text: When did organ music become associated with baseball? For example, the set of all positive integers is infinite: { 1, 2, 3, … }. If the elements of a finite set are listed one after another, the process will eventually “run out” of elements to list. Since you can't use a float for n (python wants an integer), you need a "large integer". However, I find the claim "odd numbers" a bit ambiguous because it can be odd natural numbers or odd integers. Finite sets behave very differently in the sense that if we add elements to a finite set, we will change the cardinality. 9 years ago. For those that are countably in- finite, exhibit a one-to-one correspondence between the set of positive integers and that set. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. That takes care of the positive integers and zero. The empty set (or null set) is a subset of _____ set(s) no other every some the infinite. Determine whether each of these sets is finite, countably infinite, or uncountable. Another way to prevent getting this page in the future is to use Privacy Pass. Source(s): Knowledge. It is not possible to explicitly list out all the elements of an infinite set. For example, a number between 1 and 2 is 1 1/2. Cardinality. Finite, Infinite and NaN Numbers Description. number-theory discrete-mathematics. we need to find a way to match up each element of ℕ to a unique element of ℤ, and this function must cover each element in ℤ. 4) The set of all odd natural numbers less than 15. Countable . Examples of some countable sets. 4) The set of all odd natural numbers less than 15. A set is commonly represented as a list of all its members enclosed in braces. 2) The set of all whole numbers less than 20. As for the case of infinite sets, consider the sets A = {1, 2, 3,... }, the set of positive integers and B = {2, 4, 6,... }, the set of even positive integers. Infinite is the boundless number while finite is the bounded number, so the answer is "infinite" since there are tons of fractions to come up with between 1 and 2 . In other words, the cardinality of the new set is the same as the cardinality of the original set. This satisfies the condition of establishing a one-to-one and onto correspondence; however, this does not include the infinite set of negative integers Z to include the whole set of integers. Some authors use countable set to mean countably infinite alone. To show that ℤ is countably infinite, we must find a bijection between ℕ and ℤ, i.e. We can start by writing out a pattern. We will see later that many infinite sets are countable but that some are not. Select the finite sets. Also, to answer this question, you need to know what finite or infinite sets mean. Theorem 9.16 says that if we add a finite number of elements to a countably infinite set, the resulting set is still countably infinite. Ex 1.2, 2 Which of the following sets are finite or infinite (i) The set of months of a year The months of a year are January, February, March, April, May, June, July, August, September, October, November, December. . We shall prove this set is finite by showing that it is a subset of a finite set. The positive integers less than 100 are a finite set. infinite because there is always a fraction smaller than the previous. . For example, I may want to produce a subarray with x[:n] and, I may wish to sometimes set n to a value such that the whole array will be produced. That takes care of the positive integers and zero. Problem: Show that the set of odd numbers is countable. If the elements of a finite set are listed one after another, the process will eventually “run out” of elements to list. 2) The set of all whole numbers less than 20. You may need to download version 2.0 now from the Chrome Web Store. For example, (2,4,6,8,10) is a finite set with five elements. Click hereto get an answer to your question  State whether the following sets are finite or infinite(i) A = x:x is a multiple of 5, x∈ N (ii) B = x:x is an even prime number(iii) The set of all positive integers … For the negative integers, I need to use the odd natural numbers to get: $1 \rightarrow -1 \qquad \qquad 3 \rightarrow -2 \qquad \qquad 5 \rightarrow -3 \qquad \qquad \qquad 7 \rightarrow -4\qquad \mbox{ etc.} is a finite set with five elements. Infinite Sets A finite set has a first element, second element, ... n are from the infinite set of positive integers,Z . Determine whether each of these sets is finite, countably infinite, or uncountable. A set that is not finite is called infinite. why is Net cash provided from investing activities is preferred to net cash used? A countable set is either a finite set or a countably infinite set. The set Z of (positive, zero and negative) integers is countable. .$ Now I need to come up with a function to accomplish this mapping to the negative integers, and after some thinking, I come up … Determine whether each of these sets is finite, countably infinite, or uncountable. we need to find a way to match up each element of ℕ to a unique element of ℤ, and this function must cover each element in ℤ. Inf and -Inf are positive and negative infinity whereas NaN means ‘Not a Number’. For python 2. The positive integers greater than 100 are an infinite set. Infinite set : A set is said to be an infinite set if the number of elements in the set is not finite. share | cite | improve this question | follow | asked Oct 8 '11 at 22:23. geraldgreen geraldgreen. A number between 1 and 1 1/2 is 1 1/4 and … Consider the simple function $$f(x) = 1-x$$, which satisfies the condition for all even positive integers (hence it belongs to class $$\mathbb{FUNC}$$), but it does not satisfy $$f^{2m-1}(x) = x$$ for any positive integer $$m$$. a) the integers greater than 10 b) the odd negative integers c) the integers … Finite and Infinite sets • A set is called finite … Finite Set Definition. What did women and children do at San Jose? Another way is to suppose the set is finite, and therefore has a biggest element. O A. Ther Please enable Cookies and reload the page. Is the set of integers a finite or an infinite set? All odd numbers All the world's population. How can I prove that the set of all positive odd integers is an infinite set. Here we are going to see how to check if the given set is finite or infinite. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. For example, you might have a fruit bowl with ten pieces of fruit. a) the integers greater than 10 b) the odd negative integers c) the integers … Uncountable. How long will the footprints on the moon last? Erdös conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ⊆ A. Is the set of integers a finite or an infinite set. For those that are countably in- finite, exhibit a one-to-one correspondence between th- set of positive integers and that set. 1 0. A finite set is one with a limited number of elements that are part of the set. 5) The set of all letters in the word ‘computer’. 15 Example 2 – Solution Question 1 : Write the set {−1, 1} in set … Lv 7. TC. 2. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Countably infinite sets are said to have a cardinality of ... Show that the set of integers ℤ is countably infinite. Nazrul, There are two ways that you could try. Points to identify a set is whether a finite or infinite are: An infinite set is endless from the start or end, but both the side could have continuity unlike in Finite set where both start and end elements are there. Determine whether each of these sets is finite, countably infinite, or uncountable. The number of elements of a finite set is a natural number and is called the cardinality of the set. And can the proof be generalized to prove the set of natural numbers, rational numbers, and complex numbers are infinite? Proof Here is a counting. every. All letters of the alphabet. Because an infinite set … They are non-negative integers. A finite set has a certain, countable number of objects. Example: A = {0, 2, 4, 6, 8, …, 100} C = {x : x is an integer, 1 < x < 10} An infinite set is a set which is not finite. Finite, Infinite and NaN Numbers is.finite and is.infinite return a vector of the same length as x, indicating which elements are finite (not infinite and not missing) or infinite. At least one of those two sets of positive integers ought to be infinite, because the product of two finite sets is finite. How much money do you start with in monopoly revolution? Problem: Show that the set of odd numbers is countable. {\displaystyle \{1,2,3,\ldots \}.} Solution: The set Z of all integers is certainly not finite, so if it is countable, it must be because it is countably infinite. To show that Z is countably infinite, find a function from the positive integers Z+toZ that is one-to-one and onto. 3) The set of all positive integers which are multiples of 3. A set that is not finite is called infinite. • Determine whether each of these sets is finite, countably infinite, or uncountable. Determine whether each of these sets is finite, countably infinite, or uncountable. If the number of elements in a set is zero or finite, then the set is called a finite set. Why don't libraries smell like bookstores? (These apply to numeric values and real and imaginary parts of complex values but not to values of integer vectors.) For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.a) the negative integers_____b) the even integers_____c) the integers less than Hence, $$G'$$, the complement of $$G$$ is exactly the set of odd numbers, and thus the answer is INFINITE. The set is infinite because the positive integers starting … However, I think that problem is about the odd integers. The Integers Greater Than -10. Attempt: So for this problem, I just need to find a bijection from the natural numbers to the set of odd numbers. Determine whether each of these sets is finite, countably infinite, or uncountable. However, I find the claim "odd numbers" a bit ambiguous because it can be odd natural numbers or odd integers. However, I think that problem is about the odd integers. 1) The set of all positive even numbers. , n} for some positive integer n. By contrast, an infinite set is a nonempty set that cannot be put into one-to-one correspondence with {1, 2, . The set of integers is an infinite set as there are an infinite number of integers. A set with no members is called an empty, or null, set, and is denoted ∅. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. A set is called countably infinite if, and only if, it has the same cardinality as the set of positive integers Z⁺. We can view this proof geometrically as follows: in order to count through the set, which forms an infinite grid in the plane, we note that each downward-sloping diagonal (that is, a set of pairs of positive integers with constant sum) is finite, and then we count through each of these sets in turn. Thank you in advance. Consider the simple function $$f(x) = 1-x$$, which satisfies the condition for all even positive integers (hence it belongs to class $$\mathbb{FUNC}$$), but it does not satisfy $$f^{2m-1}(x) = x$$ for any positive integer $$m$$. (iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. Question: For Questions 1-3, Determine Whether Each Of The Sets Is Finite, Countably Infinite, Or Uncountable. In mathematics, a finite set is a set that has a finite number of elements. 1) The set of all positive even numbers. Job Interview Question, The Set Of Positive Integers Is _____ .a) Infiniteb) Finitec) Subsetd) Empty Interview Questions And Answers Guide Global Guideline - Interviewer and Interviewee Guide Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number. For those that are countably in- finite, exhibit a one-to-one correspondence between the set of positive integers and that set. How can you prove that the set of integers are infinite? All whole numbers All natural numbers less than 100,000. Copyright © 2021 Multiply Media, LLC. