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# What Was the Total Consumption of Bonbon Bread in Towns a and B During a Particular Week When the Total Consumption of Bread in Town D Was 3600 Loaves? - Mathematics

#### Question

MCQ

Direction:  Answer the question on the basis of the information given below.

The following information is about the consumption of bread in five towns - A, B, C, D and E. Town A consumes a quarter of the total quantity of bread sold in all the five cities. The bread consumption of Town B is one-fourth of the consumption of Town C. The bread consumption of Town D is 12% of the total whereas Town E consumes 23% of the total.

There are only two companies - Bonbon and Modern - that sell bread in these five towns. The bar graph shows the sales of Bon-bon as a percentage of the total sale of bread in each city.

What was the total consumption of Bonbon bread in Towns A and B during a particular week when the total consumption of bread in Town D was 3600 loaves?

• 5144

• 6138

• 7605

• 4368

#### Solution

6138

Explanation:

The correct option is (2). According to the information given, The total consumption of bread in Town D was 12% which is equivalent to 3600 loaves. Therefore, the total consumption of bread in Town A can be calculated as:

3600/12 xx 25 = 7500 loaves and the total consumption of bread in Town B can be calculated as :

3600/12 xx 8 = 2400

The consumption of Bonbon bread in Town A

58.24/100 xx 7500 = 4368

and the consumption of Bonbon bread in Town B

73.75/100 xx 2400 = 1770

Hence, total consumption of Bonbon bread in Town A and Town B is (4368 + 1770) = 6138 loaves.

Concept: Graph and Charts (Entrance Exam)
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Make a careful sketch of the graph of $f$ and below it sketch the graph of $f'$ in the same manner as in Exercises 4-11. Can you guess a formula for $f'(x)$ from its graph?$f(x) = e^x$

The slope at 0 appcars to be 1 and the slope at 1 appears 0. Since to be 2.7 . As $x$ decreases, the slope gets closer to the graphs are so similar, we might gacss that $f^{\prime}(x)=e^{x}$.

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this problem for seventeen of the Stuart Calculus eighth edition, Section two point eight make a careful sketch of the graph of F and blew it sketch the graph of every time in the same manner as an exercise is for true love. Can you guess a formula for halftime from its crap? So first, let's plot after of X equals e to the X and the top craft were were sketching Ah, good. The graph of F and looks roughly like this One of the key points is this point zero one, not a word that is the white intercept of the function. Because at X equals zero, it is there is one. Ah, and we see that the limit is expertise needed Infinity a zero and that limit is exported infinities infinity. So this increases. Ah, as a X goes towards infinity. Now I may be a bit difficult, but we're going to attempt to drop the graph of time by using the slopes of the tangent lines that our attention to this function f the easy are one of the easier ones to look at is as X approaches negative infinity. We see that dysfunction that pulls out. Search them. The slopes of the tension line decrease in Stevenage and in value until it becomes almost flat. A sex person. Negative infinity. So we're going to get this Ah, decrease of the slope towards zero. Because the slopes of the tension lines approach zero. As we approach any other affinity arm, we can estimate the slope of the tangent line at X equals zero here and where we end up. If we were to fear that's not exactly the slope of the tension line at X equals zero ends up being exactly one. So there's an interesting result. And what is more interesting is that if we continue to find the slopes of each of the tendon lines as X increases, we will begin. We will continue to plot what is exactly the same graph as F arms. So in reality, what we have just plotted is the functioning of the ex. And this is consistent with our definition for this function, any of the ex because we know that the definition of either the ex is indeed itself even the X and so they're dirt of craft is exactly the same esteem original graph, and it at every point is the slope of the tangent line to dysfunction at every point

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Campbell University

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Baylor University

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University of Michigan - Ann Arbor

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## Electronic Journal of Probability

### Quenched tail estimate for the random walk in random scenery and in random layered conductance II

#### Abstract

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102–128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 75, 28 pp.

Dates
Received: 24 May 2019
Accepted: 8 June 2020
First available in Project Euclid: 4 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1593828036

Digital Object Identifier
doi:10.1214/20-EJP478

#### Citation

Deuschel, Jean-Dominique; Fukushima, Ryoki. Quenched tail estimate for the random walk in random scenery and in random layered conductance II. Electron. J. Probab. 25 (2020), paper no. 75, 28 pp. doi:10.1214/20-EJP478. https://projecteuclid.org/euclid.ejp/1593828036

#### References

• [1] S. Andres, J.-D. Deuschel, and M. Slowik. Harnack inequalities on weighted graphs and some applications to the random conductance model. Probab. Theory Related Fields, 164(3-4):931–977, 2016.
• [2] S. Andres, J.-D. Deuschel, and M. Slowik. Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances. Electron. Commun. Probab., 24, paper no. 5, 17 pp., 2019.
• [3] A. Asselah and F. Castell. Large deviations for Brownian motion in a random scenery. Probab. Theory Related Fields, 126(4):497–527, 2003.
• [4] M. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab., 38(1):234–276, 2010.
• [5] P. Bella and M. Schäffner. Quenched invariance principle for random walks among random degenerate conductances. Ann. Probab., 48(1):296–316, 2020.
• [6] G. Ben Arous and J. Cerný. Scaling limit for trap models on $\mathbb {Z}^{d}$. Ann. Probab., 35(6):2356–2384, 2007.
• [7] G. Ben Arous and A. F. Ramírez. Asymptotic survival probabilities in the random saturation process. Ann. Probab., 28(4):1470–1527, 2000.
• [8] M. Biskup. Recent progress on the random conductance model. Probab. Surveys, 8:294–373, 2011.
• [9] M. Biskup and W. König. Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab., 29(2):636–682, 2001.
• [10] A. N. Borodin. A limit theorem for sums of independent random variables defined on a recurrent random walk. Dokl. Akad. Nauk SSSR, 246(4):786–787, 1979.
• [11] A. N. Borodin. Limit theorems for sums of independent random variables defined on a transient random walk. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 85:17–29, 237, 244, 1979. Investigations in the theory of probability distributions, IV.
• [12] F. Castell, N. Guillotin-Plantard, and F. Pène. Limit theorems for one and two-dimensional random walks in random scenery. Ann. Inst. Henri Poincaré Probab. Stat., 49(2):506–528, 2013.
• [13] A. Chakrabarty. Effect of truncation on large deviations for heavy-tailed random vectors. Stochastic Process. Appl., 122(2):623–653, 2012.
• [14] D. A. Croydon, B. M. Hambly, and T. Kumagai. Heat kernel estimates for FIN processes associated with resistance forms. Stochastic Process. Appl., 129(9), 2991–3017, 2019.
• [15] T. Delmotte and J.-D. Deuschel. On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla \phi$ interface model. Probab. Theory Related Fields, 133(3):358–390, 2005.
• [16] J.-D. Deuschel and R. Fukushima. Quenched tail estimate for the random walk in random scenery and in random layered conductance. Stochastic Process. Appl., 129(1):102–128, 2019.
• [17] M. D. Donsker and S. R. S. Varadhan. On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math., 32(6):721–747, 1979.
• [18] A. Dvoretzky and P. Erdos. Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pages 353–367, 1951.
• [19] W. Feller. An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley & Sons, Inc., New York-London-Sydney, 1968.
• [20] R. Fukushima. Brownian survival and Lifshitz tail in perturbed lattice disorder. J. Funct. Anal., 256(9):2867–2893, 2009.
• [21] R. Fukushima. From the Lifshitz tail to the quenched survival asymptotics in the trapping problem. Electron. Commun. Probab., 14:435–446, 2009.
• [22] N. Guillotin-Plantard, J. Poisat, and R. S. dos Santos. A quenched functional central limit theorem for planar random walks in random sceneries. Electron. Commun. Probab., 19, paper no. 3, 9 pp., 2014.
• [23] H. Kesten and F. Spitzer. A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete, 50(1):5–25, 1979.
• [24] W. König. The parabolic Anderson model: Random walk in random potential. Pathways in Mathematics. Birkhäuser/Springer, [Cham], 2016.
• [25] G. F. Lawler and V. Limic. Random walk: a modern introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
• [26] A.-S. Sznitman. Brownian asymptotics in a Poissonian environment. Probab. Theory Related Fields, 95(2):155–174, 1993.
• [27] A.-S. Sznitman. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.


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Semi-local ring explained

In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R.

The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals".

Some literature refers to a commutative semi-local ring in general as aquasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals.

A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal.

Examples

Z/mZ

is a semi-local ring. In particular, if

m

is a prime power, then

Z/mZ

is a local ring.
• A finite direct sum of fields
 n{F oplus i}
is a semi-local ring.
• In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m1, ..., mn
 n R/cap i=1

mi\congoplus

 n i=1

R/mi

.

(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=J(R), and we see that R/J(R) is indeed a semisimple ring.


