBook Air this program took a few seconds to calculate. Adding a couple of orders of magnitude on to the number of paths would quickly make the program prohibitive to execute. Thus we have run into the first problem - that of optimising execution speed. The second is that of making the above code maintainable. Notice that there is a significant duplication of code in both the monte_carlo_call_price and monte_carlo_put_price functions. Later articles will discuss these issues in depth and provide solutions to them.


text:
## Electronic Journal of Probability

### On time reversal of piecewise deterministic Markov processes

#### Abstract

We study the time reversal of a general Piecewise Deterministic Markov Process (PDMP). The time reversed process is defined as $X_{(T-t)-}$, where $T$ is some given time and $X_t$ is a stationary PDMP. We obtain the parameters of the reversed process, like the jump intensity and the jump measure.

#### Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 13, 29 pp.

Dates
Accepted: 23 January 2013
First available in Project Euclid: 4 June 2016

http://projecteuclid.org/euclid.ejp/1465064238

Digital Object Identifier
doi:10.1214/EJP.v18-1958

Mathematical Reviews number (MathSciNet)
MR3035741

Zentralblatt MATH identifier
1294.60099

Rights

#### Citation

Löpker, Andreas; Palmowski, Zbigniew. On time reversal of piecewise deterministic Markov processes. Electron. J. Probab. 18 (2013), paper no. 13, 29 pp. doi:10.1214/EJP.v18-1958. http://projecteuclid.org/euclid.ejp/1465064238.