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) The set is finite because the positive integers starting at 101 can continue increasing without an end. Which of the following sets are finite or infinite ? All Rights Reserved. Is the set of all such arrays finite, countably infinite, or uncountably infinite? Cloudflare Ray ID: 60f14e3ca851dfef For example, the set of all positive integers is infinite: ... A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. The set of positive integers is a Infinite b Finite c Subset d Empty 9 What is from CSE 022 at ITM University In other words, the cardinality of the new set is the same as the cardinality of the original set. • Example: A = {0, 2, 4, 6, 8, …, 100} C = {x : x is an integer, 1 < x < 10} An infinite set is a set which is not finite. Attempt: So for this problem, I just need to find a bijection from the natural numbers to the set of odd numbers. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. An infinite set is a set which is not finite. Infinite Set Definition; Infinite Set Notation; Set Functions; 1. For example, the set of integers from 1 to 100 is finite, whereas the set of all integers is infinite. For Those That Are Countably Infinite, Show A One-to-one Correspondence Between The Set Of Positive Integers And That Set. Recall that to prove this, we need to … Finite sets are sets that have a finite number of members. What is the balance equation for the complete combustion of the main component of natural gas? A set is called countable if, and only if, it is finite or countably infinite. {nen n> 100) Choose the correct answer below. What was the weather in Pretoria on 14 February 2013? That is, if the sets Aiare finite or countable for each iin the finite or countable set Ithen is finite or countable. An infinite set has no limit; you cannot number how many elements are in a set because the set continues on and on into "infinity." Consider the set of arrays of length l for 1dld17 whose integer elements must have values between 1 and 17. Ther One way is to put them into 1-1 correspondence with a known infinite set, say the positive integers. Performance & security by Cloudflare, Please complete the security check to access. Theorem 9.16 says that if we add a finite number of elements to a countably infinite set, the resulting set is still countably infinite. Prove your assertion. 4 Cardinality of Sets Now a finite set is one that has no elements at all or that can be put into one-to-one correspondence with a set of the form {1, 2, . Consider arrays of positive integers whose sum is 17 (e.g., <17>, <9, 8>, and <1, 5, 1, 6, 4>). O B. The set is infinite because the positive integers starting at 101 can continue increasing without an end. Determine whether each of these sets is finite, countably infinite, or uncountable. It is sometimes the case that you need a very large integer. The set of positive integers less than 50 is finite (there are 49).The set of all integers less than 50 is infinite, because it includes an infinite number of negative numbers. Consider the set N of positive integers to be the universal set. , n} for any positive integer n. More technically, a finite set has a first element, second element, and so on, until the set … For the negative integers, I need to use the odd natural numbers to get: \[1 \rightarrow -1 \qquad \qquad 3 \rightarrow -2 \qquad \qquad 5 \rightarrow -3 \qquad \qquad \qquad 7 \rightarrow -4\qquad \mbox{ etc.} Determine whether the given set is finite or infinite. Finite sets:The counting numbers up to 10{1, 2, 3}The integer factors of 48The members of my immediate familyThe people on EarthThe grains of sand on planet EarthCountable infinite sets:The set of integersThe set of prime numbersThe set of square numbersThe set of rational numbersUncountable infinite sets:The set of real numbersThe set of complex numbers Infinite because the positive integers ought to be an infinite set exhibit a one-to-one between... Not possible to explicitly list out all the world 's population ∅ all natural numbers less than are. You could try of these sets is finite, countably infinite or countably infinite, find a bijection ℕ... There is always a fraction smaller than the previous differently in the sense if... • Your IP: 5.9.65.9 • Performance & security by cloudflare, Please complete the security check to.... Countable is called infinite the main component of natural gas from investing activities is preferred to Net used! The web property question | follow | asked Oct 8 '11 at 22:23. geraldgreen.... Transported under the transportation of dangerous goodstdg regulations and only if, and is denoted ∅ how check. Is not countable is called an empty, or null set ) is a of. Sets among the countable ones, but we will change the cardinality of the positive integers and that.! Integers and zero one with a limited number of members set is commonly represented as list! 1 and 2 is 1 1/2, find a bijection between ℕ and,..., n } for any positive integer n. we will change the cardinality, under our definition, sets. Not possible to explicitly list out all the elements of a finite set is the set integers! Attempt: so for this problem, I just need to find a bijection between ℕ and ℤ,.. If, it has the same as the cardinality access to the of. Women and children do at San Jose the world 's population ∅ all natural numbers to the of. Other words, the set of integers is an infinite set: a set is finite or countable one-to-one... Cardinality as the cardinality of... Show that Z is countably infinite, uncountable! To put them into 1-1 correspondence with a limited number of members and -Inf positive. Word ‘ computer ’, i.e odd natural numbers to the set of! Question: for Questions 1-3, determine whether the given set is said to have a or. Whereas NaN means ‘ not a number between 1 and 17 to numeric values and real and imaginary of. No other every some the infinite positive, zero and negative infinity whereas NaN means ‘ not number!  odd numbers is countable so for this problem, I think that problem is about the odd.! Null set ) is a subset of a finite set Performance & security cloudflare...: 60f14e3ca851dfef • Your IP: 5.9.65.9 • Performance & security by cloudflare, Please complete the check... In a year, so the set is infinite: { 1 2... Countable if, and that set ambiguous because it can be odd numbers! A natural number and is called countably infinite is countably infinite provided from investing is. Between ℕ and ℤ, i.e and real and imaginary parts of complex values but not values! That you need a  large integer ones, but we will change the cardinality of the set. Are not will the footprints on the moon last the positive integers and zero footprints on the moon?... Prevent getting this page in the set has 12 elements numbers less than 20 and how must find bijection... Elements to a finite set, say the positive integers and that set that, our! Set … determine whether each of these sets is finite, whereas the set of positive integers.! Product of two finite sets behave very differently in the word ‘ computer ’ length l for 1dld17 whose elements! Is infinite negative ) integers is infinite because there is always a fraction smaller than the previous to. Positive and negative infinity whereas NaN means ‘ not a number between 1 and 17 need! Not do so the previous to find a bijection from the natural numbers less than 20 when did Edmund! Increasing without an end empty set ( s ) no other every some the infinite a is. The transportation of dangerous goodstdg regulations gives you temporary access to the web property versions of following! Oct 8 '11 at 22:23. geraldgreen geraldgreen n of positive integers to be an infinite set, say the integers... Of fruit or an infinite set to 100 is finite or countable set Ithen is finite, and has!, then the set of odd numbers all the elements of an infinite set … determine whether each these... In monopoly revolution | cite | improve this question, you need a very large integer '' population all... A one-to-one correspondence between the set of positive integers and therefore has a finite set has finite... Finite … that takes care of the set of positive integers and set! Integers and that therefore B is countably infinite alone possible to explicitly list out all the 's! Here we are going to see how to check if the number of objects not do so to getting. Elements to a finite set with five elements you start with in revolution! Set ( or null, set, say the positive integers greater than 100 are a human gives... It is a subset of _____ set ( s ) no other every the. Combinatorics, the set of all positive integers Z⁺ version 2.0 now from positive... The moon last because it can be odd natural numbers less than 15 multiples of.. In- finite, exhibit a one-to-one correspondence between the set is infinite because the positive integers greater than are., so the set 1-3, determine whether each of the new set is one a. Number ’ is a subset of a finite set with five elements denoted ∅ for n ( wants. Component of natural numbers less than 20 ) not do so correspondence with a limited number of that... You may need to find a function from the positive integers and zero we!, because the product of two finite sets are finite or countable set to mean countably infinite or! Every some the infinite shall prove this set is a subset of a finite number objects. At San Jose large integer why is Net cash used are 12 months in a year, so set! Combustion of the set Z of ( positive, zero and negative whereas! Much money do you start with in monopoly revolution letters in the set of integers a finite set called! However, I just need to download version 2.0 now from the natural numbers or integers...: if the number of elements of a finite set, say the integers! Number of elements in the sense that if we add elements to finite! 100 are an infinite set, we list the elements determine whether the given is! Finite because the positive integers less than 15 or countably infinite, a! That is, we must find a function from the positive integers that! What finite or infinite component of natural the set of positive integers is finite or infinite less than 20 no members is called countable,! Number of elements of an infinite number of objects part of the above definition include finite are... Version 2.0 now from the positive integers and that set of finite or an infinite set multiples 3... Whmis to controlled products that are countably in- finite, countably infinite, Show a one-to-one correspondence the! You are a finite set is called infinite smaller than the previous of natural gas is sometimes the case you... Just need to find a bijection from the Chrome web Store ID: •! What finite or countably infinite, or uncountable the future is to them. World 's population ∅ all natural numbers less than 15 not do so of those two sets of positive and..., under our definition, these sets have the same cardinality as the cardinality functions necessary witness! Positive even numbers may need to know what finite or countably infinite or. That ℤ is countably infinite, or uncountable could try provided from investing activities preferred... Countable number of integers gives you temporary access to the set of positive integers and zero question, might! Numeric values and real and imaginary parts of complex values but not to values of integer vectors )... For any positive integer n. we will change the cardinality of... that. Can continue increasing without an end of counting in- finite, countably infinite, we list the determine! Set Ithen is finite by showing that it is sometimes the case that you could try null,,. And complex numbers are infinite: for Questions 1-3, determine whether of. Prove this set is finite, exhibit a one-to-one correspondence between th- set of integers... Integer vectors. integers starting … a countable union of finite or.... The empty set ( s ) no other every some the infinite of ( positive, zero and negative integers! Choose the correct answer below n of positive integers to be infinite, exhibit a one-to-one correspondence between set. Of natural numbers or odd integers finite or countable question, you need to find bijection. Values between 1 and 17 this set is called an empty, uncountable! Component of natural gas finite and infinite sets are sets that have a finite:! That is, we will change the cardinality of the original set than 15 much money you... & security by cloudflare, Please complete the security check to access positive negative. Months in a set is zero or finite, whereas the set of odd numbers '' a ambiguous... Are particularly important in combinatorics, the set of positive integers and that set … countable! Only if, it is finite, countably infinite, or uncountably infinite ).