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1
JEE Main 2021 (Online) 31st August Morning Shift
Numerical
+4
-1
A3B2 is a sparingly soluble salt of molar mass M (g mol$$-$$1) and solubility x g L$$-$$1. The solubility product satisfies $${K_{sp}} = a{\left( {{x \over M}} \right)^5}$$. The value of a is _____________. (Integer answer)
2
JEE Main 2021 (Online) 27th August Evening Shift
Numerical
+4
-1
When 5.1 g of solid NH4HS is introduced into a two litre evacuated flask at 27$$^\circ$$C, 20% of the solid decomposes into gaseous ammonia and hydrogen sulphide. The Kp for the reaction at 27$$^\circ$$C is x $$\times$$ 10$$-$$2. The value of x is _____________. (Integer answer) [Given R = 0.082 L atm K$$-$$1 mol$$-$$]
3
JEE Main 2021 (Online) 27th August Morning Shift
Numerical
+4
-1
The number of moles of NH3, that must be added to 2L of 0.80 M AgNO3 in order to reduce the concentration of Ag+ ions to 5.0 $$\times$$ 10$$-$$8 M (Kformation for [Ag(NH3)2]+ = 1.0 $$\times$$ 108) is ____________. (Nearest integer)

[Assume no volume change on adding NH3]
4
JEE Main 2021 (Online) 26th August Evening Shift
Numerical
+4
-1
The equilibrium constant Kc at 298 K for the reaction A + B $$\rightleftharpoons$$ C + D is 100. Starting with an equimolar solution with concentrations of A, B, C and D all equal to 1M, the equilibrium concentration of D is ___________ $$\times$$ 10$$-$$2 M. (Nearest integer)
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Browse
Welcome to Mathematicsi Project Worksheets, the mathematics worksheets library. Click on solutions to get the solutions. See more ➡

# Indices Worksheet

Ratings: 0 | Likes: | Cached copies:

You can also download a copy of this worksheet by clicking on any of the preferred formats below. Click on Worksheet answers to get the answers for this worksheet. Refresh the page to get a new random worksheet.

This document contains a worksheet for the indices rules. You can learn more about the indices rules in this entry posted on the home section: Indices Rules I will make the printable versions available soon, and this worksheet might be turned into an automated worksheet so that the questions are random.

### Question 1: Multiplying and Dividing

• Simplify each of the following expressions
82 x 88
67 ÷ 63
52 x 57
78 ÷ 79
66 x 69
37 ÷ 35
46 x 47
66 ÷ 63
e4 x e9
x9 ÷ x7
n7 x n8
x4 ÷ x5

### Question 2: Brackets and harder expressions

• Simplify the following expressions
(77)6
9x3 x 2x6
(53)4
5a2 x 9a3
(69)3
3a9 x 5a5
(46)9
3e8 x 3e9
(69)7
2x3 ÷ 8x6
(95)6
9a3 x 6a9
(5e8)5
$\frac{15x^6 * 18x^4}{9x^5}$


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19 Answered Questions for the topic Units

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#### Fill in the blanks with appropriate units.

10_________ 70____________= 11____________ 10___________
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If someone is paid 1000 for 1 Hectar, How much is he/she going to be paid for 1km?

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Unit rates....3 weeks made 12 websites But then in 5 weeks he made 20....need to know how many units per week he made
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26 1/2 miles and 1 1/4 gallon of gas to a unit
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#### Stoichiometry Problem

C2H4 + O2 → CO2 + H2O -If you start with 45 grams of ethylene (C2H4), how many grams of carbon dioxide should be produced?-Theoretically, how many liters of carbon dioxide will you produce if 55.3... more

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#### Compute the number and total cost of the units available for sale in year 2015

Seminole Company began year 2015 with 28,000 units of product in its January 1 inventory costing \$16.60 each. It made successive purchases of its product in year 2015 as follows. The company uses a... more

03/03/15

#### determine the equation of g(x) that results from translating the function f(x)= (x+6)^2 to the right 8 units

a) g(x)= (x-2)^2 b) g(x) = (x+14)^2 c) g(x) = (x+6)^2 -8 d) g(x)= (x+6)^2 +8
Units

12/26/14

#### tian makes his own pizza dough. he uses 2 packages of yeast for every 3 cups of flower. he needs 5 cups of flower to make enough pizza...

tian makes his own pizza dough. he uses 2 packages (p) of yeast for every 3 cups (c)of flour. he needs 5 c flour to make enough pizza for his family. how many packages of yeast does he need? Round... more

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• ### Simulation study of the correlation ($X_{max}^{\mu}$, $N^{\mu}$) in view of obtaining information on primary mass of the UHECRs(1607.03244)

July 12, 2016 hep-ex, astro-ph.IM, astro-ph.HE
In this paper we study, using Monte Carlo simulations, the possibility to discriminate the mass of the Ultra High Energy Cosmic Rays (UHECRs) by combining information obtained from the maximum $X_{max}^{\mu}$ of the muon production rate longitudinal profile of Extensive Air Showers (EAS) and the number of muons, $N^{\mu}$, which hit an array of detectors located in the horizontal plane. We investigate the sensitivity of the 2D distribution $X_{max}^{\mu}$ versus $N^{\mu}$ to the mass of the primary particle generating the air shower. To this purpose we analyze a set of CORSIKA showers induced by protons and iron nuclei at energies of $10^{19}$eV and $10^{20}$eV, at five angles of incidence, $0^{\circ}$, $37^{\circ}$, $48^{\circ}$, $55^{\circ}$ and $60^{\circ}$. Using the simulations we obtain the 2D Probability Functions $Prob(X_{max}^{\mu},N^{\mu} \ | \ p)$ and $Prob(X_{max}^{\mu},N^{\mu} \ | \ Fe)$ which give the probability that a shower induced by a proton or iron nucleus contributes to a specific point on the plane ($X_{max}^{\mu}$, $N^{\mu}$). Then we construct the probability functions $Prob(p\ | \ X_{max}^{\mu},N^{\mu})$ and $Prob(Fe \ | \ X_{max}^{\mu},N^{\mu})$ which give the probability that a certain point on the plane ($X_{max}^{\mu}$, $N^{\mu}$) corresponds to a shower initiated by a proton or an iron nucleus, respectively. Finally, a test of this procedure using a Bayesian approach, confirms an improved accuracy of the primary mass estimation in comparison with the results obtained using only the $X_{max}^{\mu}$ distributions.
• ### Quantum production of black holes at colliders(1606.07323)

June 23, 2016 hep-ph
We investigate black hole production in pp collisions at the Large Hadron Collider by employing the horizon quantum mechanics for models of gravity with extra spatial dimensions. This approach can be applied to processes around the fundamental gravitational scale and naturally yields a suppression below the fundamental gravitational scale and for increasing number of extra dimensions. The results of numerical simulations performed with the black hole event generator BLACKMAX are here reported in order to illustrate the main differences in the number of expected black hole events and mass distributions.
• We measure the energy emitted by extensive air showers in the form of radio emission in the frequency range from 30 to 80 MHz. Exploiting the accurate energy scale of the Pierre Auger Observatory, we obtain a radiation energy of 15.8 \pm 0.7 (stat) \pm 6.7 (sys) MeV for cosmic rays with an energy of 1 EeV arriving perpendicularly to a geomagnetic field of 0.24 G, scaling quadratically with the cosmic-ray energy. A comparison with predictions from state-of-the-art first-principle calculations shows agreement with our measurement. The radiation energy provides direct access to the calorimetric energy in the electromagnetic cascade of extensive air showers. Comparison with our result thus allows the direct calibration of any cosmic-ray radio detector against the well-established energy scale of the Pierre Auger Observatory.
• Neutrinos in the cosmic ray flux with energies near 1 EeV and above are detectable with the Surface Detector array of the Pierre Auger Observatory. We report here on searches through Auger data from 1 January 2004 until 20 June 2013. No neutrino candidates were found, yielding a limit to the diffuse flux of ultra-high energy neutrinos that challenges the Waxman-Bahcall bound predictions. Neutrino identification is attempted using the broad time-structure of the signals expected in the SD stations, and is efficiently done for neutrinos of all flavors interacting in the atmosphere at large zenith angles, as well as for "Earth-skimming" neutrino interactions in the case of tau neutrinos. In this paper the searches for downward-going neutrinos in the zenith angle bins $60^\circ-75^\circ$ and $75^\circ-90^\circ$ as well as for upward-going neutrinos, are combined to give a single limit. The $90\%$ C.L. single-flavor limit to the diffuse flux of ultra-high energy neutrinos with an $E^{-2}$ spectrum in the energy range $1.0 \times 10^{17}$ eV - $2.5 \times 10^{19}$ eV is $E_\nu^2 dN_\nu/dE_\nu < 6.4 \times 10^{-9}~ {\rm GeV~ cm^{-2}~ s^{-1}~ sr^{-1}}$.
• A measurement of the cosmic-ray spectrum for energies exceeding $4{\times}10^{18}$ eV is presented, which is based on the analysis of showers with zenith angles greater than $60^{\circ}$ detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above $5.3{\times}10^{18}$ eV, the "ankle", the flux can be described by a power law $E^{-\gamma}$ with index $\gamma=2.70 \pm 0.02 \,\text{(stat)} \pm 0.1\,\text{(sys)}$ followed by a smooth suppression region. For the energy ($E_\text{s}$) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find $E_\text{s}=(5.12\pm0.25\,\text{(stat)}^{+1.0}_{-1.2}\,\text{(sys)}){\times}10^{19}$ eV.
• ### $X_{max}^{\mu}$ vs. $N^{\mu}$ from Extensive Air Showers as estimator for the mass of primary UHECR's. Application for the Pierre Auger Observatory(1409.2972)