#### References

• Albrecher, Hansjörg; Thonhauser, Stefan. Optimality results for dividend problems in insurance. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 103 (2009), no. 2, 295–320.
• E. Altman, K. Avrachenkov, A. Kherani, and B. Prabhu. Performance Analysis and Stochastic Stability of Congestion Control Protocols, Tech. Report RR-5262, INRIA, Sophia-Antipolis, France, July 2004.
• Anick, D.; Mitra, D.; Sondhi, M. M. Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61 (1982), no. 8, 1871–1894.
• Asmussen, Søren. Stationary distributions for fluid flow models with or without Brownian noise. Comm. Statist. Stochastic Models 11 (1995), no. 1, 21–49.
• Asmussen, Søren. Applied probability and queues. Second edition. Applications of Mathematics (New York), 51. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2003. xii+438 pp. ISBN: 0-387-00211-1
• Baccelli, Francois; Kim, Ki Baek; McDonald, David R. Equilibria of a class of transport equations arising in congestion control. Queueing Syst. 55 (2007), no. 1, 1–8.
• Bar-Lev, Shaul K.; Parlar, Mahmut; Perry, David. On the EOQ model with inventory-level-dependent demand rate and random yield. Oper. Res. Lett. 16 (1994), no. 3, 167–176.
• J.-B. Bardet, A. Christen, A. Guillin, F. Malrieu, and P.-A. Zitt, phTotal variation estimates for the TCP process, ArXiv e-prints (2011).
• Bekker, R.; Borst, S. C.; Boxma, O. J.; Kella, O. Queues with workload-dependent arrival and service rates. Queueing Syst. 46 (2004), no. 3-4, 537–556.
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• Borovkov, K.; Novikov, A. On a piece-wise deterministic Markov process model. Statist. Probab. Lett. 53 (2001), no. 4, 421–428.
• Borovkov, K.; Vere-Jones, D. Explicit formulae for stationary distributions of stress release processes. J. Appl. Probab. 37 (2000), no. 2, 315–321.
• Boxma, Onno; Kaspi, Haya; Kella, Offer; Perry, David. On/off storage systems with state-dependent input, output, and switching rates. Probab. Engrg. Inform. Sci. 19 (2005), no. 1, 1–14.
• Boxma, Onno; Perry, David; Stadje, Wolfgang; Zacks, Shelemyahu. A Markovian growth-collapse model. Adv. in Appl. Probab. 38 (2006), no. 1, 221–243.
• Brockwell, P. J. Stationary distributions for dams with additive input and content-dependent release rate. Advances in Appl. Probability 9 (1977), no. 3, 645–663.
• Browne, Sid; Sigman, Karl. Work-modulated queues with applications to storage processes. J. Appl. Probab. 29 (1992), no. 3, 699–712.
• Ã‡inlar, E.; Pinsky, M. A stochastic integral in storage theory. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 227–240.
• Chafaï, Djalil; Malrieu, Florent; Paroux, Katy. On the long time behavior of the TCP window size process. Stochastic Process. Appl. 120 (2010), no. 8, 1518–1534.
• Chung, K. L.; Walsh, John B. To reverse a Markov process. Acta Math. 123 1969 225–251.
• Costa, O. L. V. Stationary distributions for piecewise-deterministic Markov processes. J. Appl. Probab. 27 (1990), no. 1, 60–73.
• Dassios, A.; Embrechts, P. Martingales and insurance risk. Comm. Statist. Stochastic Models 5 (1989), no. 2, 181–217.
• Dassios, Angelos; Jang, Jiwook. The distribution of the interval between events of a Cox process with shot noise intensity. J. Appl. Math. Stoch. Anal. 2008, Art. ID 367170, 14 pp.
• Davis, M. H. A. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. With discussion. J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
• Davis, M. H. A. Markov models and optimization. Monographs on Statistics and Applied Probability, 49. Chapman & Hall, London, 1993. xiv+295 pp. ISBN: 0-412-31410-X
• Dufour, François; Costa, Oswaldo L. V. Stability of piecewise-deterministic Markov processes. SIAM J. Control Optim. 37 (1999), no. 5, 1483–1502 (electronic).
• Dumas, Vincent; Guillemin, Fabrice; Robert, Philippe. A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34 (2002), no. 1, 85–111.
• Embrechts, Paul; Schmidli, Hanspeter. Ruin estimation for a general insurance risk model. Adv. in Appl. Probab. 26 (1994), no. 2, 404–422.
• Faggionato, A.; Gabrielli, D.; Ribezzi Crivellari, M. Non-equilibrium thermodynamics of piecewise deterministic Markov processes. J. Stat. Phys. 137 (2009), no. 2, 259–304.
• Hansen, Lars Peter; Scheinkman, José Alexandre. Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63 (1995), no. 4, 767–804.
• Harrison, J. Michael; Resnick, Sidney I. The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Oper. Res. 1 (1976), no. 4, 347–358.
• Harrison, J. Michael; Resnick, Sidney I. The recurrence classification of risk and storage processes. Math. Oper. Res. 3 (1978), no. 1, 57–66.
• P.G. Harrison and N. Thomas. Product-form solution in PEPA via the reversed process, Network Performance Engineering (D. Kouvatsos, ed.), Lecture Notes in Computer Science, vol. 5233, Springer Berlin Heidelberg, 2011, pp. 343–356.
• Hartman, Philip. Ordinary differential equations. Reprint of the second edition. BirkhÃ¤user, Boston, Mass., 1982. xv+612 pp. ISBN: 3-7643-3068-6
• Jacobsen, Martin. Point process theory and applications. Marked point and piecewise deterministic processes. Probability and its Applications. BirkhÃ¤user Boston, Inc., Boston, MA, 2006. xii+328 pp. ISBN: 978-0-8176-4215-0; 0-8176-4215-3
• Jacod, Jean; Protter, Philip. Time reversal on Lévy processes. Ann. Probab. 16 (1988), no. 2, 620–641.
• Jacod, J.; Skorohod, A. V. Jumping filtrations and martingales with finite variation. Séminaire de Probabilités, XXVIII, 21–35, Lecture Notes in Math., 1583, Springer, Berlin, 1994.
• Kelly, Frank P. Reversibility and stochastic networks. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1979. viii+230 pp. ISBN: 0-471-27601-4
• Kolmogoroff, A. Zur Theorie der Markoffschen Ketten. (German) Math. Ann. 112 (1936), no. 1, 155–160.
• bysame, Zur Umkehrbarkeit der statistischen Naturgesetze., Math. Ann. 113 (1936), 766–772 (German).
• Last, Günter. Ergodicity properties of stress release, repairable system and workload models. Adv. in Appl. Probab. 36 (2004), no. 2, 471–498.
• Borovkov, K.; Last, G. On level crossings for a general class of piecewise-deterministic Markov processes. Adv. in Appl. Probab. 40 (2008), no. 3, 815–834.
• Last, Günter; Szekli, Ryszard. Stochastic comparison of repairable systems by coupling. J. Appl. Probab. 35 (1998), no. 2, 348–370.
• Löpker, Andreas; Stadje, Wolfgang. Hitting times and the running maximum of Markovian growth-collapse processes. J. Appl. Probab. 48 (2011), no. 2, 295–312.
• Maulik, Krishanu; Zwart, Bert. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2006), no. 2, 156–177.
• M. Miyazawa. Reversibility in Queueing Models, ArXiv e-prints (2012).
• Nagasawa, Masao. Time reversions of Markov processes. Nagoya Math. J. 24 1964 177–204.
• Nagasawa, Masao. Time reversal of Markov processes and relativistic quantum theory. Chaos Solitons Fractals 8 (1997), no. 11, 1711–1772.
• Norris, J. R. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3
• Ogata, Y.; Vere-Jones, D. Inference for earthquake models: a self-correcting model. Stochastic Process. Appl. 17 (1984), no. 2, 337–347.
• Palmowski, Zbigniew; Rolski, Tomasz. A technique for exponential change of measure for Markov processes. Bernoulli 8 (2002), no. 6, 767–785.
• Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, Volker; Teugels, Jozef. Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1999. xviii+654 pp. ISBN: 0-471-95925-1
• M. SchÃ¤l, phSprungprozesse, Lecture Notes, University of Bonn (1997) (English).
• Schäl, Manfred. On piecewise deterministic Markov control processes: control of jumps and of risk processes in insurance. The interplay between insurance, finance and control (Aarhus, 1997). Insurance Math. Econom. 22 (1998), no. 1, 75–91.
• E. Schrödinger, phÃœber die Umkehrung der Naturgesetze., (1931) (German).
• Tanaka, Hiroshi. Time reversal of random walks in one-dimension. Tokyo J. Math. 12 (1989), no. 1, 159–174.
• E.A. Van Doorn and W.R.W. Scheinhardt, phAnalysis of birth-death fluid queues, Memorandum, University of Twente (1996).
• Löpker, Andreas H.; van Leeuwaarden, Johan S. H. Transient moments of the TCP window size process. J. Appl. Probab. 45 (2008), no. 1, 163–175.
• van Leeuwaarden, J. S. H.; Löpker, A. H.; Ott, T. J. TCP and iso-stationary transformations. Queueing Syst. 63 (2009), no. 1-4, 459–475.
• D. Vere-Jones. On the variance properties of stress release models., Austral. J. Statist. 30A (1988), 123–135.
• Walsh, John B. Time reversal and the completion of Markov processes. Invent. Math. 10 1970 57–81.
• Weiss, Gideon. Time-reversibility of linear stochastic processes. J. Appl. Probability 12 (1975), no. 4, 831–836.
• Wobst, Reinhard. On jump processes with drift. Dissertationes Math. (Rozprawy Mat.) 202 (1983), 51 pp.
• Yeh, J. Lectures on real analysis. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xvi+548 pp. ISBN: 981-02-3936-X.
• Zheng, Xiao Gu. Ergodic theorems for stress release processes. Stochastic Process. Appl. 37 (1991), no. 2, 239–258.