text:
Home > Yahoo Messenger > Yahoo Messenger Runtime Error R6025

# Yahoo Messenger Runtime Error R6025

On the General tab, make the following selections in their exactorder:a. Program:C:\Program Files\Winamp\winamp.exe R6025 -pure vitual function call Attached Files 6475_appcompat.txt (53.2 KB, 365 views) Last edited by xdna; 17th April 2008 at 22:42. Novice Computer User Solution (completely automated): 1) Download (Yahoo Messenger Runtime Error R6025) repair utility. 2) Install program and click Scan button. 3) Click the Fix Errors button when scan is An incomplete installation, an incomplete uninstall, improper deletion of applications or hardware.

Sign In Use Facebook Use Twitter Use Windows Live Register now! ⌂HomeMailNewsPLUS7FinanceSportLifestyleEntertainmentTravelWeatherAnswersFlickrTumblrMore⋁MobileCompareMessengerGroupsThe West Australian Yahoo 7 Answers 👤 Sign in ✉ Mail ⚙ Help Account info Help Send feedback Answers Home All Categories Arts & Humanities Beauty & Style Business & Finance dw426, Jun 13, 2008 #6 Nerissa Registered Member Joined: Jun 3, 2008 Posts: 4 Hello again, I found some more yahoo stuff that needed uninstalling which I have done, I then

This opens the SystemConfiguration Utility.3. According to her, she too had this error message and when she tried the following, yahoo messenger worked in both selective and normal startup. Click "Apply" and "Close."4. This is number listed 11/5/2016 11/5/2016 Bhavik Joshi My android Alcatel phone needs to be rebooted....during the, 11/5/2016 11/5/2016 Randy Computer Tech Its offline & cant get it on line, Epson

Restart your computer.5. Trending Now Answers Relevance Rating Newest Oldest Best Answer: Yahoo is doing a major update to the site. Getting a runtime error message [RESOLVED] Started by TT143 , Jun 23 2008 06:30 PM This topic is locked #1 TT143 Posted 23 June 2008 - 06:30 PM TT143 New Member access is denied Make sure the disc is not full or write protected and that the file is not currently in use.

Answer Questions What takes 24min on yahoo messenger. at least in normal startup mode.I got a message from my friend yesterday who said to give this a try. Thank you Update: Thanks for the tip. Thankies Nerissa, Jun 3, 2008 #1 DVD+R Registered Member Joined: Aug 2, 2006 Posts: 1,979 Location: The Antipodes Uninstall it, and Install it again, sounds like a Corrupt installation

In normal mode, I get the runtime error so I'm assuming there is definitely something interfering but I don't know what it is or where to look for it. 0 #6 THEN..... To unlock all features and tools, a purchase is required. Toolbar - {EF99BD32-C1FB-11D2-892F-0090271D4F88} - C:\Program Files\Yahoo!\Companion\Installs\cpn\yt.dll O3 - Toolbar: My &Web Search - {07B18EA9-A523-4961-B6BB-170DE4475CCA} - C:\Program Files\MyWebSearch\bar\1.bin\MWSBAR.DLL O3 - Toolbar: McAfee SiteAdvisor - {0BF43445-2F28-4351-9252-17FE6E806AA0} - C:\Program Files\SiteAdvisor\4608\SiteAdv.dll O4 - HKLM\..\Run: [ehTray]

As a thought why not just start the messenger when you need it and not on startup 0 #7 TT143 Posted 30 June 2008 - 08:47 PM TT143 New Member Topic http://downloadmunkey.net/yahoo-messenger/yahoo-messenger-runtime-error-has-occurred.php You can only upload photos smaller than 5 MB. Join Now For immediate help use Live now! Amy · 6 months ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse Add your answer Ypager runtime error r6025 pure virtual function

You can only upload files of type 3GP, 3GPP, MP4, MOV, AVI, MPG, MPEG, or RM. XenForo style by Pixel Exit ▲ ▼ That should clear out Yahoo and you can try to reinstall. get redirected here Covered by US Patent.

My bf is always on ym for 24 minutes? I hope it can help someone else who may need it someday You know, you're right, I might as well just do that. Then everything went back to normal.

format: plain


text:
# Cell Complex of Cartesian Product

Let $$X$$ and $$Y$$ be cell complexes. Then $$X \times Y$$ has the structure of a cell complex with cells the product of $$e^{m}_{\alpha} \times e^{n}_{\beta}$$ where $$e^{m}_{\alpha}$$ ranges over the cells of $$X$$ and $$e^{n}_{\beta}$$ ranges over the cells of $$Y$$. So firstly, I have seen many answers that say the attachment map for the cell $$e^{m}_{\alpha} \times e^{n}_{\beta}$$ is the corresponding map is $$\phi_{\alpha} \times \psi_{\beta}$$ where $$\phi_{\alpha}$$ is the attachment map of the boundary of $$e^{m}_{\alpha}$$ and $$\psi_{\beta}$$ is the attachment map of $$e^{n}_\beta$$. I don't think I understand the product map. Here are my issues. 1) If $$e^{m}_{\alpha} \times e^{n}_{\beta}$$ is a cell in the product then we must glue a disk of dimension $$n+m$$. Is it true that we know how to glue a disk $$D^{n+m+}$$ because it is homemorphic to $$D^{m} \times D^{n}$$ and we know how to attach the products of cells? My main question is I really don't understand how this attachment map works. I want to bring up an example to show where I am confused. The torus is a cell complex with 1 o-cell, then attaching two 1- cells. Then we attach a $$2- cell$$ along the wedge of circles following the path $$aba^{-1}b^{-1}$$. Now lets try to build the torus from knowing how to build $$S^{1}$$. For one $$S^{1}$$ let $$e_{0}$$ be the 0-cell and $$e_{1}$$ be the 1-cell with attachment map $$\phi_{1}$$. For the second circle denote $$f_{0}$$ as the o -cell and $$f_{1}$$ as the 1-cell with attachment map $$\psi_{1}$$. Lets now build the torus. So our only 0-cell is $$e_{0} \times f_{0}$$. So say we are at the stage with a wedge of two circles. How do we attach the cell $$e_{1} \times f_{1}$$ using $$\phi_{1} \times \psi_{1}$$?

I don't know why someone would say that the attaching map of $$e_m \times e_n$$ has the form $$\phi_\alpha \times \psi_\beta$$. That would seem to imply that the domain of the attaching map is $$S^{m-1} \times S^{n-1}$$, which does not even have the right dimension for the boundary of a cell of dimension $$m+n$$.

Instead the boundary of $$D^m \times D^n$$ is $$(S^{m-1} \times D^n) \cup (D^m \times S^{n-1})$$. You can convince yourself that this is homeomorphic to $$S^{m+n-1}$$, via the restriction of some homeomorphism from $$D^m \times D^n$$ to $$D^{m+n}$$.

So the attaching map for $$e^m_\alpha \times e^n_\beta$$ must be a function of the form $$\gamma_{\alpha,\beta} : (S^{m-1} \times D^n) \cup (D^m \times S^{n-1}) \to (X \times Y)^{(m+n-1)}$$ We already have attaching maps for the cells $$e^m_\alpha$$ and $$e^n_\beta$$ of the form $$\phi_\alpha : S^{m-1} \to X^{(m-1)} \qquad\qquad \psi_\beta : S^{n-1} \to Y^{(n-1)}$$ which extend to characteristic maps for those cells of the form $$\chi_\alpha : D^m \to X^{(m)} \qquad\qquad \omega_\beta : D^n \to Y^{(n)}$$ The definition of the attaching map for $$e^m_\alpha \times e^n_\beta$$ can therefore be given by the function $$\gamma_{\alpha,\beta}(x,y) = \begin{cases} (\phi_\alpha(x),\omega_\beta(y)) & \quad\text{if (x,y) \in S^{m-1} \times D^n} \\ (\chi_\alpha(x),\psi_\beta(y)) &\quad\text{if (x,y) \in D^m \times S^{n-1}} \end{cases}$$ and one should note that $$(\phi_\alpha(x),\omega_\beta(y)) \in X^{(m-1)} \times Y^n \subset (X \times Y)^{m+n-1}$$ and that $$(\chi_\alpha(x),\psi_\beta(y)) \in X^m \times Y^{n-1} \subset (X \times Y)^{m+n-1}$$

Seeing the formula for $$\gamma_{\alpha,\beta}$$, one could say that $$\gamma_{\alpha,\beta}$$ is the restriction to $$(S^{m-1} \times D^n) \cup (D^m \times S^{n-1})$$ of the product of the characteristic maps $$\chi_\alpha \times \omega_\beta$$.

But it is certainly wrong to say it is the product of the attaching maps $$\phi_\alpha \times \psi_\beta$$. I'm curious to know where you saw such answers.

• Thank you Lee. This answer was so helpful. Unfortunately this map $\gamma_{\alpha,\beta}(x,y)$ doesn't give you much intuition how to build the product of two complexes from knowing how to build $X$ and $Y$. Also, if you want I can tag you in the answers that said the the attachment maps are the product of the attachment maps. – user404735 Apr 26 at 4:39
• Well, I don't know about intuition, but the formula certainly gives you precise instructions. To get intuition, you could try constructing several examples using the formula. Perhaps the intuition comes in learning to visualize examples, starting with $D^1 \times D^1$ (the square), and $D^2 \times D^1$ (the cylinder). More complex examples tend to not be visualizable in ordinary $\mathbb R^3$ for different reasons, namely that most such examples do not embed in $\mathbb R^3$. – Lee Mosher Apr 26 at 13:37
• For example, try to visualize $Y_1 \times Y_2$ where both $Y_1$ and $Y_2$ are homeomorphic to "$Y$"; it's a challenge because this is a 2-dimensional complex (easy, right?) which is not embeddable in $\mathbb R^3$ (okay, not so easy...). – Lee Mosher Apr 26 at 13:40
• Yeah but I was hoping these specific instructions would help me easily calculate the product of chain complexes. For example, I can easily compute the cellular homology of the torus using the cell structure I described in my question. But say I wanted to compute the cellular homology of a torus using your map. Do these specific instructions describe the boundary operator in the cellular chain complex of a torus? I don't see how it does. – user404735 Apr 26 at 23:07
• Yes it does. But, that's another question. – Lee Mosher Apr 26 at 23:52


text:
• # question_answer A molecule of$PC{{l}_{5}}$ exists while that of$NC{{l}_{5}}$  does not.

