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## simplify complex numbers square root

In ⓑ, we added under the radical sign first and then found the square root. ... Finding the square root of z is logically the same as solving the equation (x+I*y)**2 = z. When working with complex numbers it is important to rewrite the expression in terms of "i" (which is defined as ) before doing anything else: Now we can simplify the remaining square roots and then multiply or we can do it the other way around. The maximum number of decimal places can be chosen between 0 and 10. The square roots of other numbers are not whole numbers. The complex number calculator is also called an imaginary number calculator. Square root is an inverse operation of the squaring a number.. Complex numbers are the numbers which are expressed in the form of a+ib where ‘i’ is an imaginary number called iota and has the value of (√-1).For example, 2+3i is a complex number, where 2 is a real number and 3i is an imaginary number. Then, i 2 = -1. (Again, i is a square root, so this isn’t really a new idea. I do believe that you are ready to get acquainted with imaginary and complex numbers. Then, rewrite the square root as a multiplication problem under the square root sign. The complex symbol notes i. Calculate the positive principal root and negative root of positive real numbers. Therefore, the combination of both the real number and imaginary number is a complex number.. Positive and negative are not atttributes of complex numbers as far as I know. To plot a complex number, we use two number lines, crossed to form the complex plane. Turn on complex numbers if you want to be able to evaluate the square root of a complex number. Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers. Complex Numbers and Operations : Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and i is the square root of -1. Login. Join Now. Tutorial Imaginary Unit where This is the definition of an imaginary number. imaginary part 0), "on the imaginary axis" (i.e. )When the numbers are complex, they are called complex conjugates.Because conjugates have terms that are the same except for the operation between them (one is addition and one is subtraction), the i terms in the product will add to 0. Courses. Why? Just as and are conjugates, 6 + 8i and 6 – 8i are conjugates. So far we have only worked with square roots of perfect squares. Example 3 – Simplify the number √-3.54 using the imaginary unit i. Negative 4, if I take a square root, I'm going to get an imaginary number. Geometric representation of the 2nd to 6th roots of a complex number z, in polar form re i φ where r = |z | and φ = arg z. Square roots of negative numbers can be simplified using and And you would be right. Up to now, you’ve known it was impossible to take a square root of a negative number. Therefore, both 13 and −13 are square roots of 169. This website uses cookies to ensure you get the best experience. LEARNING APP; ANSWR; CODR; XPLOR; SCHOOL OS; STAR; answr. Can we simplify $$\sqrt{−25}$$? For example, if you're trying to find the square root of 98, the smallest prime number possible is 2. The first value is the complex number, where, = +/-. So, in this note I will explain you on how to find square root of a given complex number easily without using the above formulae. Any positive number squared is positive, and any negative number squared is also positive. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.. Let’s look at a numerical example. Notice (−13) 2 = 169 also, so −13 is also a square root of 169. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. Simplifying Square Roots – Techniques and Examples. Let z1 = x1 + iy1 be the given complex number and we have to obtain its square root. If you divide 98 by 2, you get 49. SymPy and square roots of complex numbers. Simplifying Imaginary Numbers . Let x + iy = (x1 + iy1)½ Squaring , => x2 – y2 + 2ixy = x1 + iy1 => x1 = x2 – y2 and y1 = 2 xy => x2 – y12 /4x2 … Continue reading "Square Root of a Complex Number & Solving Complex Equations" Asked on December 26, 2019 by Kavitha Rajora. real part 0). They include all real and imaginary numbers, as well as the sum of real and imaginary numbers. Square roots of numbers that are not perfect squares are irrational numbers. The square root of any negative number can be written as a multiple of i. The calculator allows you to manipulate complex numbers in their algebraic form , it can simplify an expression composed of complex numbers as does the site's complex number calculator . If z is real, φ = 0 or π. Complex numbers are made from both real an imaginary numbers. The answer you come up with is a valid "zero" or "root" or "solution" for "ax 2 + bx + c = 0", because, if you plug it back into the quadratic, you'll get zero after you simplify. maths. We might conclude that the square roots of numbers between 4 4 and 9 9 will be between 2 2 and 3, 3, and Free simplify calculator - simplify algebraic expressions step-by-step . Principal square root of a complex number. The square root of the complex number has two values. Find the square root, or the two roots, including the principal root, of positive and negative real numbers. Imaginary numbers allow us to take the square root of negative numbers. Imaginary numbers are represented as ki, where i = . Enter the complex number whose square root is to be calculated. I will explain it through different examples. The square root of any negative number can be written as a multiple of $$i$$. $(\;)^{2} = -25?