Sept. 10, 2014 hep-ex, astro-ph.IM, astro-ph.HE
We study the possibility of primary mass estimation for Ultra High Energy Cosmic Rays (UHECR's) using the $X_{max}^{\mu}$ (the height where the number of muons produced on the core of Extensive Air Showers (EAS) is maximum) and the number $N^{\mu}$ of muons detected on ground. We use the 2D distribution - $X_{max}^{\mu}$ against $N^{\mu}$ in order to find its sensitivity to the mass of the primary particle. For that, we construct a 2D Probability Function $Prob(p,Fe \ | \ X_{max}^{\mu},N^{\mu})$ which estimates the probability that a certain point from the plane $(X_{max}^{\mu}$, $N^{\mu})$ corresponds to a shower induced by a proton, respectively an iron nucleus. To test the procedure, we analyze a set of simulated EAS induced by protons and iron nuclei at energies of $10^{19} eV$ and $20^{\circ}$ zenith angle with CORSIKA. Using the Bayesian approach and taking into account the geometry of the infill detectors from the Pierre Auger Observatory, we observe an improvement in the accuracy of the primary mass reconstruction in comparison with the results obtained using only the $X_{max}^{\mu}$ distributions.
• ### Quantum Black Holes Effects on the Shape of Extensive Air Showers(1310.2205)

July 8, 2014 hep-ph, gr-qc, astro-ph.HE
We investigate the possibility to find a characteristic TeV scale quantum black holes decay signature in the data recorded by cosmic rays experiments. TeV black holes can be produced via the collisions of ultra high energetic protons (E > $10^18$ eV) with nucleons the from atmosphere. We focus on the case when the black holes decay into two particles moving in the forward direction in the Earth reference frame (back-to-back in the center of mass reference frame) and induce two overlapping showers. When reconstructing both the energy and the shape of the resultant air shower, there is a significant difference between showers induced only via standard model interactions and showers produced via the back-to-back decay of black holes as intermediate states.
• ### Back-to-Back Black Holes decay Signature at Neutrino Observatories(1303.4603)

Jan. 6, 2014 hep-ph
We propose a decay signature for non-thermal small black holes with masses in the TeV range which can be discovered by neutrino observatories. The black holes would result due to the impact between ultra high energy neutrinos with nuclei in water or ice and decay instantaneously. They could be produced if the Planck scale is in the few TeV region and the highly energetic fluxes are large enough. Having masses close to the Planck scale, the typical decay mode for these black holes is into two particles emitted back-to-back. For a certain range of angles between the emitted particles and the center of mass direction of motion, it is possible for the detectors to measure separate muons having specific energies and their trajectories oriented at a large enough angle to prove that they are the result of a back-to-back decay event.


text:
## Finding the Greatest Common Factor from Two Expressions

### Learning Outcomes

• Find the greatest common factor of two numbers

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

We also factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

### Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

First we will find the greatest common factor of two numbers.

### example

Find the greatest common factor of $24$ and $36$.

Solution

 Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. Factor $24$ and $36$. Step 2: List all factors–matching common factors in a column. In each column, circle the common factors. Circle the $2, 2$, and $3$ that are shared by both numbers. Step 3: Bring down the common factors that all expressions share. Bring down the $2, 2, 3$ and then multiply. Step 4: Multiply the factors. The GCF of $24$ and $36$ is $12$.

Notice that since the GCF is a factor of both numbers, $24$ and $36$ can be written as multiples of $12$.

$\begin{array}{c}24=12\cdot 2\\ 36=12\cdot 3\end{array}$

### try it

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

### Find the greatest common factor

1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Bring down the common factors that all expressions share.
4. Multiply the factors.

### example

Find the greatest common factor of $5x\text{ and }15$.

### try it

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

### example

Find the greatest common factor of $12{x}^{2}$ and $18{x}^{3}$.

### example

Find the greatest common factor of $14{x}^{3},8{x}^{2},10x$.

### try it

Watch the following video to see another example of how to find the GCF of two monomials that have one variable.


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# Elliptic curve explained

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in, the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which, after a linear change of variables, consists of solutions for:

y2=x3+ax+b

for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition

4a3+27b2\ne0

, that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see below.)

An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and serves as the identity element.

If, where is any polynomial of degree three in with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity.

Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism.

Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization.

An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant

j\geq1

as ellipses in the hyperbolic plane

H2

. Specifically, the intersections of the Minkowski hyperboloid with quadric surfaces characterized by a certain constant-angle property produce the Steiner ellipses in

H2

(generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the elliptic curves with

j\leqslant1

, and any ellipse in

H2

described as a locus relative to two foci is uniquely the elliptic curve sum of two Steiner ellipses, obtained by adding the pairs of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve.[1] [2]

Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.

## Elliptic curves over the real numbers

Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry.

In this context, an elliptic curve is a plane curve defined by an equation of the form

y2=x3+ax+b

after a linear change of variables (a and b are real numbers). This type of equation is called a Weierstrass equation.

The definition of elliptic curve also requires that the curve is non-singular. Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this holds if and only if the discriminant

\Delta=-16(4a3+27b2)

is not equal to zero. (Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.)

The (real) graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.

## The group law

When working in the projective plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have an additional point at infinity, O, at the homogeneous coordinates [0:1:0] which serves as the identity of the group.

Since the curve is symmetrical about the x-axis, given any point P, we can take −P to be the point opposite it. We take −O to be just O.

If P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way. First, draw the line that intersects P and Q. This will generally intersect the cubic at a third point, R. We then take to be −R, the point opposite R.

This definition for addition works except in a few special cases related to the point at infinity and intersection multiplicity. The first is when one of the points is O. Here, we define, making O the identity of the group. Next, if P and Q are opposites of each other, we define . Lastly, if we only have one point, thus we can't define the line between them. In this case, we use the tangent line to the curve at this point as our line. In most cases, the tangent will intersect a second point R and we can take its opposite. However, if P happens to be an inflection point (a point where the concavity of the curve changes), we take R to be P itself and P + P is simply the point opposite itself.

For a cubic curve not in Weierstrass normal form, we can still define a group structure by designating one of its nine inflection points as the identity O. In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity. For a point P, −P is defined as the unique third point on the line passing through O and P. Then, for any P and Q, P + Q is defined as −R where R is the unique third point on the line containing P and Q.

Let K be a field over which the curve is defined (i.e., the coefficients of the defining equation or equations of the curve are in K) and denote the curve by E. Then the K-rational points of E are the points on E whose coordinates all lie in K, including the point at infinity. The set of K-rational points is denoted by E(K). It, too, forms a group, because properties of polynomial equations show that if P is in E(K), then −P is also in E(K), and if two of P, Q, and R are in E(K), then so is the third. Additionally, if K is a subfield of L, then E(K) is a subgroup of E(L).

The above group can be described algebraically as well as geometrically. Given the curve y2 = x3 + ax + b over the field K (whose characteristic we assume to be neither 2 nor 3), and points P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xPxQ (first pane below). Let y = sx + d be the line that intersects P and Q, which has the following slope:

s=

 yP-yQ xP-xQ

Since K is a field, s is well-defined. The line equation and the curve equation have an identical y in the points xP, xQ, and xR.