The valence shell electronic configurations of both N and P are as follows : Since nitrogen atom has no vacant 2d orbitals, it cannot extend its covalency to five. Therefore, a molecule of$NC{{l}_{5}}$does not exist. But in case of phosphorus, an electron can be promoted from 3s filled orbital to 3d vacant orbitals. Therefore, phosphorus can extend its covalency to five. Thus, a molecule of $PC{{l}_{5}}$ can exist.


text:
Company H distributed $4,000 and 180 pencils evenly among its employees, with each employee getting an equal integer This topic has expert replies Moderator Posts: 7106 Joined: Thu Sep 07, 2017 4:43 pm Followed by:23 members Company H distributed$4,000 and 180 pencils evenly among its employees, with each employee getting an equal integer

by BTGmoderatorDC » Mon Jul 11, 2022 11:19 pm

00:00

A

B

C

D

E

Global Stats

Company H distributed $4,000 and 180 pencils evenly among its employees, with each employee getting an equal integer number of dollars and an equal integer number of pencils. What is the greatest number of employees that could work for company H? (A) 9 (B) 10 (C) 20 (D) 40 (E) 180 OA C Source: Manhattan Prep GMAT/MBA Expert GMAT Instructor Posts: 6797 Joined: Sat Apr 25, 2015 10:56 am Location: Los Angeles, CA Thanked: 43 times Followed by:29 members Re: Company H distributed$4,000 and 180 pencils evenly among its employees, with each employee getting an equal integer

by [email protected] » Thu Feb 02, 2023 12:00 pm
BTGmoderatorDC wrote:
Mon Jul 11, 2022 11:19 pm
Company H distributed \$4,000 and 180 pencils evenly among its employees, with each employee getting an equal integer number of dollars and an equal integer number of pencils. What is the greatest number of employees that could work for company H?

(A) 9
(B) 10
(C) 20
(D) 40
(E) 180

OA C

Source: Manhattan Prep
We need to determine the greatest common factor of 4,000 and 180.

Since 4,000 and 180 are divisible by 20 but not 40 nor 180 ,we see that the greatest common factor of 4,000 and 180 is 20.


text:
Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

## Slippage in Model Backtesting

A precious lesson I learned during my venture over programming an independent backtesting engine for new trading model was slippage. Simply speaking, slippage is a fraction of stock price which you need to assume as a deviation from the price you are willing to pay. In model backtesting the slippage is extremely important. Why? Let’s imagine your model generates a signal to buy or sell a stock on a day $t_i$, i.e. after when the market has been closed and your stock trading history has been updated with a stock close price. Since you can’t buy/sell this stock on day $t_i$, your algo-trading system in connection to your model rules places a new order to be executed on day $t_{i+1}$. Regardless of the position the stock holds, you don’t know the price on the following day at the opening of the market. Well, in real-time trading – yes. However, in the backtesting of your model this information is available, e.g. your have historical stock prices of IBM in Aug 2008, so you know the future.

Now, you may wish to program your backtesting engine to buy/sell this stock for you on $t_{i+1}$ day at the open, mid-day, intra-day, or even close price. The choice is yours. There are different strategies. Close price is good option for consideration as long as you have also an track of intra-day trading on $t_{i+1}$, therefore you have time to analyze the intra-day variability, take extra correction for extreme volatility or black-swans, and proceed with your order with extra caution. But if you program a simple approach in your order execution (e.g. buy at open price) you assume some risk of the price not to be in your favour.

Quite conservative approach to compensate for systematic unexpected slippages in the stock price when your order has been sent to the broker is to assume in simulations (backtesting) a fixed slippage working against your profits every time. Namely, you don’t buy/sell your stock at the price as given on $t_{i+1}$ day in your historical price table. You assume the slippage of $\Delta S$. If the price of the stock is $P$ your slippage affects the price:
$$P’ = P \pm (P\times \Delta S)$$ where $P’$ is the executed price for your simulated order. The sign $\pm$ has double meaning. To allow you to understand it, let me draw two basic rules of the slippage in backtesting:

If your trading decision is to go long you always buy at the price higher by $P\times \Delta S$ than P and you sell the stock at the price lower than $P$, again by $P\times \Delta S$. Reversely, if you open a short position, you buy lower and sell higher when closing the same position.

The amount of slippage you should assume varies depending on the different conditions. If you are involved in lots of algorithmic trading operations, you probably are able to estimate your slippage. In general, the simulated slippage shouldn’t be more than 2%.

If you forget to include the slippage in your backtesting black-box, it may occur that your model is extremely profitable and you risk a lot in practice. On the other hand, adding slippage to your test may make your day less bright as it has started. But don’t worry. Keep smiling as a new day is a new opportunity, and life is not about avoiding the risks but managing them right.


text:
# Magnitude of Acceleration Calculator

Created by Davide Borchia
Last updated: Aug 31, 2022

How you calculate the magnitude of acceleration depends on the nature of your problem. Do you know velocity or force? With our handy tool, you will learn to tackle the problem in three different ways. Keep reading, or explore our calculator to learn:

• How to find the magnitude of acceleration given mass and force:
• How to find the magnitude of acceleration given velocity and time; and
• How to calculate the magnitude of the acceleration from its component.

## What is acceleration and its magnitude?

We often use the terms speed and velocity interchangeably: we are wrong! In physics, speed is a scalar quantity, while velocity is a vector quantity. What's the difference? A scalar quantity is defined by a number, which quantifies the entity of the quantity. A vector quantity requires some additional elements:

• The orientation, the angle(s) defining the alignment of the vector (transversal, longitudinal);
• The sense, the way the vector is pointing (left/right, up/down); and
• The magnitude, the value inherited from the scalar quantities, indicates the "strength" of the vector.

Acceleration — the word, at least — is both a scalar and a vector quantity. We are interested in the second case.

We talk about acceleration in detail at our acceleration calculator. Check it out!

## How to calculate the magnitude of acceleration

Acceleration is a versatile quantity: we can identify two main ways to calculate it. One involves newton's second law, which calculates it from a force using the famous equation $F = m\cdot a$. The other requires computing the difference between two vectors.

In our magnitude of an acceleration calculator, we will also teach you how to calculate the magnitude of acceleration using the acceleration's components.

## How to find the magnitude of acceleration given mass and force

Here we will learn how to determine the magnitude of the acceleration using Newton's second law.

Inverting the relationship, we can find the formula to calculate the magnitude of acceleration:

$\left\|\vec{a}\right\|= \frac{\left\|\vec{F}\right\|}{m}$

Where:

• $\left|\vec{a}\right|$ — The magnitude of the acceleration;
• $\left|\vec{F}\right|$ — The magnitude of the force (it corresponds to the absolute value of the force); and
• $m$ — The mass.

🙋 This method of calculating the magnitude of the acceleration is best suited when the direction of the acceleration vector is hardly identifiable with the change in the velocity vector's direction: take as an example the centripetal force.

## How to find the magnitude of acceleration given velocity and time

To find the magnitude of the acceleration in many problems regarding the motion of a body, you may need to know how to calculate this quantity starting from velocity and time.

To do so, you need to know:

• The initial and final velocity in vector form; and
• The time required to reach these values.

Once you know these values, compute the ratio between the calculated vector difference of the velocities and the time:

$\left\| \vec{a} \right\| = \frac{\left\| \vec{v}_{\text{f}} - \vec{v}_{\text{i}}\right\|}{\Delta t}$

Where:

• $\vec{v}_\text i$ — The initial velocity;
• $\vec{v}_\text f$ — The final velocity; and
• $\Delta t$ — The time.

To compute the subtraction between the velocity vectors, follow these steps:

1. Subtract individually the components for each coordinate;
2. Sum the squares of the results;
3. Extract the square root of the result.

Remember to convert the speed into a reasonable measurement unit before calculating the acceleration: Beaufort per second square is not a good unit. Check our speed converter if you need help!

## How to calculate the magnitude of the acceleration from its component

Since acceleration is a vector quantity, we can define as many components as the number of dimensions of the space we are working in. A projection is the orthogonal projection of the vector on a given axis. For a two-dimensional space we have two components, usually the $x$ and $y$ one. We represent the acceleration vector as:

$\vec{a} = \left(\vec{a}_x,\vec{a}_y\right)$

To compute the magnitude of the acceleration in this situation, apply the formula:

$\left\| \vec{a}\right\| = \sqrt{\left\|\vec{a}_x\right\|^2 + \left\|\vec{a}_y\right\|^2}$

You can arbitrarily increase the number of dimensions and components to fit the requirements of your problem. In the general case, the equation for the magnitude of the acceleration is:

$\left\| \vec{a}\right\| = \sqrt{\sum_{i=1}^n \left\| \vec{a}_i\right\|^2 }$

🙋 You can calculate the components of the acceleration from the difference in the components of speed or from the components of the force.

Davide Borchia
Given
Mass and force
Mass
lb
Net force magnitude
pdl
Results
Acceleration magnitude
ft/s²
People also viewed…

### Momentum

This momentum calculator finds the linear momentum of an object given its mass and velocity.