$ None of the numbers that we have dealt with so far have a square that is −25. Square Root of a Negative Number. In exact mode the square root of an integer is not evaluated if it would result in an approximate number. Answer. The calculator uses different methods to simplify mathematical expressions: it uses function parity to simplify certain results. Define "a positive square root of a complex number". Instead, the square root of a negative number is an imaginary number--a number of the form , where k < 0. Click the Simplify button. Remember that when a number is multiplied by itself, we write and read it “n squared.” For example, reads as “15 squared,” and 225 is called the square of 15, since . They have attributes like "on the real axis" (i.e. Here ends simplicity. To plot a complex number, we use two number lines, crossed to form the complex plane. Click hereto get an answer to your question  Find the square root of complex number - 8 - 6i. So we're essentially going to get two complex numbers when we take the positive and negative version of this root. Complex number $$Re\;$$ $$Im\;$$ Square root Decimal places Calculate the square root of a complex number. When the Formula gives you a negative inside the square root, you can now simplify that zero by using complex numbers. But here you will learn about a new kind of number that lets you work with square roots of negative numbers! For example, = 5i and = i. The second value is the complex number , where, = -/+. Square root calculator and perfect square calculator. But we can find a fraction equivalent to by multiplying the numerator and denominator by .. Now if we need an approximate value, we divide . In floating point mode the square root of any number is evaluated. From … The horizontal axis is the real axis, and the vertical axis is the imaginary axis. 12 $\begingroup$ Oh, and I am afraid I must object to the assertion that going with De Moivre's formula would be the easiest way. Is there a number whose square is −25? Imaginary numbers result from taking the square root of a negative number. So, every positive number has two square roots—one positive and one negative. Solution: For this one, we will skip some of the intermediate steps and go straight to simplifying the number by replacing the negative sign under the square root with the imaginary unit i in front of the square root sign. $\begingroup$ @user1374 There is also a consensus about which square root of a complex number is the principal square root--at least for almost every complex number ... See my answer. So when the negative signs can be neutralized before taking the square root, it becomes wrong to simplify to an imaginary number. Square root of complex number (a+bi) is z, if z 2 = (a+bi). Complex numbers have the form a + bi, where a and b are real numbers and i is the square root of −1. To represent a complex number, we use the algebraic notation, z = a + ib with i ^ 2 = -1 The complex number online calculator, allows to perform many operations on complex numbers. Also tells you if the entered number is a perfect square. To simplify a square root, start by dividing the square root by the smallest prime number possible. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. Principal roots are shown in black. The square root of a number x is denoted with a radical sign √x or x 1/2.A square root of a number x is such that, a number y is the square of x, simplify written as y 2 = x.. For instance, the square root of 25 is represented as: √25 = 5. Learn more Accept. Imaginary numbers have the form bi and can also be written as complex numbers by setting a = 0. In the following description, $$z$$ stands for the complex number… The nature of problems solved these days has increased the chances of encountering complex numbers in solutions. This is true, using only the real numbers. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. The imaginary unit i, is the principal square root of -1. $\endgroup$ – Did Jun 10 '11 at 5:55. What if we only wanted the positive square root of a positive number? I will take you through adding, subtracting, multiplying and dividing complex numbers as well as finding the principle square root of negative numbers. All real numbers can be written as complex numbers by setting b = 0. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula. Search. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Ask Question Asked 3 years, 11 months ago. $\endgroup$ – Did Jun 10 '11 at 6:35. Estimate Square Roots. Find the square root of complex number − 8 − 6 i. But there is a very easy trick to find the square root of a complex number. Simplify functions thanks to their properties. Donate Login Sign up. The only two roots of this quadratic equation right here are going to turn out to be complex, because when we evaluate this, we're going to get an imaginary number. We have not been able to take the square root of a negative number because the square root of a negative number is not a real number. Simplify complex numbers. Simplify Expressions with Square Roots. Is not evaluated if it would simplify complex numbers square root in an approximate number tells you the... Tutorial imaginary unit i, is the simplify complex numbers square root number ( a+bi ) is z, you!.Kasandbox.Org are unblocked mode the square roots of perfect squares the possible values the. Answer to your Question  find the square root of 98, the easiest way is probably to with... 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# Engaging students: Compound interest

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Lydia Rios. Her topic, from Precalculus: compound interest.