text:
# Differential equation RLC

I'm trying to find the current at $$t=\frac{\pi}{4}$$

The series RLC circuit has a voltage source given by $$E(t)=\sin(100t)$$ with a capacitor of 2 F, a resistor of 0.02 ohms, and an inductor of 0.001H. If the initial current and the initial charge on the capacitor are both zero, determine the current in the circuit for $$t=\frac{\pi}{4}$$

Based on the information given I have the equation as follows:

$$\frac{d^2I}{dt^2}+20\frac{dI}{dt}+500I=100000\cos(100t)$$

so the homogeneous equation associated is: $$r^2+20r+500=(r+10)^2+20^2=0$$

whose roots are $$-10 \pm 20i$$. Hence, the solution to the homogeneous equation is:

$$I_h=C_1e^{-10t}\cos(20t)+C_2e^{-10t}\sin(20t)$$

To find a particular solution, I use the method of undetermined coefficients.

$$I_p=-10.080 \cos(100t)+2.122 \sin(100t).$$

And after finding $$C_1$$ and $$C_2$$ I have:

$$I(t)=e^{-10t}(10.080 \cos(100t) -5.570 \sin(20t))-10.080\cos(20t)+2.122\sin(20t)$$

But $$I(\frac{\pi}{4})=0.080111$$, and the answer the professor give was 10.076. Am I doing something wrong?

• $0.001I_h''+0.02I_h'+500I_h \equiv I_h''+20I_h'+500000I_h=0$ – Cesareo Nov 23 '18 at 1:22
• @Cesareo, it wa a mistake. Now it's fixed. – Lexie Walker Nov 23 '18 at 1:28
• @Moo I didn't know if writing everything was crucial. I wrote cosine becuase with that form of equation is the derivate of E(t). – Lexie Walker Nov 23 '18 at 2:37
• What did you use for the initial conditions of $I$? We know the initial charge and the initial current, so isn't it better so find $q$ first? – Dylan Nov 24 '18 at 13:46

Hint.

We have

$$E(t) = \frac 1C \int_0^t i(\tau)d\tau+Ri(t) + L\frac{di}{dt}$$

or calling $$\int_0^t i(\tau)d\tau = q(t)$$

$$E(t) = \frac{q}{C}+R\frac{dq}{dt}+L\frac{d^2q}{dt^2}$$

Solving this DE with the initial conditions

$$q(0) = 0\\ \frac{dq}{dt}(0) = 0$$

we will obtain the desired result.

NOTE

Calling $$\alpha = \frac{R}{2L}$$ and $$\beta = \sqrt{\frac{C^2R-4L}{4L^2C}}$$ we have

$$q_h = (C_1\sin\beta t+C_2\cos\beta t)e^{-\alpha t}$$

and calling $$E(t) = E_0\cos\omega t$$

$$q_p = \frac{C E_0 \left(\left(1-C L \omega ^2\right) \cos (\omega t)+C \omega R \sin (\omega t)\right)}{C^2 \omega ^2 R^2+\left(CL \omega ^2-1\right)^2}$$

then

$$q = q_h + q_p$$

and now knowing that $$i(t) = q'$$ we can establish the $$C_1, C_2$$ constants.