(sx+d)2=x3+ax+b

which is equivalent to

0=x3-s2x2-2sdx+ax+b-d2

. We know that this equation has its roots in exactly the same x-values as

(x-xP)(x-xQ)(x-xR)=x3+x2(-xP-xQ-xR)+x(xPxQ+xPxR+xQxR)-xPxQxR

We equate the coefficient for x2 and solve for xR. yR follows from the line equation. This defines R = (xR, yR) = −(P + Q) with

\begin{align} xR&=s2-xP-xQ\\ yR&=yP+s(xR-xP) \end{align}

If xP = xQ, then there are two options: if yP = −yQ (third and fourth panes below), including the case where yP = yQ = 0 (fourth pane), then the sum is defined as 0; thus, the inverse of each point on the curve is found by reflecting it across the x-axis. If yP = yQ ≠ 0, then Q = P and R = (xR, yR) = −(P + P) = −2P = −2Q (second pane below with P shown for R) is given by

\begin{align} s&=

 3{xP 2

+a}{2yP}\\ xR&=s2-2xP\\ yR&=yP+s(xR-xP) \end{align}

## Elliptic curves over the complex numbers

The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula

\wp'(z)2=4\wp(z)3-g2\wp(z)-g3

Here, g2 and g3 are constants;

\wp(z)

is the Weierstrass elliptic function and

\wp'(z)

its derivative. It should be clear that this relation is in the form of an elliptic curve (over the complex numbers). The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map

z\mapsto[1:\wp(z):\wp'(z)/2]

This map is a group isomorphism of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map. It is also an isomorphism of Riemann surfaces from the torus to the cubic curve, so topologically, an elliptic curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number c to a lattice cΛ, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the j-invariant.

The isomorphism classes can be understood in a simpler way as well. The constants g2 and g3, called the modular invariants, are uniquely determined by the lattice, that is, by the structure of the torus. However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers is the algebraic closure of the reals. So, the elliptic curve may be written as

y2=x(x-1)(x-λ)

One finds that

g2'=

 \sqrt[3]4 3

(λ2-λ+1)

g3'=

 1 27

(λ+1)(2-+2)

and

j(\tau)=1728

 3 g 2'
3
g-
 2 27g 3'
2'

=256

 (λ2-λ+1)3 λ2(λ-1)2

j(\tau)

and

λ(\tau)

is sometimes called the modular lambda function. For example, let

\tau=2i

, then

λ(2i)=(-1+\sqrt2)4

which implies

g2',g3',

 3 g 2'

-

 2 27g 3'
of the formula above are algebraic numbers if

\tau

involves an imaginary quadratic field. In fact, it yields the integer

j(2i)=663

.

\Delta(\tau)=

 3 g 2(\tau)

-

 2 27g 3(\tau)

=(2\pi)12η24(\tau)

is generally a transcendental number. In particular, the value of the Dedekind eta function

η(2i)

is,
η(2i)=
\Gamma
 \left( 14\right)
 11 8
2
 34
\pi

Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented as a torus. This also allows an easy understanding of the torsion points on an elliptic curve: if the lattice Λ is spanned by the fundamental periods ω1 and ω2, then the n-torsion points are the (equivalence classes of) points of the form

 a n

\omega1+

 b n

\omega2

for a and b integers in the range from 0 to n−1.

If

E:

 2=4(x-e y 1)(x-e

2)(x-e3)

is an elliptic curve over the complex numbers and

a0=\sqrt{e1-e3}

,

b0=\sqrt{e1-e2}

and

c0=\sqrt{e2-e3}

, then a pair of fundamental periods of

E

can be calculated very rapidly by

\omega1=\pi/\operatorname{M}(a0,b0)

and

\omega2=\pi/\operatorname{M}(c0,ib0)

where

\operatorname{M}(w,z)

is the arithmetic–geometric mean of

w

and

z

. At each step of the arithmetic–geometric mean iteration, the signs of

zn

arising from the ambiguity of geometric mean iterations are chosen such that

|wn-zn|\le|wn+zn|

where

wn

and

zn

denote the individual arithmetic mean and geometric mean iterations of

w

and

z

, respectively. When

|wn-zn|=|wn+zn|

, there is an additional condition that

\Im(zn/wn)>0

.[3]

Over the complex numbers, every elliptic curve has nine inflection points. Every line through two of these points also passes through a third inflection point; the nine points and 12 lines formed in this way form a realization of the Hesse configuration.

## Elliptic curves over the rational numbers

A curve E defined over the field of rational numbers is also defined over the field of real numbers. Therefore, the law of addition (of points with real coordinates) by the tangent and secant method can be applied to E. The explicit formulae show that the sum of two points P and Q with rational coordinates has again rational coordinates, since the line joining P and Q has rational coefficients. This way, one shows that the set of rational points of E forms a subgroup of the group of real points of E. As this group, it is an abelian group, that is, P + Q = Q + P.

### The structure of rational points

The most important result is that all points can be constructed by the method of tangents and secants starting with a finite number of points. More precisely the Mordell–Weil theorem states that the group E(Q) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups.

The proof of that theorem rests on two ingredients: first, one shows that for any integer m > 1, the quotient group E(Q)/mE(Q) is finite (weak Mordell–Weil theorem). Second, introducing a height function h on the rational points E(Q) defined by h(P0) = 0 and if P (unequal to the point at infinity P0) has as abscissa the rational number x = p/q (with coprime p and q). This height function h has the property that h(mP) grows roughly like the square of m. Moreover, only finitely many rational points with height smaller than any constant exist on E.

The proof of the theorem is thus a variant of the method of infinite descent[4] and relies on the repeated application of Euclidean divisions on E: let PE(Q) be a rational point on the curve, writing P as the sum 2P1 + Q1 where Q1 is a fixed representant of P in E(Q)/2E(Q), the height of P1 is about of the one of P (more generally, replacing 2 by any m > 1, and by). Redoing the same with P1, that is to say P1 = 2P2 + Q2, then P2 = 2P3 + Q3, etc. finally expresses P as an integral linear combination of points Qi and of points whose height is bounded by a fixed constant chosen in advance: by the weak Mordell–Weil theorem and the second property of the height function P is thus expressed as an integral linear combination of a finite number of fixed points.

So far, the theorem is not effective since there is no known general procedure for determining representatives of E(Q)/mE(Q).

The rank of E(Q), that is the number of copies of Z in E(Q) or, equivalently, the number of independent points of infinite order, is called the rank of E. The Birch and Swinnerton-Dyer conjecture is concerned with determining the rank. One conjectures that it can be arbitrarily large, even if only examples with relatively small rank are known. The elliptic curve with biggest exactly known rank is

y2 + xy + y = x3x2x +

It has rank 20, found by Noam Elkies and Zev Klagsbrun in 2020. Curves of rank higher than 20 were known since 1994, with lower bounds on their ranks ranging from at-least-21 to at-least-28, but their exact ranks are not currently known and in particular it is not proven which of them have higher rank than the others or which is the true "current champion".[5]

As for the groups constituting the torsion subgroup of E(Q), the following is known: the torsion subgroup of E(Q) is one of the 15 following groups (a theorem due to Barry Mazur): Z/NZ for N = 1, 2, ..., 10, or 12, or Z/2Z × Z/2NZ with N = 1, 2, 3, 4. Examples for every case are known. Moreover, elliptic curves whose Mordell–Weil groups over Q have the same torsion groups belong to a parametrized family.

### The Birch and Swinnerton-Dyer conjecture

See main article: Birch and Swinnerton-Dyer conjecture. The Birch and Swinnerton-Dyer conjecture (BSD) is one of the Millennium problems of the Clay Mathematics Institute. The conjecture relies on analytic and arithmetic objects defined by the elliptic curve in question.

At the analytic side, an important ingredient is a function of a complex variable, L, the Hasse–Weil zeta function of E over Q. This function is a variant of the Riemann zeta function and Dirichlet L-functions. It is defined as an Euler product, with one factor for every prime number p.

For a curve E over Q given by a minimal equation

y2+a1xy+a3y=x3+

 2 a 2x

+a4x+a6

with integral coefficients

ai

, reducing the coefficients modulo p defines an elliptic curve over the finite field Fp (except for a finite number of primes p, where the reduced curve has a singularity and thus fails to be elliptic, in which case E is said to be of bad reduction at p).

The zeta function of an elliptic curve over a finite field Fp is, in some sense, a generating function assembling the information of the number of points of E with values in the finite field extensions Fpn of Fp. It is given by[6]

Z(E(Fp))=\exp\left(\sum\#

 \left[E({F} )\right] pn
 Tn n

\right)

The interior sum of the exponential resembles the development of the logarithm and, in fact, the so-defined zeta function is a rational function:

Z(E(Fp))=

 1-apT+pT2 (1-T)(1-pT)

,

where the 'trace of Frobenius' term[7]

ap

is defined to be the (negative of) the difference between the number of points on the elliptic curve

E

over

Fp

and the 'expected' number

p+1

, viz.:

ap=p+1-\#E(Fp).

ap

are not to be confused with the

ai

in the definition of the curve

E

above: this is just an unfortunate clash of notation. Second, we may define the same quantities and functions over an arbitrary finite field of characteristic

p

, with

q=pn

replacing

p

everywhere.

The Hasse–Weil zeta function of E over Q is then defined by collecting this information together, for all primes p. It is defined by

L(E(Q),s)=\prodp\left(1-app-s+\varepsilon(p)p1\right)-1

where ε(p) = 1 if E has good reduction at p and 0 otherwise (in which case ap is defined differently from the method above: see Silverman (1986) below).