How did people’s conception of this topic change over time?

While this concept is tied with business which is something that started rapidly changing in the early nineteen hundreds, we have understand that there is an accrued interest on loans long before then. People would loan out seeds or cattle and the interest would be paid after a harvest or with the young of the cattle. Of course now we use this concept mathematically but the concept still holds. We understand that there is a base fee and you must return that fee along with a little more. We then started using this with loose change and then as our currency changed from the gold standard we adapted to a new understanding of compound interest. Today we use the equation $A = P \left(1 + \frac{r}{n} \right)^{nt}$ , where $A$ is the amount accumulated, $P$ is the principal, $r$ is interest rate, $n$ is the compound period and $t$ is the number of periods.

Compound Interest Is Responsible for Modern Civilization (businessinsider.com)

What are the contributions of various cultures to this topic?

We have all experienced trade over the years. Native Americans would trade corn for other goods and offered payment plus interest with their corn harvest. The Silks Roads was a network of trading routes where China and other countries would trade textiles and other materials. They established the concept of payment and interest for purchases. Banks in America and other countries also have a set principal and a interest, whether this be in reference to your savings account or the billed interest on your credit card purchases. Even the invention of cars played a part on this and how our interest can decrease with the deterioration of the car. Over the years your interest payment can go down as the worth of the car goes down.

How have different cultures throughout time used this topic in their society?

# Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Morgan Mayfield. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent. How could you as a teacher create an activity or project that involves your topic? I want to provide some variety for opportunities to make this an engaging opportunity for Precalculus students and some Calculus students. Here are my three thoughts: IDEA 1: For precalculus students in a regular or advanced class, have them derive this formula in groups. After students are familiar with the Pythagorean identities and with angle sum identities, group students and ask them to derive a formula for double angles Sin(2θ), Cos(2θ), Tan(2θ). Let them struggle a bit, and if needed give them some hints such as useful formulas and ways to represent multiplication so that it looks like other operations. From here, encourage students to simplify when they can and challenge students to find the other formulas of Cos(2θ). Ask students to speculate instances when each formula for Cos(2θ) would be advantageous. This gives students confidence in their own abilities and show how math is interconnected and not just a bunch of trivial formulas. Lastly, to challenge students, have them come up with an alternative way to prove Tan(2θ), notably Sin(2θ)/Cos(2θ). This would make an appropriate activity for students while having them continue practicing proving trigonometric identities. IDEA 2: This next idea should be implemented for an advanced Precal class, and only when there is some time to spare. Euler was an intelligent man and left us with the Euler’s Formula: $e^{ix}=\cos x + i \sin x$. Have Precalculus students suspend their questions about where it comes from and what it is used for. This is not something they would use in their class. Reassure them that for what they will do, all they need to understand is imaginary numbers, multiplying imaginary numbers, and laws of exponents. Have them plug in x = A + B and simplify the right-hand side of the equation so that we get: $\cos(A+B)+i\sin(A+B)= a + bi$ where $a$ and $b$ are two real numbers. The goal here is to get $\cos(A+B)+i\sin(A+B)= \cos \theta \cos \theta - \sin \theta \sin \theta + (\sin \theta \cos \theta + \cos \theta \sin \theta)i$. All the steps to get to this point is Algebra, nothing out of their grasp. Now, the next part is to really get their brains going about what meaning we can make of this. If they are struggling, have them think about the implications of two imaginary numbers being equal; the coefficient of the real parts and imaginary parts must be equal to each other. Lastly, ask them if these equations seem familiar, where are they from, and what are they called…the angle sum formulas. From here, this can lead into what if x=2A? Students will either brute force the formula again, and others will realize x = A + A and plug it in to the equation they just derived and simplify. This idea is a 2-in-1 steal for the angle sum formulas and double angle formulas. It’s biggest downside is this is for Sin(2θ) and Cos(2θ).   IDEA 3: Take IDEA 2, and put it in a Calculus 2 class. Everything that the precalculus class remains, but now have the paired students prove the Euler’s Formula using Taylor Series. Guide them through using the Taylor Series to figure out a Taylor Series representation of $e^x$, $sin x$,  and $cos x$. Then ask students to find an expanded Taylor Series of to 12 terms with ellipses, no need to evaluate each term, just the precise term. Give hints such as $i^2= -1$ and to consider $i^3=i^2 \cdot i = -i$ and other similar cases. Lastly, ask students to separate the extended series in a way that mimics $a + bi$ using ellipses to shows the series goes to infinity. What they should find is something like this: Look familiar? Well it is the addition of two Taylor Series that represent Sin(x) and Cos(x). This is the last connection students need to make. Give hints to look through their notes to see why the “a” and “b” in the imaginary number look so familiar. This, is just one way to prove Euler’s Formula, then you can continue with IDEA 2 until your students prove the angle sum formulas and double angle formulas. How does this topic extend what your students should have learned in previous courses? Students in Texas will typically be exposed to the Pythagorean Theorem in 8th grade. At this stage, students use $a^2+b^2=c^2$ to find a missing side length. Students may also be exposed to Pythagorean triples at this stage. Then at the Geometry level or in a Trigonometry section, students will be exposed to the Pythagorean Identity. The Identity is $\sin^2 \theta + \cos^2 \theta = 1$.  I think that this is not fair for students to just learn this identity without connecting it to the Pythagorean Theorem. I think it would be a nice challenge student to solve for this identity by using a right triangle with hypotenuse c so that Sin (θ) = b/c and cos (θ) = a/c, one could then show either $c^2 \sin^2 \theta + c^2 \cos^2 \theta = c^2$ and thus $c^2(\sin^2 \theta + \cos^2 \theta) = c^2$ or one could show $(a/c)^2 + (b/c)^2 = (c/c)^2 = 1$ (using the Pythagorean theorem). From here, students learn about the angle addition and subtraction formulas in Precalculus. This is all that they need to derive the double angle formulas.