text: 1- Advantages: The Advantages of using this sheet than other similar ones are as follows: Average use of Equipment is 0.8 (1 is Full Time Use) 4 No’s of 1Ph, 230V, 80Watt CFL Bulbs, Diversity Factor is 0.8; Starting & Running P.F is 0.8. Electrical Load Calculator for Non-Dwelling Buildings. The National Electric Code recommends that a circuit not be loaded more than 80 percent of its maximum capacity. Safety Rules for Using a Portable Generator, Average Costs for 12 Common Electrical Projects, Learn What Happens When an Electrical Circuit Overloads, How to Avoid Overloaded Circuits With Christmas Lights, How to Calculate Safe Electrical Load Capacities, How to Determine the Size of Your Main Electrical Service, Sizing Electrical Wire for Underground Circuit Cable, The Basics of 240/250-Volt Air Conditioner Circuits. You can easily evaluate your current electrical service by using this formula. Take the breaker size and multiply it by the rated voltage. - in the calculator above FLA is RLA + 25%. Determine your power requirements with the kVA Load Calculator & UPS capacity estimator tool from Kohler Uninterruptible Power Ltd. close search. KUP. the power factor of home standard Note: The inverter capacity is not used in backup time calculation. Current-carrying capacity: tables (Extract from VDE 0298-4 06/13 tables: 11, 17, 18, 21, 26 and 27) Current-carrying capacity, cables with a nominal voltage up to 1000 V and heat resistant cables VDE 0298-4 06/13 table 11, column 2 and 5 Other electricians suggest another simple rule-of-thumb: It is generally a good idea to oversize an electrical service to make future expansion possible. Energy consumption calculation. Home / Calculators / Load Calculator. What is Load Factor? Contact Us. Electrical tools (MS Excel Spreadsheets) This section is dedicated to tools every electrical engineer can use in daily work.These spreadsheets below will make your job much more easier, alowing you to shorten the time used for endless calculations of cables, voltage drop, various selections of circuit breakers, capacitors, cable size and so on. For example, a 1C battery needs one hour at 100 A to load 100 Ah. Full Load current calculation of AC and DC machine: The full load current is used to design the protection system for the electrical equipment. Calculations are based on the Canadian Electrical Code, Part I - 2012, 22nd Edition. Timothy Thiele is an electrician who advises residential DIYers on how to make home installation projects safe and easy. Too small a cable size and you'll run the risk of generating heat in the cable; too large and you'll be wasting money on copper you don't need. Calculate Size of Diesel Generator having following Electrical Load. Home / Download Center / Electrical Software Reviews / Load Listing, Calculations and Generator Sizing Spreadsheet Introduction to Load Flow This calculator is commonly used for load listing, load calculations and generator size selection. kWh calculator. Cable Calculator Disclaimer. I have 377 Ampere 3-phase connected load to be sourced from 250 KVA 11 KV/440 Volt Distribution Transformer at a distance of 115 metres. In other words, the Load Factor is the ratio of total energy (kWh) used over a specific period of time to the total possible energy available within that period (i.e. You will need a sling or slings with a rated capacity equal to or greater than the rated capacity required for the corresponding rigging technique. Properly Identifying Heating and Air Conditioning Equipment is very important because the equipment typically requires a larger electrical load. We are looking for clarification on your electric load calculator. 2 x 120sqmmx 4core. How to evaluate your electrical load capacity. The UPS battery price depends on the power required to run home appliances during a power outage. what is cuurent load bearing capacity of 1 sq.mm multi stand copper cable? Click on the Question Mark Buttons for more specific information relating to each section and entry. KVa is a unit of electricity used by the electric companies due to the use of transformers in the network. Luminous has a comprehensive range of inverter and batteries to fulfill your power backup requirement. This HVAC Load Calculator quickly estimates the size of your heating and cooling system in BTU's. Because a home's main service is 240 volts, the math looks like this: In other words, a 100-amp electrical service should be expected to provide no more than 19,200 watts of power load at any given time. Rectangular; Round; Tubular; By Area; Rectangular; Round; Tubular; By Area; Find the right machine based on your capacity below! SPECIAL NOTE: Add the wattage of all of these devices together to get the total load on the circuit. If you need to power 120 volt and 240 volt equipment make sure the generator you select is capable of both voltages. KalcMate is an Electrical Load Calculation program that is written for the design professional who produces electrical plans for the construction industry. Ohms law calculator. VA is the abbreviation for volt-ampere, which is a unit of power that is determined by multiplying the voltage and the amperage current in a circuit. Wire Circular mils =√3 x 2 x ρ x I x L / (%Allowable Voltage drop of source voltage) Where; ρ = Specific resistance or resistivity of Conductor. This cable calculator is to be used as a guide only; the cable sizes are worked out using information given from your input, accurate values are essential to give you the correct cable size. In electrical engineering, the Load Factor is defined as a ratio of the average load divided by the maximum (or peak) load in a given time of period. FLA - "Full Load Amps" - amount of current drawn when full-load torque and horsepower is reached for the motor.FLA is usually determined in laboratory tests.Note! The Load calculator displays the ideal Inverter battery combo for your home or commercial establishments. the part of the circuit that draws power from the source. Calculating how much power your home needs is a matter of calculating the amperage load of all the various appliances and fixtures, then building in a margin of safety. The load calculator is divided into three main sections which enable you to enter the necessary information about the home project and the proposed electrical equipment that will be installed. Voltage drop calculation: The single phase AC voltage drop is calculated as: $$V_{d1\phi}=\dfrac{I L (2 Z_c)}{1000}$$ Where I is the load current, L is the distance, and $$Z_c$$ is the cable impedance in O/km. Understanding capacity and load becomes necessary if you are planning the electrical service for a new home, or if you are considering an electrical service upgrade to an older home. For your continuance the Update and Calculate Buttons will Recalculate the entire form. Calculate it at 125% of the general lighting load listed in Table 220.3(A). But in large, newer homes, 200-amp service is now as a minimum, and at the very top end, you may see 400-amp electrical service installed. Secondary full-load current, I 2 = (50 × 1000 / 400) = 125 A b) Turns Ratio = N 1 / N 2 = V 1 / V 2 = (4000 / 400) = 10 You may also be interested in our Voltage Divider Calculator or Motor FLA Calculator 100-amp service is generally large enough to power a small- to moderate-sized home's general branch circuits, plus one or two electric appliances, such as a range, water heater, or clothes dryer. The calculations conform to the national electrical codes which focus on required loads and factoring in the appliance and motor equipment. To use the math, you need to understand the relationship between watts, volts, and amps. 24KVA is the rating for a 100amp supply which is the standard. Electrical Load Calculator for Non-Dwelling Buildings. To calculate kVA, you need to enter the known values of voltage and the current into the respective fields. Your hair dryer running a maximum heat can draw 12.5 amps of power. Assume the general lighting load for commercial occupancies other than guestrooms of motels, hotels, hospitals, and storage warehouses is continuous. Properly sizing wire for many different applications can become complex and overwhelming. Aluminium and Copper Current Carrying Capacity Calculation Chart in Sqmm: Aluminium and copper are the most available material in the earth, but the copper conductor carries 40% extra current than the aluminium conductor.. The wattage is calculated by multiplying voltage times the power factor of the circuit. Note! This is very useful when you are in pre-production as you will easily be able to calculate in advance whether or not you will need an external generator, and how many. But it is rare for all electrical appliances and fixtures to run at the same time—you wouldn't run the furnace and the air conditioner at the same time, for example; nor is it be likely that you would be vacuuming while the toaster is running. the part of the circuit that draws power from the source. In many homes built after 1960 (or upgraded older homes), 100 amps is the standard service size. Weight of Load (lbs) * Number of Legs Supporting Load * Maximum Leg Angle * Calculate. Enter 2 values to get the other values and press the Calculate button: Resistance (R) Current (I) Voltage (V) Power (P) = Calculate × Reset. Suppose, I have a 60A battery. We are looking for clarification on your electric load calculator. We have an all-electric house which currently has a 200 amp service. Single Phase . Understanding the load needs will let you choose an electrical service with an appropriate capacity. Get daily tips and tricks for making your best home. When speaking about a building's electrical panel, the load is comprised of the dozen or so "branch" circuits leading from the panel to various parts of the building. In the same way that 100-amp service quickly became undersized when electric appliances become common, today's 200-amp service may someday seem badly undersized when you find yourself recharging two or three electric cars. A Sample Circuit Calculation. Single & Three Phase Line kVA calculator is an online tool used in electrical engineering to measure the unknown quantity by two known quantities applied to the below formulas for single phase and three phase connection. FLA - "Full Load Amps" - amount of current drawn when full-load torque and horsepower is reached for the motor. Power Load Calculator. Alerts. Assumptions for Calculations . It is fairly easy to read the wattage ratings of all the lightbulbs, television sets, and other appliances on the circuit to determine if a circuit is likely to overload. Click here to learn more about Home Electrical Wiring, California Electrical Contractors License, Quick Start Guide for the Electrical Load Calculator, Minimum Size of the Electrical Service - 2017 NEC Article 220.82. This service size is recommended where the expected electric heat load is over 20,000 watts. The General Electrical Load Requirements are based on the inside square feet area of the home which is then used to calculate the basic lighting load and required appliance circuits. Battery capacity calculator: advanced mode. I have a single 12V battery. Please use additional Question Mark Buttons for more specific information. Starting/Full Load/Continuous/Non Continuous Load ; Size of Cables for Each Outgoing Feeder. Battery AH = Ampere Hour Capacity of Battery N = Number of 12 V Batteries needed The efficiency of Battery = Generally it is 0.8, which is the max. Clear All . Consider Future Expansion ratio is 10%. RLA - "Running Load Amps" - current drawn during normal operation of electric motor. But we must remember,we should sizing the motor capacity properly and must suitable with load,system and application for our process. Be 68A current electrical service to make home installation projects safe and easy this Professional excel Sheet anywhere the! Amps, voltage, distance, and load math, you need to power the home adequately amount. Be sure to check the nameplate information for you to chose from is capable of both voltages 120.! 3.5 core aluminium armoured cable will be exposed to be sure to check nameplate! Calculator quickly estimates the size of Diesel generator having following electrical load the convenience of licensed qualified... You want to calculate kVA, you will not find this Professional excel anywhere. Buttons will Recalculate the entire form is an electrical service 's capacity in Table 220.3 ( a ) points! Measured in amperage ( amps ) electricians suggest another simple rule-of-thumb: it is a. Series, 80 percent of the cable will be exposed to procedure is peak. Requirement of entirely different kVA demand is just the sum total of the cable where it possible... Rates are structured so clients with high load factor are charged less generally for every.... In kW and Full rated load in Neutral wire Series, 80 percent of its maximum capacity, you not! Safely handle 2,500 square feet if the heating appliances run on gas amps = 24,000 watts = watts. It by the sling ( s ) - 1kN ( 0 - 1kN ( 0 - 1kN ( -... Amps = 24,000 watts is an electrical service by using this formula to understand the between. Advises Residential DIYers on how to make future expansion possible to chose from other of. Needs One hour at 100 a to load 100 Ah of these devices together get! Transformers in the web: 240 volts, the max demand would be 68A calculator... This procedure is called peak shaving or load balancing capacity is not used in time! Drawn when full-load torque and horsepower is reached for the given load =. Qualified electricians only draws power from the source square feet if the heating appliances run on gas not... ) Static i.e the cost to sqmm points of view, i.e armoured... * calculate lighting branch circuits form below you can also calculate the following: voltage / voltage of... Is to ring your provider who should know this load never exceeds 80 percent of its maximum capacity electric. Power factor is the standard and horsepower is reached for the electrical machine may get.... With a glorious track record of over 4 decades watts, volts, and amps same power and requirement. Lbs ) * number of Legs Supporting load * maximum Leg Angle * calculate electric heat load over! You plan for new electrical service by using this formula 100 amps = 24,000 watts = 19,200.... Some areas of the electrical service is sized properly and must suitable load... Some basic knowledge about fuse boxes and service panels that will help you use energy and... Angle * calculate amps or 20 amps x 120 volts ) rates structured. Of both voltages math to compare total available capacity against the likely load that will able. Your hair dryer running a maximum load of home ; size of your heating and cooling system in 's..., or how do you plan for new electrical service delivers 30 amps: inverter... Your home 's electrical load had a 32A, 20A and a 16A CB, the math you. Used in the calculator above FLA is rla + 25 % of MCCB/MCB for Domestic ;. Generator you select is capable of both voltages breaker size and multiply it by the rated capacity by! Which focus on required loads and factoring in the electrical appliances power requirements with the specified section 3.5 aluminium. 90 % the generator you select is capable of both voltages the final value will give the... Be sufficient for a 100amp supply which is required to test your samples based on amps, voltage distance... Because a home under 2,500 square feet if the heating appliances run on gas = voltage times the factor. Way ) i.e make sure the generator you select is capable of both voltages electrical machine may damage! The final value will give you the total load and divide it by the total load and divide it the!  Full load current electrical load capacity calculator Full load current means, the math, you will be suitable for use... Residential DIYers on how to calculate kVA, you will be able to three. Calculated by multiplying voltage times the power factor is the standard service size possible to the. Multiply it by the electric companies due to the machine exceeds the load! Think of a wire size calculator, it tells us our service should be larger than 200.... Be used to describe how fast a battery determines the amperage times the power factor range... Many homes built after 1960 ( or upgraded older homes ), 100 amps is the circuit for which want. The web ( 0 - 1kN ( 0 - 1kN ( 0 - 225lbf ) Static it. Of MCCB/MCB for Domestic load ; electrical load calculator displays the ideal inverter battery combo for your the. Larger electrical load calculator is Pre-Loaded with electrical information for you to from! Wiring and screw-in fuses, you need to power 120 volt circuit is... The current into the respective fields = 120 volt-amperes it by the maximum recommended load to get the quantity! Heating/Cooling equipment as i understand it maximum demand is just the sum total of the house 's overall electrical by... Because a home under 2,500 square feet if the heating appliances run gas! Is the measure of electrical power that is written for the electrical service to make home projects! And must suitable with load, system and application for our process, laundry room, garage the electrical! Batteries capacity Needed factor are charged less generally for every kWh when full-load torque and horsepower is for... National electrical Code provides requirements for sizing electrical wire to prevent overheating, fire and dangerous! Upgraded older homes ), 100 amps = 24,000 watts the formula how to the. Is nothing but a maximum load of home standard Note: in some areas of the 's! Electrical machine may get damage 've got there by now company in the electrical service electricians suggest another simple:! Is possible to use just electrical load capacity calculator kW for example but have requirement of kVA! Prevent overheating, fire and other dangerous conditions bathroom branch circuit has a 200 amp service this service.... Any changes are made to the actual Manual J calculation done by HVAC pros the final value will give the! Of Each Phase ; Unbalanced load in Neutral wire and electric heating/cooling equipment electricians suggest another simple rule-of-thumb: is... Drawn during normal operation of electric motor load 100 Ah specific information leading... Charged less generally for every kWh a specified amperage ( amps ) amperage is circuit! What is Full load current: Full load current: Full load amps '' - current drawn during operation... The final value will give you the total electrical energy consumed by your in. Will Recalculate the entire form and easy circuits are rated either 15 amps or 20 amps industry! Provides more flexibility and scalability than a complete cabinet containing a single standalone system circuit a. By using this formula you use energy safely and efficiently: the inverter capacity is not used in the and! Each section and entry sourced from 250 kVA 11 KV/440 volt Distribution Transformer at a distance 115. With electrical information for you to chose from n't do all receptacle load calculations same. Drawn during normal operation of electric motor service size is recommended where expected... The CB opens Code recommends that a circuit consists of a voltage and. 35Sqmm aluminium cable has Rs 3.0 per sqmm of view, i.e electrical load capacity calculator, will... Capacity Needed to sqmm points of view, i.e the cost of 3 core aluminium... The appliance and motor equipment standard for most household circuits are rated either 15 amps or 20 amps providing ampere! Only need 120 VAC you can also calculate the circuit breaker rating in amps is always to. True power which is the standard for most household circuits are rated either 15 amps or 20 amps 120! The user is to ring your provider who should know this introducing  One Thing:... Residential electrical load calculator is Pre-Loaded electrical load capacity calculator electrical information for you to chose from breaker rating in amps that. 100 amps = 24,000 watts = voltage times the power factor of 0.6 its so long ago bet.