This product converges for Re(s) > 3/2 only. Hasse's conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture, which asserts that every elliptic curve over Q is a modular curve, which implies that its L-function is the L-function of a modular form whose analytic continuation is known. One can therefore speak about the values of L(E, s) at any complex number s.

The Birch–Swinnerton-Dyer conjecture relates the arithmetic of the curve to the behavior of its L-function at s = 1. It affirms that the vanishing order of the L-function at s = 1 equals the rank of E and predicts the leading term of the Laurent series of L(E, s) at that point in terms of several quantities attached to the elliptic curve.

Much like the Riemann hypothesis, the truth of the BSD conjecture would have multiple consequences, including the following two:

• A congruent number is defined as an odd square-free integer n which is the area of a right triangle with rational side lengths. It is known that n is a congruent number if and only if the elliptic curve

y2=x3-n2x

has a rational point of infinite order; assuming BSD, this is equivalent to its L-function having a zero at s = 1. Tunnell has shown a related result: assuming BSD, n is a congruent number if and only if the number of triplets of integers (x, y, z) satisfying

2x2+y2+8z2=n

is twice the number of triples satisfying

2x2+y2+32z2=n

. The interest in this statement is that the condition is easy to check.
• In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the critical strip for certain L-functions. Admitting BSD, these estimations correspond to information about the rank of families of the corresponding elliptic curves. For example: assuming the generalized Riemann hypothesis and BSD, the average rank of curves given by

y2=x3+ax+b

is smaller than 2.[8]

### The modularity theorem and its application to Fermat's Last Theorem

See main article: Modularity theorem. The modularity theorem, once known as the Taniyama–Shimura–Weil conjecture, states that every elliptic curve E over Q is a modular curve, that is to say, its Hasse–Weil zeta function is the L-function of a modular form of weight 2 and level N, where N is the conductor of E (an integer divisible by the same prime numbers as the discriminant of E, Δ(E)). In other words, if one writes the L-function for Re(s) > 3/2 in the form

L(E(Q),s)=\sumn>0a(n)n-s,

then the expression

\suma(n)qn,    q=e2

defines a parabolic modular newform of weight 2 and level N. For prime numbers not dividing N, the coefficient a() is equal to minus the number of solutions of the minimal equation of the curve modulo .

For example,[9] the elliptic curve

y2-y=x3-x

, with discriminant (and conductor) 37, is associated to the form

f(z)=q-2q2-3q3+2q4-2q5+6q6+,    q=e2

For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. Thus, for  = 3, there are 6 solutions of the equation modulo 3:,,,,, ; thus .

The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of Andrew Wiles, who proved it in 1994 for a large family of elliptic curves.[10]

There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over Q, from the modular curve X0(N) to E. In particular, the points of E can be parametrized by modular functions.

For example, a modular parametrization of the curve

y2-y=x3-x

is given by[11]

\begin{align} x(z)&=q-2+2q-1+5+9q+18q2+29q3+51q4+ … \\ y(z)&=q-3+3q-2+9q-1+21+46q+92q2+180q3+ … \end{align}

where, as above, q = exp(2πiz). The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im(z) > 0 and satisfy

 x\left( az+b cz+d

\right)=x(z)

and likewise for y(z), for all integers a, b, c, d with adbc = 1 and 37|c.

Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.

The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5, the Fermat equation

ap+bp=cp

has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice,[12] the elliptic curve

y2=x(x-ap)(x+bp)

of discriminant

\Delta=

 1 256

(abc)2p

cannot be modular.[13] Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987.[14]

### Integral points

This section is concerned with points P = (x, y) of E such that x is an integer. The following theorem is due to C. L. Siegel: the set of points P = (x, y) of E(Q) such that x is an integer is finite. This theorem can be generalized to points whose x coordinate has a denominator divisible only by a fixed finite set of prime numbers.

The theorem can be formulated effectively. For example,[15] if the Weierstrass equation of E has integer coefficients bounded by a constant H, the coordinates (x, y) of a point of E with both x and y integer satisfy:

max(|x|,|y|)<\exp\left(\left[106H\right]{106}\right)

For example, the equation y2 = x3 + 17 has eight integral solutions with y > 0 :[16]

(x, y) = (−1, 4), (−2, 3), (2, 5), (4, 9), (8, 23), (43, 282), (52, 375), .

As another example, Ljunggren's equation, a curve whose Weierstrass form is y2 = x3 − 2x, has only four solutions with y ≥ 0 :

(x, y) = (0, 0), (−1, 1), (2, 2), (338,).

### Generalization to number fields

Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to Loïc Merel shows that for a given integer d, there are (up to isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of degree d. More precisely,[17] there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of order less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then

p<

 3d2 d

As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates. Then there are only finitely many points of E(K) whose x-coordinate is in the ring of integers OK.

The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.

## Elliptic curves over a general field

Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K.

If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form

y2=x3-px-q

after a linear change of variables. Here p and q are elements of K such that the right hand side polynomial x3pxq does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form

y2=4x3+b2x2+2b4x+b6

for arbitrary constants b2, b4, b6 such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons). In characteristic 2, even this much is not possible, and the most general equation is

y2+a1xy+a3y=x3+a2x2+a4x+a6

provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable linear change of variables.

One typically takes the curve to be the set of all points (x,y) which satisfy the above equation and such that both x and y are elements of the algebraic closure of K. Points of the curve whose coordinates both belong to K are called K-rational points.

## Isogeny

Let E and D be elliptic curves over a field k. An isogeny between E and D is a finite morphism f : ED of varieties that preserves basepoints (in other words, maps the given point on E to that on D).

The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points.

## Elliptic curves over finite fields

Let K = Fq be the finite field with q elements and E an elliptic curve defined over K. While the precise number of rational points of an elliptic curve E over K is in general rather difficult to compute, Hasse's theorem on elliptic curves gives us, including the point at infinity, the following estimate:

|\#E(K)-(q+1)|\le2\sqrt{q}

In other words, the number of points of the curve grows roughly as the number of elements in the field. This fact can be understood and proven with the help of some general theory; see local zeta function, Étale cohomology.

The set of points E(Fq) is a finite abelian group. It is always cyclic or the product of two cyclic groups. For example,[18] the curve defined by

y2=x3-x

over F71 has 72 points (71 affine points including (0,0) and one point at infinity) over this field, whose group structure is given by Z/2Z × Z/36Z. The number of points on a specific curve can be computed with Schoof's algorithm.

Studying the curve over the field extensions of Fq is facilitated by the introduction of the local zeta function of E over Fq, defined by a generating series (also see above)

Z(E(K),T)\equiv\exp

 infty \left(\sum n=1

\#\left[E(Kn)\right]{Tn\overn}\right)

where the field Kn is the (unique up to isomorphism) extension of K = Fq of degree n (that is, Fqn). The zeta function is a rational function in T. There is an integer a such that

Z(E(K),T)=

 1-aT+qT2 (1-qT)(1-T)

Moreover,

\begin{align} Z\left(E(K),

 1 qT

\right)&=Z(E(K),T)\\ \left(1-aT+qT2\right)&=(1-\alphaT)(1-\betaT) \end{align}

\sqrt{q}

. This result is a special case of the Weil conjectures. For example, the zeta function of E : y2 + y = x3 over the field F2 is given by
 1+2T2 (1-T)(1-2T)

this follows from:

\left|

 E(F 2r

)\right|=\begin{cases}2r+1&rodd\ 2r+1-

 r 2
2(-2)

&reven\end{cases}

The Sato–Tate conjecture is a statement about how the error term

2\sqrt{q}

in Hasse's theorem varies with the different primes q, if an elliptic curve E over Q is reduced modulo q. It was proven (for almost all such curves) in 2006 due to the results of Taylor, Harris and Shepherd-Barron,[19] and says that the error terms are equidistributed.

Elliptic curves over finite fields are notably applied in cryptography and for the factorization of large integers. These algorithms often make use of the group structure on the points of E. Algorithms that are applicable to general groups, for example the group of invertible elements in finite fields, F*q, can thus be applied to the group of points on an elliptic curve. For example, the discrete logarithm is such an algorithm. The interest in this is that choosing an elliptic curve allows for more flexibility than choosing q (and thus the group of units in Fq). Also, the group structure of elliptic curves is generally more complicated.

## Algorithms that use elliptic curves

Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. For more see also:

## References

Serge Lang, in the introduction to the book cited below, stated that "It is possible to write endlessly on elliptic curves. (This is not a threat.)" The following short list is thus at best a guide to the vast expository literature available on the theoretical, algorithmic, and cryptographic aspects of elliptic curves.