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

$\tan(\alpha + \beta) = \displaystyle \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

$\tan(\alpha - \beta) = \displaystyle \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

This would be a good challenge exercise for students to do in pairs. Sin(2θ) = Sin(θ + θ), Cos(2 θ) = Cos(θ + θ), Tan(2θ) = Tan(θ + θ). Now we can apply the angle sum formula where both angles are equal: Sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ) Cos(2θ) = cos(θ)cos(θ) – sin(θ)sin(θ) =  (We use a Pythagorean Identity here) Tan(2θ) = $\displaystyle \frac{\tan \theta + \tan \theta}{1 - \tan^2 \theta} = \frac{2 \tan \theta}{1-\tan^2 \theta}$ Bonus challenge, use Sin(2θ) and Cos(2θ) to get Tan(2θ). Well, if $\tan \theta = \displaystyle \frac{\sin \theta}{\cos \theta}$, then

$\tan 2\theta = \displaystyle \frac{\sin 2\theta}{\cos 2\theta}$

$= \displaystyle \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}$

$= \displaystyle \frac{ \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} }{ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} }$

$= \displaystyle \frac{2 \tan \theta}{1 - \tan^2 \theta}$

The derivations are straight forward, and I believe that many students get off the hook by not being exposed to deriving many trigonometric identities and taking them as facts. This is in the grasp of an average 10th to 12th grader. What are the contributions of various cultures to this topic? I have included four links that talk about the history of Trigonometry. It seemed that ancient societies would need to know about the Pythagorean Identities and the angles sum formulas to know the double angle formulas. Here is our problem, it’s hard to know who “did it first?” and when “did they know it?”. Mathematical proofs and history were not kept as neatly written record but as oral traditions, entertainment, hobbies, and professions. The truth is that from my reading, many cultures understood the double angle formula to some extent independently of each other, even if there was no formal proof or record of it. Looking back at my answer to B2, it seems that the double angle formula is almost like a corollary to knowing the angle sum formulas, and thus to understand one could imply knowledge of the other. Perhaps, it was just not deemed important to put the double angle formula into a category of its own. Many of the people who figured out these identities were doing it because they were astronomers, navigators, or carpenters (construction). Triangles and circles are very important to these professions. Knowledge of the angle sum formula was known in Ancient China, Ancient India, Egypt, Greece (originally in the form of broken chords theorem by Archimedes), and the wider “Medieval Islamic World”. Do note that that Egypt, Greece, and the Medieval Islamic World were heavily intertwined as being on the east side of the Mediterranean and being important centers of knowledge (i.e. Library of Alexandria.) Here is the thing, their knowledge was not always demonstrated in the same way as we know it today. Some cultures did have functions similar to the modern trigonometric functions today, and an Indian mathematician, Mādhava of Sangamagrāma, figured out the Taylor Series approximations of those functions in the 1400’s. Greece and China for example relayed heavily on displaying knowledge of trigonometry in ideas of the length of lines (rods) as manifestations of variables and numbers. Ancient peoples didn’t have calculators, and they may have defined trigonometric functions in a way that would be correct such as the “law of sines” or a “Taylor series”, but still relied on physical “sine tables” to find a numerical representation of sine to n numbers after the decimal point. How we think of Geometry and Trigonometry today may have come from Descartes’ invention of the Cartesian plane as a convenient way to bridge Algebra and Geometry. References: https://www.mathpages.com/home/kmath205/kmath205.htm https://en.wikipedia.org/wiki/History_of_trigonometry https://www.ima.umn.edu/press-room/mumford-and-pythagoras-theorem