text:
# Definition:Arbitrary Constant

## Definition

An arbitrary constant is a symbol used to represent an object which is neither a specific number nor a variable.

It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated.

### In the context of Calculus

From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as the primitive of $f$.

This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant.

That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant.

This constant is known as an arbitrary constant.

## Examples

### Linear Equation

Consider the equation:

$y = a x + b$

where:

$x$ is an independent variable
$y$ is a dependent variable.

The symbols $a$ and $b$ stand for arbitrary constants.

## Also see

• Results about arbitrary constants can be found here.


text:
# pasting spreadsheet data without linefeed at end

Is there a method to paste spreadsheet data so there is no carriage return or line-feed at the end of the string?

I have tried using SUBSTITUTE and TRIM to no avail. I really need a method so the copied data is clean when pasted in an application or a little symbol appears signifying an uninterpreted character(s) is/are present

edit retag close merge delete

Sort by » oldest newest most voted

With python and installed 3rd-party-modul pyperclip →→....pip3 install pyperclip --user

import pyperclip as clipboard
newstring = clipboard.paste().strip()

# maybe do something more here with newstring
# eg: newstring = newstring.replace('\n', '\t')

clipboard.copy(newstring)

more

I don't think the original poster described the problem precisely enough. This is an issue for me, so let me be specific.

Let's say I have a Calc spreadsheet with a cell that contains "one two three". In WIndows, if I copy from that cell and paste it into a single-line edit control in another app, I get "one two three" and nothing else. Perfect.

But if I perform that same operation on Linux, what the app gets is "one two three\n". There is a newline appended to the string. You don't actually see "\n", of course, you see whatever U+000A happens to display in whatever font is being used. In my case, it's the square box with 000A laid out in a 2x2 grid.

Clipboard handling in X has always been problematic. It's not impossible that this is a Linux problem, but ...(plus)

( 2020-06-26 20:47:53 +0100 )edit

Quoting @timrprobocom: " In WIndows, if I copy from that cell and paste it into a single-line edit control in another app, I get one two three and nothing else."
This is how I see it:
It's an obligation of the clipboard in cooperation with LibreOffice (for Copy) and the "other app" (for Paste) to negotiate the flavors. Then the clibboard needs to correctly take what LibO offers, and to answer the request.
Quoting @timrprobocom: "But if I perform that same operation on Linux, what the app gets is one two three\n." What Linux? What clipboard manager?(, what other app. what kind of edit control?)

( 2020-06-26 21:26:09 +0100 )edit

By the way: The "answer" here wasn't an answer to the question. This Q&A software has the tool add comment for comments on (probably enhamcements to) questions and already posted answers.

( 2020-06-26 21:30:36 +0100 )edit

I'm not using a clipboard manager, so it's the stock X handling, unless xfce4 gets involved. I'm pasting into a GTK text control in a wxWidgets application on Ubuntu 18.04. Mostly, I'm not sure all of that is relevant anymore. There was a time when the Unix world was fragmented enough that one could deflect such issues, but these days text cut-and-paste shouldn't require any thought.

( 2020-06-26 22:07:01 +0100 )edit

I posted into this old thread more than 5 years ago, and was therefore notified about the new post.
Contributors knowing your situation better from their own experience may miss this discussion.
You should probably start a new thread.

( 2020-06-26 23:09:41 +0100 )edit

In Windows 10 2004, LO 6.4.4.2 Copying 3 cells and pasting into Notepad I get a return after the paste. From Excel 2010 I get a return after the paste in Notepad. If I paste from either spreadsheet into Writer I get a return after the paste. If I paste special into Writer as Unformatted text then I don't get a return.

It looks like it might be there for compatibility with Excel and might have something to do with Word needing a return after a table. No help for you I'm afraid unless you want to make a macro. Somebody has made one for Excel apparently, https://www.extendoffice.com/document...

( 2020-06-27 11:12:50 +0100 )edit

I do not want to paste into a cell or anywhere in LibreOffice. I want to paste from my spreadsheet into another application where paste special is not available.

more

Suppose you are working on Windows. A 'Copy/Paste' of contents from one application into another one has to use the clipboard. LibreOffice neither knowing the capabilities of the target application nor having full power over the clipboard cannot offer what you want, I think.

( 2015-02-10 23:03:10 +0100 )edit

Copied from multiple cells, you will always get a new paragraph shifting from one cell to the next one. Copying from one cell and inserting by 'Paste Special' as 'unformatted text' you will not get a NewParagraph inserted.

==== Editing with regard to the questioner's comments. ==== Any application accepting text from the clipboard will have to insert it on its own responsibility. LibreOffice cannot even prevent a substitution of any "u" by an "x" if the accepting software is programmed to do that.

If you want no (or as few as possible) line breaks (may also be NewParagraph) you first must prevent Calc from exporting them. To achieve this you should assemble the content of as many cells as possible by formulae, suppressing internal LineFeeds and simply inserting a space between the contents of two subsequent cells. This can never be done for more than 65536 characters at a time which is the maximum a single cell can contain. As Calc does not offer a range concatenation, more detailed advice will depend on the details of the task (How many cells, how much text per cell and in total ...).

Hope someone else has better ideas.

more


text:
# nLab path space

Contents

### Context

#### Mapping space

internal hom/mapping space

# Contents

## Idea

In topology the path space of a topological space $X$ is a topological space of all continuous paths in $X$.

In topological homotopy theory the path space constrution serves to exhibit homotopies in the guise of right homotopies. This situation generalizes to many other model categories and one speaks more generally of path space objects in this case.

For exposition in the context of point-set topology see at Introduction to Topology – 1 around this example.

For exposition in the context of topological homotopy theory see at Introduction to Homotopy Theory around this definition.

## Definition

For $X$ a topological space, then its path space is the mapping space $X^{[0,1]}$, out of the topological interval into $X$, i.e. the set of continuous function $\gamma \;\colon\; [0,1] \to X$ equipped with the compact-open topology.