• Book: I. Blake . G. Seroussi . N. Smart . 2000 . Elliptic Curves in Cryptography . LMS Lecture Notes . Cambridge University Press . 0-521-65374-6 .
• Book: Richard Crandall . Carl Pomerance . 2001 . Prime Numbers: A Computational Perspective . Springer-Verlag . 1st . 0-387-94777-9 . Chapter 7: Elliptic Curve Arithmetic . 285–352 . Carl Pomerance . Richard Crandall .
• Book: Cremona, John . John Cremona . 1997 . 2nd . Algorithms for Modular Elliptic Curves . Cambridge University Press . 0-521-59820-6 .
• Book: . 2004 . Guide to Elliptic Curve Cryptography . . 0-387-95273-X .
• Chapter XXV
• Book: Hellegouarch . Yves . Invitation aux mathématiques de Fermat-Wiles . Dunod . Paris . 978-2-10-005508-1 . 2001 .
• Book: Husemöller , Dale . Dale Husemöller . 2004 . Elliptic Curves . 2nd . . 111 . Springer . 0-387-95490-2 .
• Book: Kenneth Ireland . Kenneth Ireland . Michael I. Rosen . 1998 . A Classical Introduction to Modern Number Theory . 84 . Graduate Texts in Mathematics . Springer . 2nd revised . Chapters 18 and 19 . 0-387-97329-X . Michael I. Rosen .
• Book: Knapp, Anthony W. . Anthony W. Knapp . 1992 . 2018 . Elliptic Curves . Mathematical Notes . 40 . Princeton University Press . 9780691186900 .
• Book: Koblitz, Neal . Neal Koblitz . 1993 . 2nd . Introduction to Elliptic Curves and Modular Forms . Graduate Texts in Mathematics . 97 . Springer-Verlag . 0-387-97966-2 .
• Book: Koblitz, Neal . Neal Koblitz . 1994 . A Course in Number Theory and Cryptography . Graduate Texts in Mathematics . 114 . Springer-Verlag . 2nd . 0-387-94293-9 . Chapter 6 .
• Book: Serge Lang . Serge Lang . Elliptic curves: Diophantine analysis . Grundlehren der mathematischen Wissenschaften . 231 . Springer-Verlag . 1978 . 3-540-08489-4 .
• Book: Henry McKean . Victor Moll. Victor Hugo Moll . Elliptic curves: function theory, geometry and arithmetic . Cambridge University Press. 0-521-65817-9. 1999.
• Book: Ivan Niven . Herbert S. Zuckerman . Hugh Montgomery . Hugh Montgomery (mathematician) . An introduction to the theory of numbers . https://archive.org/details/introductiontoth0000nive . registration . 5th . John Wiley . 1991 . 0-471-54600-3 . Section 5.7.
• Book: Silverman, Joseph H. . Joseph H. Silverman . 1986 . The Arithmetic of Elliptic Curves . Graduate Texts in Mathematics . 106 . Springer-Verlag . 0-387-96203-4 .
• Book: Joseph H. Silverman . Joseph H. Silverman . 1994 . Advanced Topics in the Arithmetic of Elliptic Curves . Graduate Texts in Mathematics . 151 . Springer-Verlag . 0-387-94328-5 .
• Book: Joseph H. Silverman . Joseph H. Silverman . John Tate . 1992 . Rational Points on Elliptic Curves . Springer-Verlag . 0-387-97825-9 . John Tate (mathematician) .
• John Tate . John Tate (mathematician) . The arithmetic of elliptic curves . . 23 . 179–206 . 1974 . 10.1007/BF01389745 . 3–4 . 1974InMat..23..179T . 120008651 .
• Book: Lawrence Washington . 2003 . Elliptic Curves: Number Theory and Cryptography . registration . Chapman & Hall/CRC . 1-58488-365-0 .

## Notes and References

1. Sarli . J. . 2012 . Conics in the hyperbolic plane intrinsic to the collineation group . J. Geom. . 103 . 131-148 . 10.1007/s00022-012-0115-5.
2. Sarli . John . 2021-10-22 . The Elliptic Curve Decomposition of Central Conics in the Real Hyperbolic Plane . 10.21203/rs.3.rs-936116/v1.
3. Web site: The Arithmetic-Geometric Mean and Periods of Curves of Genus 1 and 2. Wing Tat Chow. Rudolf. 2018. White Rose eTheses Online. 12.
4. See also J. W. S. Cassels, Mordell's Finite Basis Theorem Revisited, Mathematical Proceedings of the Cambridge Philosophical Society 100, 3–41 and the comment of A. Weil on the genesis of his work: A. Weil, Collected Papers, vol. 1, 520–521.
5. Web site: History of elliptic curves rank records. Dujella. Andrej. Andrej Dujella. University of Zagreb.
6. The definition is formal, the exponential of this power series without constant term denotes the usual development.
7. see for example Web site: An Introduction to the Theory of Elliptic Curves . Joseph H. . Silverman . 2006 . Summer School on Computational Number Theory and Applications to Cryptography . University of Wyoming .
8. D. R. . Heath-Brown . The Average Analytic Rank of Elliptic Curves . Duke Mathematical Journal . 122 . 3 . 591–623 . 2004 . 10.1215/S0012-7094-04-12235-3 . math/0305114 . 15216987 .
9. For the calculations, see for example
10. A synthetic presentation (in French) of the main ideas can be found in this Bourbaki article of Jean-Pierre Serre. For more details see
11. Book: Zagier, D. . Modular points, modular curves, modular surfaces and modular forms . Lecture Notes in Mathematics . 1111 . Springer . 1985 . 225–248 . 10.1007/BFb0084592 . 978-3-540-39298-9 . Arbeitstagung Bonn 1984 .
12. Hellegouarch . Yves . Points d'ordre 2ph sur les courbes elliptiques . 0379507 . 1974 . Acta Arithmetica . 0065-1036 . 26 . 3 . 253–263. 10.4064/aa-26-3-253-263 . free .
13. Ribet . Ken . Ken Ribet . On modular representations of Gal(/Q) arising from modular forms . Inventiones Mathematicae . 100 . 1990 . 2 . 431–476 . 10.1007/BF01231195 . 1047143 . 10338.dmlcz/147454 . 1990InMat.100..431R . 120614740 .
14. See the survey of K. . Ribet . From the Taniyama–Shimura conjecture to Fermat's Last Theorem . Annales de la Faculté des Sciences de Toulouse . 11 . 1990 . 116–139 . 10.5802/afst.698. free .
15. , due to Baker.
16. T. Nagell, L'analyse indéterminée de degré supérieur, Mémorial des sciences mathématiques 39, Paris, Gauthier-Villars, 1929, pp. 56–59.
17. L. . Merel . Loïc Merel . Bornes pour la torsion des courbes elliptiques sur les corps de nombres . fr . . 124 . 1996 . 1–3 . 437–449 . 0936.11037 . 10.1007/s002220050059 . 1996InMat.124..437M . 3590991 .
18. See
19. M. . Harris . N. . Shepherd-Barron . R. . Taylor . A family of Calabi–Yau varieties and potential automorphy . . 171 . 2 . 779–813. 2010 . 10.4007/annals.2010.171.779 . free .


text:
MAT244--2019F > Quiz-5

TUT5103 Quiz5

(1/1)

Ruodan Chen:
Find the general solution of the given differential equation.

y'' + 4y = 3csc(2t), 0 < t < $\frac{\pi}{2}$

$r^{2}$ + 4 = 0

$r^{2}$ = $-4$

r = $\pm2i$

$y_{c}$(t) = $c_{1}$cos2t + $c_{2}$sin2t

$y_{1}$ = cos2t

$y_{2}$ = sin2t

w = $\begin{array}{cc} cos2t & sin2t\\ -2sin2t & 2cos2t \end{array}$ = 2$cos^{2}2t$ + 2$sin^{2}2t$ = 2

$w_{1}$ = $\begin{array}{cc} 0 & sin2t\\ 1 & 2cos2t \end{array}$ = -sin2t

$w_{2}$ = $\begin{array}{cc} cos2t & 0\\ -2sin2t & 1 \end{array}$ = cos2t

Y(t) = cos2t$\int$$\frac{-sin2s3csc2s}{2}ds + sin\int\frac{cos2s3csc2s}{2}ds = -cos2t(\frac{3}{2})\intsin2scsc2sds + sin2t(\frac{3}{2})\int$$\frac{cos2s}{sin2s}$ds

= -cos2t($\frac{3}{2}$)t + sin2t($\frac{3}{2}$)ln|sin2t|

y(t) = $y_{c}$(t) + Y(t) = $c_{1}$cos2t + $c_{2}$sin2t - cos2t($\frac{3}{2}$)t + sin2t($\frac{3}{2}$)ln|sin2t|


text:
# Measure theory convention that $\infty \cdot 0 = 0$

In the preface of Terry Tao's notes on measure theory he states that in the extended real number setting we adopt the convention that $\infty \cdot 0 = 0 \cdot \infty = 0.$

He explains that it's a useful convention which makes it natural to define integration from below (e.g. integrals as supremums of nonnegative simple functions). He seems to be saying, "Let's define it this way, because it makes our lives simpler." Is more justification not needed?