# Engaging students: Computing the cross product of two vectors

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Chi Lin. Her topic, from Precalculus: computing the cross product of two vectors.

How could you as a teacher create an activity or project that involves your topic?

I found one of the real-life examples of the cross product of two vectors on a website called Quora. One person shares an example that when a door is opened or closed, the angular momentum it has is equal to $r \times p$, where $p$ is the linear momentum of the free end of the door being opened or closed, and $r$ is the perpendicular distance from the hinges on which the door rotates and the free end of the door. This example gives me an idea to create an example about designing a room. I try to find an example that closes to my idea and I do find an example. Here is the project that I will design for my students. “If everyone here is a designer and belongs to the same team. The team has a project which is to design a house for a client. Your manager, Mr. Johnson provides a detail of the master room to you and he wants you to calculate the area of the master room to him by the end of the day. He will provide every detail of the master room in three-dimension design paper and send it to you in your email. In the email, he provides that the room ABCD with $\vec{AB} = \langle -2,2,5 \rangle$ and $\vec{AD} = \langle 5,6,3 \rangle$. Find the area of the room  (I will also draw the room (parallelogram ABCD) in three dimensions and show students).”

Reference:

https://www.quora.com/What-are-some-daily-life-examples-of-dot-and-cross-vector-products

https://www.nagwa.com/en/videos/903162413640/

How does this topic extend what your students should have learned in previous courses?

This topic is talking about computing cross product of two vectors in three dimensions. First, students should have learned what a vector is. Second, students should know how to represent vectors and points in space and how to distinguish vectors and points. Notice that when students try to write the vector in space, they need to use the arrow. Next, since we are talking about how to distinguish the vectors and the points, here students should learn the notations of vectors and what each notation means. For example, $\vec{v} = 1{\bf i} + 2 {\bf j} + 3 {\bf k}$. Notice that $1{\bf i} + 2 {\bf j} + 3 {\bf k}$ represents the vectors in three dimensions. After understanding the definition of the vectors, students are going to learn how to do the operation of vectors. They start with doing the addition and scalar multiplication, and magnitude. One more thing that students should learn before learning the cross product which is the dot product. However, students should understand and master how to do the vector operation before they learn the dot product since the dot product is not easy. Students should have learned these concepts and do practices to make sure they are familiar with the vector before they learn the cross products.

References:

How did people’s conception of this topic change over time?

Most people have the misconception that the cross product of two vectors is another vector. Also, the majority of calculus textbooks have the same misconception that the cross product of two vectors is just simply another vector. However, as time goes on, mathematicians and scientists can explain by starting from the perspective of dyadic instead of the traditional short‐sighted definition. Also, we can represent the multiplication of vectors by showing it in a geometrical picture to prove that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. Also, by using the way that the limitation of such an entity to exactly a three‐dimensional space does not allow for one of the three metric motions (reflection in a mirror). We can understand that the intrinsic difference between true vectors and pseudo‐vectors.

Reference:

https://www.tandfonline.com/doi/abs/10.1080/0020739970280407

# Engaging students: Exponential Growth and Decay

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Ashlyn Farley. Her topic, from Precalculus: exponential growth and decay.

The most current example of exponential growth and decay is with the global pandemic, Covid 19. One example is that The Washington Post wrote an article stating that “The spread of coronavirus boils down to a simple math lesson.” The article goes on to explain what exponential growth is and how that applies to Covid 19.  Another website, ourworldindata.org, has a graph of the daily new cases of Covid 19. This graph allows one to see the information for multiple countries, and starts on January 28th 2020 until Today, whatever day that you may be viewing it. Many other news sources also have graphs and information on the growth, and decay in some cases, of the pandemic situation. Teachers can use this information to easily make a connection from math class to the real world.