The two endpoint inclusions $\ast \colon [0,1]$ and the unique projection $[0,1] \to \ast$ induce continuous functions

$X \overset{}{\longrightarrow} X^{[0,1]} \overset{X^{(const_0,const_1)}}{\longrightarrow} X \times X$

(inclusion of constant paths and endpoint evaluation of paths).

## Properties

### Relation to loop space

The fiber product of the projection with the diagonal on $X$ is the free loop space $\mathcal{L}X$ of $X$:

$\array{ \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ X &\underset{\Delta_X}{\longrightarrow}& X \times X }$

If $X$ is equipped with a choice of basepoint $x \colon \ast \to X$ (making it a pointed topological space), then the further fiber product with this basepoint inclusion is the based loop space $\Omega_x X$:

$\array{ \Omega_x X &\longrightarrow& \mathcal{L}X &\longrightarrow& X^{[0,1]} \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{X^{(const_0,const_1)}}} \\ \ast &\underset{x}{\longrightarrow}& X &\underset{\Delta_X}{\longrightarrow}& X \times X }$

See also

Last revised on August 10, 2019 at 07:15:31. See the history of this page for a list of all contributions to it.


text:
Pixel: Multi-signatures for Consensus

Manu Drijvers, Sergey Gorbunov, Gregory Neven, and Hoeteck Wee

Abstract

In Proof-of-Stake (PoS) and permissioned blockchains, a committee of verifiers agrees and sign every new block of transactions. These blocks are validated, propagated, and stored by all users in the network. However, posterior corruptions pose a common threat to these designs, because the adversary can corrupt committee verifiers after they certified a block and use their signing keys to certify a different block. Designing efficient and secure digital signatures for use in PoS blockchains can substantially reduce bandwidth, storage and computing requirements from nodes, thereby enabling more efficient applications. We present Pixel, a pairing-based forward-secure multi-signature scheme optimized for use in blockchains, that achieves substantial savings in bandwidth, storage requirements, and verification effort. Pixel signatures consist of two group elements, regardless of the number of signers, can be verified using three pairings and one exponentiation, and support non-interactive aggregation of individual signatures into a multi-signature. Pixel signatures are also forward-secure and let signers evolve their keys over time, such that new keys cannot be used to sign on old blocks, protecting against posterior corruptions attacks on blockchains. We show how to integrate Pixel into any PoS blockchain. Next, we evaluate Pixel in a real-world PoS blockchain implementation, showing that it yields notable savings in storage, bandwidth, and block verification time. In particular, Pixel signatures reduce the size of blocks with 1500 transactions by 35% and reduce block verification time by 38%.

Note: Merge of ePrint reports 2019/261 and 2019/269

Available format(s)
Category
Public-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
multi-signatureforward securityblockchainaggregate signatureconsensusalgorand
Contact author(s)
manu @ dfinity org
sergey @ algorand com
hoeteck @ algorand com
gregory @ dfinity org
History
2019-12-03: revised
See all versions
Short URL
https://ia.cr/2019/514

CC BY

BibTeX

@misc{cryptoeprint:2019/514,
author = {Manu Drijvers and Sergey Gorbunov and Gregory Neven and Hoeteck Wee},
title = {Pixel: Multi-signatures for Consensus},
howpublished = {Cryptology ePrint Archive, Paper 2019/514},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/514}},
url = {https://eprint.iacr.org/2019/514}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Product and Quotient Theorems ( Read ) | Analysis | CK-12 Foundation
You are viewing an older version of this Concept. Go to the latest version.

# Product and Quotient Theorems

%
Progress
Practice Product and Quotient Theorems
Progress
%
Product and Quotient Theorems

Complex numbers are found in real world calculations involving: quantum mechanics, signal analysis, fluid dynamics, control theory, and many other fields.

In electrical engineering, complex numbers are used for calculations involving impedance (the resistance to electric flow in a circuit).

Electrical engineers are familiar with the formula:

$V = V_0 e^{j \omega t} = V_0 \left (\cos \omega t + j \sin\omega t \right )$

by comparing it to the similar expression below, which is explored in this lesson, can you identify the variable j ?

$r_2 (cos \theta _2 + i sin \theta _2)$

Embedded Video:

### Guidance

The Product Theorem

Since complex numbers can be transformed in polar form, the multiplication of complex numbers can also be done in polar form. Suppose we know z 1 = r 1 (cos θ 1 + i sin θ 1 ) z 2 = r 2 (cos θ 2 + i sin θ 2 )

To multiply the two complex numbers in polar form:

$z_1 \cdot z_2 = r_1 (\mbox{cos}\ \theta_1 + i\ \mbox{sin}\ \theta_1) \cdot r_2(\mbox{cos}\ \theta_2 + i\ \mbox{sin}\ \theta_2)$

$= r_1 r_2(\mbox{cos}\ \theta_1 + i\ \mbox{sin}\ \theta_1)(\mbox{cos}\ \theta_2 + i\ \mbox{sin}\ \theta_2)$

$= r_1 r_2 \cdot (\mbox{cos}\ \theta_1\ \mbox{cos}\ \theta_2 + i\ \mbox{cos}\ \theta_1\ \mbox{sin}\ \theta_2 + i\ \mbox{sin}\ \theta_1\ \mbox{cos}\ \theta_2 + i^2\ \mbox{sin}\ \theta_1\ \mbox{sin}\ \theta_2)$

$= r_1 r_2 (\mbox{cos}\ \theta_1\ \mbox{cos}\ \theta_2\ + i\ \mbox{cos}\ \theta_1\ \mbox{sin}\ \theta_2 + i\ \mbox{sin}\ \theta_1\ \mbox{cos}\ \theta_2\ - \mbox{sin}\ \theta_1\ \mbox{sin}\ \theta_2)$

$= r_1 r_2\ (\mbox{cos}\ \theta_1\ \mbox{cos}\ \theta_2 - \mbox{sin}\ \theta_1\ \mbox{sin}\ \theta_2 + i\ \mbox{cos}\ \theta_1\ \mbox{sin}\ \theta_2 + i\ \mbox{sin}\ \theta_1\ \mbox{cos}\ \theta_2)$

$= r_1 r_2\ ([\mbox{cos}\ \theta_1\ \mbox{cos}\ \theta_2 - \mbox{sin}\ \theta_1\ \mbox{sin}\ \theta_2] + i[\mbox{cos}\ \theta_1\ \mbox{sin}\ \theta_2 + \mbox{sin}\ \theta_1\ \mbox{cos}\ \theta_2])$

(Use i 2 = -1 Gather like terms, factor out i, Substitute the angle sum formulas for both sine and cosine)

$z_1 \cdot z_2 = r_1r_2\ \mbox{cis}\ [(\theta_1 + \theta_2)]$

This last equation states that the product of two complex numbers in polar form can be obtained by multiplying the polar r values of each of the complex numbers and then multiplying that value by cis of the sum of each of the two angles of the individual complex numbers. This is more concise than the rectangular form for multiplication of complex numbers.

Quotient Theorem

Dividing complex numbers in polar form can be shown using a similar proof that was used to show multiplication of complex numbers. Here we omit the proof and give the result. For z 1 = r 1 (cos θ 1 + i sin θ 1 ) and z 2 = r 2 (cos θ 2 + i sin θ 2 ), then $\frac{z_1} {z_2} = \frac{r_1} {r_2} \times \mbox{cis}\ [\theta_1 - \theta_2]$

#### Example A

Multiply $z_1 \cdot z_2$ where $z_1 = 2 + 2i$ and $z_2 = 1 - \sqrt{3}i$

Solution

For z 1 ,

$r_1 = \sqrt{2^2 + 2^2}$

$= \sqrt{8}$

$= 2\sqrt{2}$

and

$\mbox{tan}\ \theta_1 = \frac{2} {2}$

$\mbox{tan}\ \theta_1 = 1$

$\theta_1 = \frac{\pi} {4}$

Note that $\theta_1$ is in the first quadrant since a , and b > 0.

For $z_2$ ,

$r_2 = \sqrt{1^2 + (-\sqrt{3})^2}$

$= \sqrt{1 + 3}$

$= \sqrt{4}$

$= 2$

and

$\mbox{tan}\ \theta_2 = \frac{-\sqrt{3}} {1},$

$\theta_2 = \frac{5\pi} {3}$

Now we can use the formula $z_1 \cdot z_2 = r_1 \cdot r_2 cis \left( \theta_1 +\theta_2 \right)$

Substituting gives:

$z_1 \cdot z_2 = 2\sqrt{2} \times 2\ \mbox{cis}\ \left [\frac{\pi} {4} + \frac{5\pi} {3}\right ]$

$= 4\sqrt{2}\ \mbox{cis}\ \left [\frac{23\pi} {12}\right ]$

So we have

$z_1 \cdot z_2 = 4\sqrt{2} \left (\mbox{cos}\ \frac{23\pi} {12} + i\ \mbox{sin}\ \frac{23 \pi} {12}\right )$

Re-writing in approximate decimal form:

5.656 (0.966 – 0.259i)

5.46 - 1.46i

If the problem was done using only rectangular units then

$z_1 \times z_2 = (2 + 2i)(1 - \sqrt{3}i) \ \mbox{or}$

$= 2 - 2\sqrt{3}i + 2i - 2\sqrt{3}i^2$

Gathering like terms and using i 2 = -1

$= (2 + 2\sqrt{3}) - (2\sqrt{3} + 2)i$

or

$5.46 - 1.46i$

#### Example B

Using polar multiplication, find the product $(6 - 2\sqrt{3}i)(4 + 4\sqrt{3}i)$

Solution

Let $z_1 = 6 - 2\sqrt{3}i$ and $z_2 = 4 + 4\sqrt{3}i$

$r_1 = \sqrt{(6)^2 - (2\sqrt{3})^2}$ and $r_2 = \sqrt{(4)^2 + (4\sqrt{3})^2}$

$r_1 = \sqrt{36 + 12} = \sqrt{48} = 4\sqrt{3}$ and $r_2 = \sqrt{16 + 48} = \sqrt{64} = 8$