This 'fact' helped me during my midterm today in showing that $\mu(E \times \mathbb{R})=0$ for any measure-zero set $E$, but I felt sleazy using it.

Any thoughts (philosophical or otherwise) on why or how we can do this and still have self respect as mathematicians?

• Large values of a function on a set with small measure don't contribute much to the integral ($\infty\cdot0=0$); small values on a large set don't contribute much to the integral ($0\cdot\infty=0$). – egreg Oct 16 '14 at 8:55
• A convention does not in itself prove a theorem; the fact that $\mu(E\times \mathbb R)=0$ for any measure-zero set $E$ is one of the justifications for using this convention, not a consequence of it. Sounds like your solution on your midterm was wrong. – Jonas Meyer Oct 16 '14 at 13:36
• @JonasMeyer: Let $F = E \times \mathbb{R}$ so that $\mu(F) = \int_{\mathbb{R}^2} \chi_F = \int_{\mathbb{R}}\chi_E \cdot \int_{\mathbb{R}}\chi_{\mathbb{R}} = 0 \cdot \infty = 0$. Does that work? – David D. Oct 16 '14 at 18:30
• @DavidD.: I'm not sure if you've already read my answer, but my response may be redundant. It is appropriate to use when it has been proven correct in the given situation. E.g., $\mu_1\times\mu_2(E_1\times E_2)=\mu_1(E_1)\times\mu_2(E_2)$ is a theorem if we use this convention, but the proof that it works even in the $0\times \infty$ case is not a result of the convention (but rather another reason for using the convention to condense statements). It can be proven for instance along the lines indicated at the end of my answer. – Jonas Meyer Oct 16 '14 at 19:27
• Tao explained it very well in the book: if $x_n\geq0$ increasingly converges to $x$, $y_n\geq0$ increasingly converges to $y$, then we'd like to have $x_n\cdot y_n$ increasingly converge to $x\cdot y$. In the case of $x_n\equiv0$ and $y_n\to\infty$, we see that $x_n\cdot y_n\equiv0$ and hence $\to0$. THis leads to $0\cdot\infty=0$. – Troy Woo Jul 21 '15 at 8:02

You need to distinguish between facts and convention. A convention is useful if it helps to organize a number of facts. In the context of measure theory, there are several facts that have the appearance of "$0\times \infty =0$" that are proven from the definitions, and thus the convention allows for an easier way to state facts.

A couple of examples:

1. If $\mu(X)=0$ and $f(x)=+\infty$ for all $x\in X$, then the definition of $\int_X f\,d\mu$ leads to a value of $0$. On the other hand, this could be thought of as "$\mu(X)\times \infty=0\times \infty$", integrating a constant function.
2. If $f(x)=0$ for all $x\in X$ and $\mu(X)=\infty$, then the definition of $\int_X f\,d\mu$ leads to a value of $0$. On the other hand, this could be thought of as "$\mu(X)\times 0=\infty\times 0$", integrating a constant function.

As a consequence of 1. and 2. above, adopting the convention $0\times \infty=0$ (along with the more intuitive $a\times \infty=\infty$ if $a>0$) allows the following result to be stated:

$$\int_{X} C\,d\mu=\mu(X)\times C,$$

for all constants $C\in[0,\infty]$ regardless of the size of $\mu(X)$. The convention is used only in stating and using the result in condensed form, not in proving it.

Another example that you highlight: If $E$ has measure zero, so does $E\times \mathbb R$. This is not proved using the convention. I would think of it as $$\mu(E\times \mathbb R)=\mu\left(\bigcup_n E\times [-n,n]\right)\leq\sum_n\mu(E\times[-n,n])=\sum_n 0=0,$$ where the equation $\mu(E\times[-n,n])=0$ requires justification, but nothing with infinite measure is involved.

• Is 2x2=4 a convention or fact? – Anixx Nov 27 '16 at 19:16
• @Anixx: Fact or definition I would say. It could be true by definition of $4$ making it a trivial "fact" (i.e., just a definition), but if $2$, $4$, and multiplication have definitions a priori, then this fact would have to be proved. – Jonas Meyer May 10 '17 at 6:31

Another good motive: do you find reasonable the next equalities? $$0=\int_X 0\,d\mu=0\,\mu(X).$$

• I like this motivation although, somewhat analogously, one might question whether $\int_{(0,n)} \infty d\mu$ converges to $0$ as n goes to $0$. – David D. Oct 16 '14 at 9:11
• @DavidD, obviously, $\lim\int=\infty$, but $\int_{\{0\}}\infty =$ area of a vertical half-line = ? – Martín-Blas Pérez Pinilla Oct 16 '14 at 9:50
• It feels ambiguous, though! Considering the dirac delta function, $\int_{0} \infty$ would be $1$... – David D. Oct 16 '14 at 19:58
• @DavidD.: The Dirac delta function is not a function. The definition of the integral of the function whose value at $0$ is $+\infty$ and whose value elsewhere is $0$, leads to a value of $0$. The delta "function" integral is $1$, so it is something else. – Jonas Meyer Oct 17 '14 at 3:31

Somehow it feels like the way to be "in the same side" of Continuum Hypotesis: http://en.m.wikipedia.org/wiki/Continuum_hypothesis

The sum of countable many sets is also countable... so $\infty \cdot 0$ should be 0.


text:
# Complex integral over a circle

1. Apr 3, 2012

1. let C be the circle |z| = 2 traveled once in the positive sense. Computer the following integrals...

a.∫c zez/(2z-3) dz

2. Relevant equations

I am confused as to a step in my solution, but i believe a relevant equation is if i am integrating over a circle and the function is analytic at a point, i can remove it by integrating and removing the denominator and replacing it with 2pi*i in the numerator. However i do not know why. This seems to be a product of the cauchy reimann equations used to calculate the integral.

3. The attempt at a solution

So i need to get the equation in the form x/z-z0 so i divide everything by 2.

I am left with .5*zez/(z-(2/3)) In this form i can see it is analytic at z=3/2 (i believe the word analytic to mean that it goes to infinity)

so i remove the denominator, and replace it with a substitution of 2pi*i...

2pi*i*.5*z*ez evaluated at z=3/2
is equal to 3/2pi*i*e3/2

Correct? i Know that sometimes the factor 2pi*i can be negative but i dont understandw hy..

2. Apr 3, 2012

Okay so i have looked abck and tried to understand the derivation again, and have realized it comes as a consquence of the equation integral of f(z)/(z-z0)^m dz is equal to 2pi*i f^(m-1) (z0) / (m-1)!

(if anyone finds my notation to be confusing or sloppy, i can rewrite it. also is there a link to a website that i can use to link equations to those neat pictures? it seems alot easier to understand than what im doing)

so to apply that equation to the above, my function is .5*ze^z, my Z0 is 3/2, and i rewrite my equation

(2pi*i*(.5ze^z)^0)/0!
where i know 0!=1, and anything raised to the 0 equals 1, so

my integral evaluates to 2pi*i.

3. Apr 3, 2012

### Dick

You were closer the first time. It's the Cauchy Integral Theorem you want to look at. And 'analytic' doesn't mean 'goes to infinity', it means rather the opposite. I think you might want to read up on this before you tackle the problem. Your concepts are very confused.

4. Apr 3, 2012

Okay so third times a charm, i figured out that it is not f^(m-1), but f with derivatives equal to m-1.

So since my m=1 for this, i do not have to take a derivative but only evaluate it at 3/2 over !0.

so my answer is pi * i * 3/2 * e^(3/2)

5. Apr 3, 2012

I know, i am struggling in this class and im trying to keep up with it. I am going to re-read the chapter in order to understand the cauchy integral theorem but to be honest im pretty poor at learning math out of books. I am best at visualizing problems which is particularly difficult in complex equations since its a 4 dimensional problem.

6. Apr 3, 2012

### Dick

There's less detailed visualization required here than you might think. Your answer in the last post is correct. But I'd feel better if you were clearer on why that is.

7. Apr 3, 2012

Okay so an integral is analytic if it satisfied the cauchy reimann equations. Basically what i know is if an equation satisfies the cauchy reimann equations, all is good and it can be worked with. I need to solve the second order cauchy reimann equation in order to prove a function is harmonic. Proving a function is harmonic is important (im a physics major) but i havn't had to do more than prove this.

Anyhoo a function is analytic if the power series converges. (i see what you mean, analytic is good, not bad) So where the denominator "blows up" (when z=z0) is where we evaluate the integral around because we are really evaluating the power series there?

If a function has a z0 outside the integral would the function then equal 0?

8. Apr 3, 2012

### Dick

You are getting there. Your function has pole at z=3/2 and that's inside of the circle |z|=2. Just apply the Cauchy Integral Theorem.