One idea of teaching graphing exponential functions so that it is engaging is to use a project over the zombie apocalypse. The spread of a disease is a common and great example of exponential functions, so although this disease is pretend, the idea can be applied in the real world, like with a global pandemic. Three examples of projects are:

• News Reporters
• This project has the students analyzing data they received to best report to the people who are dealing with the outbreak. It allows students to learn how to read the graphs of exponential functions, understand the functions, integrate technology into the class by creating news reports, and practice an actual career.
• Government Officials
• This project has the students running a simulation of their city. They are to use the statistics of a city to see what the impact of a zombie outbreak would be. After finding the best and worst case scenario, they are to write a letter to the mayor of the city that explains the scenarios so that government can implement plans to keep the outbreak to a minimum. This allows the students a chance to practice analyzing exponential functions, modifying exponential functions, and informing others of the meaning of the functions and modifications.
• Scientists
• This project has the students predicting the outcome of a zombie outbreak, finding a cure, and determining at what point is the zombie population controlled. The students will get practice with the exponential functions, making changes to the functions, finding the point of “control”, as well as creating an action plan.

Each of these projects can be used separately or can be combined to create one major project to learn about exponential functions and their graphs. The goal is to get students excited about learning math instead of dreading it. Math is used daily, even if the students don’t realize it, so the understanding of real-life implications is very important for a teacher to bring into the classroom.

Of the many websites, one key website for educators trying to make lessons engaging is YouTube. YouTube has songs, such as the Exponential Function Music Video, explanatory videos, such as from Kahn Academy, and allows students to create their own videos about the topics. Explanatory videos may help students get a specific idea they didn’t quite understand in class, music is very catchy allowing quick memorization of information, and creating videos shows that the students truly have an understanding of the material. By giving the students multiple types of representation of the material, allows all types of learners a chance to understand the material. Multiple representations is very important in keeping students engaged in the class and having them truly learn the material.

Resources:

https://www.teacherspayteachers.com/Product/Zombie-Apocalypse-Exponential-Function-Pandemics-21st-Century-Math-Project-767712?epik=dj0yJnU9UnRuNHVLLUxrV0JkTVJQc1ZFY0szb3JJNXRyenQwb2omcD0wJm49aEQ2UjFHVUcyYm5FakE1ZXhSXzhpQSZ0PUFBQUFBR0ZnTWRB

https://medium.com/innovative-instruction/math-mini-project-idea-the-zombie-apocalypse-5ddd0e6af389#.sph1x08k8

https://www.teacherspayteachers.com/Product/Exponential-Growth-and-Decay-Activity-Exponential-Functions-Zombie-Apocalypse-2609226

https://ourworldindata.org/coronavirus/country/united-states

https://www.washingtonpost.com/weather/2020/03/27/what-does-exponential-growth-mean-context-covid-19/

# Engaging students: Finding the domain and range of a function

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Sydney Araujo. Her topic, from Precalculus: finding the domain and range of a function.

How can this topic be used in your students’ future courses in mathematics or science?

Expanding on finding the domain, this topic is frequently seen in calculus classes. Students need to understand the domain to understand and find limits of functions. Continuity directly expands on domain & range and how it works. We also see domain and range when students are exploring projectile motion. This makes since because when we think about projectile motion, we think about parabolas. With projectile motion there is a definite start, end, and peak height of the projectile. So we can use the domain to show how far the projectile travels and the range to show how high it travels. Looking even further ahead when students start to explore different functions and sets, they start to learn about a codomain and comparing it to the range which is a very valuable concept when you start to learn about injective, surjective, and bijective functions.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Desmos is a great website for students to use when exploring domain and ranges. Desmos has premade inquiry-based lessons for students to explore different topics. Teachers also have the option of creating their own lessons and visuals for their students to interact with. Desmos can also animate functions by showing how they change with a sliding bar or actually animate and show it move. This would be a great tool to use for students to visually understand domain and ranges as well as how they are affected when asymptotes and holes appear. This would also be great for ELLs because instead of focusing on just math vocabulary, they can actually visually see how it connects to the graph and the equation. For example, https://www.desmos.com/calculator/vz4fjtugk9, this ready-made desmos activity actually shows how restricting the domain and range effects the graph and what parts of the graph are actually included with the given domain and range.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Like I discussed earlier, domain and range is directly used in calculus. In the movie Stand and Deliver, they directly discuss the domain and range of functions. The movie Stand and Deliver is about a Los Angeles high school teacher, Jaime Escalante, who takes on a troublesome group of students in a not great neighborhood and teaches them math. He gets to the point where he wants to teach them calculus so they can take the advanced placement test. If they pass the advanced placement test then they get college credit which would motivate them to actually go to college and make a better life for themselves. However through great teaching and intensive studying, the students as a whole ace the exam but because of their backgrounds they are accused of cheating and must retake the exam. There is a few scenes, but one in particular where the students are finally understanding key concepts in calculus and Mr. Escalante is having them all say the domain of the function repeatedly.