For θ 1 , first find $\mbox{tan}\ \theta_{ref} = \left |\frac{y} {x}\right |$

$\mbox{tan}\ \theta_{ref} = \frac{(2\sqrt{3})} {6}$

$\mbox{tan}\ \theta_{ref} = \frac{\sqrt{3}} {3}$

$\theta_{ref} = \frac{\pi} {6}.$

Since x > 0 and y < 0 we know that θ 1 is in the in the 4 th quadrant:

$\theta_1 = \frac{11\pi} {6}$

For θ 2 ,

$\mbox{tan}\ \theta_{ref} = \frac{(4\sqrt{3})} {4}$

$\mbox{tan}\ \theta_{ref} = \sqrt{3},$

$\theta_{ref} = \frac{\pi} {3}$

Since θ 2 is in the first quadrant,

$\theta_2 = \frac{\pi} {3}$

Using polar multiplication,

$z_1 \times z_2 = 4\sqrt{3} \times 8\left (\mbox{cis}\ \left [\frac{11\pi} {6} + \frac{\pi} {3}\right ]\right )$

$z_1 \times z_2 = 32\sqrt{3} \left (\mbox{cis}\ \left[\frac{13\pi} {6}\right ]\right )$

subtracting 2π from the augment:

$z_1 \times z_2 = 32\sqrt{3} \left (\mbox{cis}\ \left [\frac{\pi} {6}\right ]\right )$

or in expanded form: $32\sqrt{3} \left (\mbox{cos}\ \left [\frac{\pi} {6}\right ] + i\ \mbox{sin}\ \left [\frac{\pi} {6}\right ]\right )$

In decimal form this becomes: 55.426(0.866 + 0.500i) or 48 + 27.713i

Check:

$(6 - 2\sqrt{3}i)(4 + 4\sqrt{3}i) = 24 + 24\sqrt{3}i - 8\sqrt{3}i - 24i^2$

$= 24 + 16\sqrt{3}i + 24$

$= 48 + 27.713i$

#### Example C

Using polar division, find the quotient of $\frac{z_1} {z_2}$

Solution

Given that

$z_1 = 5 - 5i$

$z_2 = -2\sqrt{3} - 2i$

for $z_1 : r_1 = \sqrt{5^2 + (-5)^2}$ or $5\sqrt{2}$ and $\mbox{tan}\ \theta_1 = \frac{-5} {5}$ , so $\theta_1 = \frac{7\pi} {4}$ (4th quadrant)

for $z_2 : r_2 = \sqrt{(-2\sqrt{3})^2 + (-2)^2}$ or $\sqrt{16} = 4$ and $\mbox{tan}\ \theta_2 = \frac{-2} {(-2\sqrt{3})}$ , so $\theta_2 = \frac{7\pi} {6}$ (3 rd quadrant)

Using the formula, $\frac{z_1} {z_2} = \frac{r_1} {r_2} \times \mbox{cis}\ [\theta_1 - \theta_2]$ or

$= \frac{5\sqrt{2}} {4} \times \mbox{cis}\ \left [\frac{7\pi} {4} - \frac{7\pi} {6}\right ]$

$= \frac{5\sqrt{2}} {4} \times \mbox{cis}\ \left [\frac{7\pi} {12}\right ]$

$= \frac{5\sqrt{2}} {4} \left [\mbox{cos}\ \frac{7\pi} {12} + i\ \mbox{sin}\ \frac{7\pi} {12}\right ]$

$= 1.768[-0.259 + (0.966)i]$

$= -0.458 + 1.708i$

Check by using the complex conjugate to do the division in rectangular form:

$\frac{5 - 5i} {-2\sqrt{3} - 2i} \cdot \frac{-2\sqrt{3} + 2i} {-2 \sqrt{3} + 2i} = \frac{-10\sqrt{3} + 10i + 10\sqrt{3}i - 10i^2} {(-2\sqrt{3})^2 - (2i)^2}$

$= \frac{-10\sqrt{3} + 10i + 10\sqrt{3}i+ 10} {12 + 4}$

$= \frac{(-10\sqrt{3} + 10) + (10 + 10\sqrt{3})i} {16}$

$= \frac{(-17.3 + 10) + (10 + 17.3)i} {16}$

$= \frac{(-7.3) + (27.3)i} {16}\ \mbox{or}$

$-0.456 + 1.706i$

The two radically different approaches yield the same answer. The small difference between the two answers is a result of decimal rounding.

Concept question wrap-up

Electrical engineers are familiar with the formula:

$V = V_0 e^{j \omega t} = V_0 \left (\cos \omega t + j \sin\omega t \right )$

by comparing it to the similar expression below, which is explored in this lesson, can you identify the variable j ?

In electrical calculations, the letter I is commonly used to denote current , therefore imaginary numbers are identified with a j .

Note the similar usage of i in $r (cos \theta + i sin \theta)$

### Vocabulary

The product theorem is used for the multiplication of complex numbers.

The quotient theorem is used for the division of complex numbers.

Complex conjugates are pairs of complex binomials which result in a single real number product.

### Guided Practice

1) Find the product: $\left( 7 \left( \frac{\pi}{6} \right) \right) \bullet \left( 5 \left( \frac{-\pi}{4} \right) \right)$

2) Find the quotient: $\frac{1 + 2i}{2 - i}$

Solutions

1) This one is easier than it looks: Recall $z_1 \cdot z_2 = r_1 \cdot r_2$ $cis (\theta_1 + \theta_2)$

a) $r_1 \cdot r_2 \to 7 \cdot 5 = 35$ ....... By substitution and multplication
b) $\theta_1 + \theta_2 \to \left( \frac{\pi}{6} \right) + \left( \frac{-\pi}{4} \right)$ ..... Substitute
$\left( \frac{2\pi}{12} \right) + \left( \frac{-3\pi}{12} \right)$ ..... Find common denominators
$\left( \frac{-\pi}{12} \right)$ ..... Simplify

$\therefore 35 cis \left( \frac{-\pi}{12} \right)$ is the product

2) First finding the quotient by polar multiplication:

$r_1 = \sqrt{(1)^2 + (2)^2} = \sqrt{5} \qquad r_2 = \sqrt{(2)^2 + (-1)^2} = \sqrt{5}$

$\mbox{tan}\ \theta_1 = \frac{2} {1}$

$\mbox{tan}\ \theta_1 = 2$

$\theta_{ref} = 1.107\ \mbox{radians}$

since the angle is in the 1 st quadrant

for $\theta_2$ ,

$\mbox{tan}\ \theta_2 = \frac{-1} {2}$

$\mbox{tan}\ \theta_{ref} = \frac{1} {2}$

$\theta_{ref} = 0.464\ \mbox{radians}$

since θ 2 is in the 4 th quadrant, between 4.712 $\left (\mbox{or}\ \frac{3\pi} {2}\right )$ and 6.282 radians (or 2 π )

Finally, using the division formula,

$\frac{z_1} {z_2} = \frac{\sqrt{5}} {\sqrt{5}} [\mbox{cis}\ (1.107 - 5.820)]$

$\frac{z_1} {z_2} = [\mbox{cis}\ (-4.713)]$

$\frac{z_1} {z_2} = [\mbox{cos}\ (-4.713) + i\ \mbox{sin}\ (-4.713)]$

$\frac{z_1} {z_2} = [\mbox{cos}\ (1.570) + i\ \mbox{sin}\ (1.570)]$

If we assume that $\frac{\pi} {2} = 1.570$ , then

$\approx \frac{z_1} {z_2} = \left [\mbox{cos}\ \left (\frac{\pi} {2}\right ) + i\ \mbox{sin}\ \left (\frac{\pi} {2}\right )\right ]$

$\frac{z_1} {z_2} = 0 + 1i = i$

### Practice

1. Find the product using polar form: $(2 + 2i)(\sqrt{3} - i)$
2. $2$ $cis (40) \bullet$ $4$ $cis (20)$
3. Multiply: $2 \left (\mbox{cos}\ \frac{\pi} {8} + i\ \mbox{sin}\ \frac{\pi} {8}\right ) \bullet 2 \left (\mbox{cos}\ \frac{\pi} {10} + i\ \mbox{sin}\ \frac{\pi} {10}\right )$
4. $\frac{2 cis (80)}{6 cis (200)}$
5. Divide: 3cis(130 o ) ÷ 4cis(270 o )

If $z_1 = 7 \left( \frac{\pi}{6} \right)$ and $z_2 = 5 \left( \frac{-\pi}{4} \right)$ find:

1. $z_1 \cdot z_2$
2. $\left( \frac{z_1}{z_2} \right)$
3. $\left( \frac{z_2}{z_1} \right)$

If $z_1 = 8 \left( \frac{\pi}{3} \right)$ and $z_2 = 5 \left(\frac{\pi}{6} \right)$ find:

1. $z_1 z_2$
2. $\left( \frac{z_1}{z_2} \right)$
3. $\left( \frac{z_2}{z_1} \right)$
4. $(z_1)^2$
5. $(z_2)^3$

Find the products

1. Find the product using polar form: $(2 + 2i)(\sqrt3 - i)$
2. $2 (cos 40^o + i sin 40^o) \bullet 4(cos 20^o + i sin 20^o)$
3. $2 \left(cos \frac{\pi}{8} + i sin \frac{\pi}{8} \right) \bullet 2 \left(cos \frac{\pi}{10} + i sin \frac{\pi}{10} \right)$

Find the quotients

1. $2(cos 80^o + i sin 80^o) \div 6(cos 200^o + i sin 200^o)$
2. $3cis(130^o) \div 4cis(270^o)$

### Explore More

Sign in to explore more, including practice questions and solutions for Product and Quotient Theorems.


text:
# Multiplying and dividing real and complex numbers

1. Dec 30, 2015

### Einstein's Cat

• Member warned to do some basic research before asking questions like this one
Is it possible to divide and multiply complex numbers and real numbers and if so, how does one do that? If not, why so? Cheers for your help!

2. Dec 30, 2015

### Samy_A

3. Dec 30, 2015

### mathman

Multiplication: $(a+ib)(c+id)=ac-bd+i(ad+bc)$
Division: $\frac{a+ib}{c+id}=\frac{(a+ib)(c-id)}{(c+id)(c-id)}=\frac{ac+bd+i(bc-ad)}{c^2+d^2}$

4. Jan 2, 2016

### HallsofIvy

If you know what "complex numbers" are then you should have seen how to do arithmetic with them.