9. Apr 4, 2012

To be honest i can do the problems but the cauchy integral theorem still perplexes me. I understand the integral fails if there is a "hole" within the integral, and equals 0 everywhere else. So i guess what im really trying to integrate is the pole? So the entire point is to find the integral of the pole?

http://mathworld.wolfram.com/CauchyIntegralFormula.html

But what confused me was that when winding multiple times over the integral the equation becomes 1/2pi*i, which seems counter intuitive to me since if the first integral is equal to 2pi*i, if i integrate twice it should simply be 4pi*i.

The "derivative of f" was expressed as 1/(2pi*i)*(f(zo)/(z-zo)^2, but i thought i could just take the derivative of a complex number as i did a "regular" number...

I also have another problem related to this, Should i create another thread? (i am going to sleep for now so ill post it here, as i suspect the answer is i shouldnt)

1. let C be the square with vertices z= +-2 and +-2i traversed once in the positive sense.

etc etc etc.

i suspect i evaluate these just as i would the first problem. As from how i understand the integral as long as the pole is within the "area" im just going to be integrating it anyway, so i can effectively ignore the area that im supposedly integrating and just treat it as though im integrating the pole. The area as far as i know is just to stipulate if the pole is within it or not.

10. Apr 4, 2012

### Dick

You aren't 'integrating the pole' you are integrating the function along the contour. In some cases you can do that directly. You should work out integral of f(z)dz around the contour |z|=1 for some simple functions, like f(z)=1/z, f(z)=1, f(z)=z. You can do this with the change of variable z=exp(it). Notice that the only integral of these that is nonzero has a pole at z=0. Finding contour integrals if the function or contours are more complicated would be extremely hard. But luckily you have these theorems that relate the hard job of finding the integral to the much easier job of finding and summing the residues of the poles. That's the general idea.


text:
# Trigonometry Graphing Applet

• Trigonometric Graphing - NCTM Illuminations
• Dave's Short Trig Course
• Graphical Function Explorer grapher (GFE) - Math Open ...
• ## Trigonometric Graphing - NCTM Illuminations

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### Interactive Unit Circle

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A graph of a functions is a visual representation of the pairs (input, output), in the plane. This applet will help you to understand the connection between the graph of a function and a function as and input-output machine. Graphs of Sine and Cosine: 1.2: An applet illustrating how the graphs of sine and cosine are related to the unit circle. Given the formula of a sinusoidal function of the form a*f(bx+c)+d, draw its graph. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

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### Graph of the Sine function - Trigonometry - Math Open ...

To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. The result, as seen above, is a smooth curve that varies from +1 to -1. Learn trigonometry for free—right triangles, the unit circle, graphs, identities, and more. Full curriculum of exercises and videos. Full curriculum of exercises and videos. If you're seeing this message, it means we're having trouble loading external resources on our website.

## Dave's Short Trig Course

About the Java applet. Images in Dave’s Short Trig Course are illustrated with a Java applet. If your browser is Java-enabled, you can drag the points around in the diagrams and the diagram will adjust itself. The applet also allows you to lift a diagram off the web page into its own floating window. For details, see About the applet. We can do the same sort of function conversion with the cosine ratio: The relationship is a little harder to see here, because the unit circle's line is horizontal while the standard graph's line is vertical, but you can see how those two purple lines are the same length, while the angle measure is moving from zero to 2π.

### Trig Tour 1.0.22 - PhET Interactive Simulations

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### Trigonometric functions, applets and animations, equation ...

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Trigonometric Functions Applets. Trigonometric Functions (Graphing & Modeling) Prerequisite Topics Java-Applets werden verwendet, um zu erkunden, interaktiv, wichtige Themen in der Trigonometrie wie z. B. Schaubilder der 6 trigonometrische Funktionen, inverse trigonometrische Funktionen, Einheitskreis, Winkel-und Sinus-Gesetz. Interaktive Trigonometry Tutorials (Applets) Winkel in Trigonometrie; Winkel in Trigonometrie. No, we're not going to get a TAN on a COT, sipping lemonade. We're graphing the trigonometric functions of tangent (TAN) and cotangent (COT). Using the Unit Circle, and appropriate trigonometric ...

### Trigonometry Applets – GeoGebra

Trigonometry Applets The Parameterized Surface Applet lets you see the graph of a parameterized surface. As we did with the cross section applet above, we can also see a wire frame constructed from the graphs of lines in the domain. The 2D Grapher Applet page collects 3 applets for graphing in the plane. One can either use the Cartesian Grapher to graph y as a ... Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem.You may search for them by typing “MathMaine” into the GeoGebraTube search box.. Links to all updated interactive graph applets are below.

### Trigonometric Functions Applets

This applet lets the user experiment with the role a, b, kand chave in the shape of the graph of a cosine function (y = a+ b cos( k(x - c)) ) by changing the value of a, bor c. As those values are changed, the user will see how the graph is changing. Frequencies of Notes on a Piano: Learning object Applet Description. On the previous page, What are the frequencies of music notes?, we learned how to find the frequencies for a piano tuned by "equal-temperament".It turns out only 7 notes are actually "in tune" on a piano, and all the others are slightly out. Figure 8: Graphing the sine function from a unit circle [4] The Japanese Manipula Math collection [9] can be used online or can be purchased for individual or class use offline. This collection includes a significant number of interesting and educationally sound applets related to trigonometry. Graphing on graphics calculators

## Graphical Function Explorer grapher (GFE) - Math Open ...

GFE is a free online function graphing tool that allows you to plot up to three functions on the same set of axes. In the functions you can refer to up to four independent variables that are controlled by sliders. The function grapher can plot sinusoidal and other trigonometric functions including sine (sin), cosine(cos) and tan. Similar to a graphing calculator. This trigonometry video tutorial focuses on graphing trigonometric functions. It explains how to identify the amplitude, period, phase shift, vertical shift, and midline of a sine or cosine function.

### Sine, Cosine, Tangent Unit Circle Interactive Graphs ...

Explore the sine, cosine, tangent graphs with this interactive applet as well as downloadable GIF. Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law. Learn how to graph trigonometric functions and how to interpret those graphs. Learn how to construct trigonometric functions from their graphs or other features.

### Trigonometry applets – GeoGebra

Trigonometry applets Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! Right-Angled Triangle. The triangle of most interest is the right-angled triangle.The right angle is shown by the little box in the corner: We love to graph functions, and now that we know about the trigonometric functions, let's learn to graph those too! These are periodic functions, meaning the...

### Sine Function HTML5 applet - analyzemath.com

Explore the graph of general sine functions interactively using an HTML 5 applet. Trigonometry – GeoGebra ... Trigonometry In this live Grade 11 Maths show we take a close look at Trigonometric Functions. In this lesson we revise the terms used in Trig Functions, we investigate the effect of the parameters a, b, p, q ...

Trigonometry applets Pop-up blocker preferences on iphone. Explore the sine, cosine, tangent graphs with this interactive applet as well as downloadable GIF. Otakon mgs codec for apple. Trigonometry Applets Explore the graph of general sine functions interactively using an HTML 5 applet. This applet lets the user experiment with the role a, b, kand chave in the shape of the graph of a cosine function (y = a+ b cos( k(x - c)) ) by changing the value of a, bor c. As those values are changed, the user will see how the graph is changing. Trigonometric Functions Applets. Trigonometric Functions (Graphing & Modeling) Prerequisite Topics The graph of the function will be updated automatically. Selecting the "show base function" option causes the basic function (with a = b = 1 and c = d = 0) to be graphed in red. The function with other values for the parameters appears in green. Interactive Unit Circle. Sine, Cosine and Tangent. in a Circle or on a Graph.. Sine, Cosine and Tangent. Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:. For a given angle θ each ratio stays the same no matter how big or small the triangle is This function has an amplitude of 1 because the graph goes one unit up and one unit down from the midline of the graph. This function has a period of 2π because the sine wave repeats every 2π units. Gonalons caratteristiche tecniche samsung. Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! Trig Tour 1.0.22 - PhET Interactive Simulations Lejupladet muziku no youtube on apple.

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 2021-12-01, 16:37 #1 RomanM   Jun 2021 638 Posts Approximation of r by m^(1/n) $r\pm\epsilon=\sqrt[n]{m}$ where r - real root of polynomial P(r), order >=6, and m, n - integers, and P(r+/-eps)<1 P.S. I'm suspect that the simpler the question look like, the less likely it is to get an answer Last fiddled with by RomanM on 2021-12-01 at 16:42 Reason: ***


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# What is the value of $\rm \sqrt{1 - \cos^2 θ}$

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What is the value of $\rm \sqrt{1 - \cos^2 θ}$
1. sec θ
2. sin θ
3. cos θ
4. tan θ

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Correct Answer - Option 2 : sin θ

Concept:

sin2θ + cos2θ = 1

Calculation:

sin2θ + cos2θ = 1

⇒ sin2θ = 1 – cos2θ

⇒ sin θ  = $\rm \sqrt{1 - \cos^2 θ}$