# Parabolic Properties from Pieces of String

I am pleased to announce that my latest paper, “Parabolic Properties from Pieces of String,” has now been published in Math Horizons. This was a really fun project for me. As I describe in the paper, I started wondering if it was possible to convince a student who hadn’t learned calculus yet that string art from two line segments traces a parabola. Not only was I able to come up with a way of demonstrating this without calculus, but I was also able to (1) prove that a quadratic polynomial satisfies the focus-directrix property of a parabola, which is the reverse of the usual logic when students learn conic sections, and (2) prove the reflective property of parabolas. I was really pleased with the final result, and am very happy that this was accepted for publication.

Due to copyright restrictions, I’m not permitted to freely distribute the final, published version of my article. However, I am able to share the following version of the article.

The above PDF file is an Accepted Manuscript of an article published by Taylor & Francis in College Mathematics Journal on February 24, 2022, available online: Full article: Parabolic Properties from Pieces of String (tandfonline.com)

# Engaging students: The quadratic formula

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Sydney Araujo. Her topic, from Algebra: the quadratic formula.

D4. What are the contributions of various cultures to this topic?

The quadratic formula can be traced all the way back to the Ancient Egyptians. The ancient Egyptians knew how to calculate the area of different shapes but did not know how to calculate the length of the sides of a shape. Moving forward, it is speculated that the Babylonians developed the completing the square method to solve problems involving areas. The Babylonians used a more similar number system to the one we use today. Instead, they used hexagesimal which made addition and multiplication easier. We can also see a similar method used by the Chinese around the same time. Pythagoras and Euclid were some of the first to attempt to find a more general formula to solve quadratic equations, both using a geometric approach. They’re ideas differ slightly, Pythagoras observed that the value of a square root is not always an integer but he refused to allow for proportions that were not rational. Whereas Euclid proposed that irrational square roots are also possible. At the time, the ancient Greeks did not use the same number system that we use, so it was impossible to calculate square roots by hand. It wasn’t until the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic formula. This is because Indian mathematics used the decimal system as well as zero which had a massive advantage over the Egyptians and Greeks. Brahmagupta was the one that recognized that there are two roots in the solution to the quadratic equation and described the quadratic formula.

A2. How could you as a teacher create an activity or project that involves your topic?

One of my fondest high school memories is from my junior year physics class. It was the famous Punkin’ Chunkin’ project. Students were put in groups and asked to build a trebuchet or catapult that could launch a pumpkin across a field. The only requirement was for the device to work, the distance was just fun extra credit. For this project we had to predict the pumpkins trajectory using different variables like the pumpkin’s weight, force, momentum, etc. However, by the time we were juniors, we had either taken Algebra 2 or were currently in it. So, our physics and algebra teacher were working together so that by the time this project came around we were working on quadratic equations in algebra. As the shape of the trajectory of a pumpkin was a parabola. Because of this experience, I can create an activity or even a similar project with the physics teacher. This way students see the different applications of quadratic equations and have a tangible real world math experience.

B2. How does this topic extend what your students should have learned in previous courses?

As the quadratic formula is taught in Algebra 1, students have only seen linear equations prior to that point. Students recognize that when they are solving these equations, they are looking for one solution, no solution, or infinitely many solutions. The one solution being a singular ordered pair and then they are done. What students then must extend on when they reach quadratic equations, and the quadratic formula is that they’re now looking for two separate solutions. So, at this point they know how to solve for x and understand inverses which is important when it comes to quadratic equations. During the solving process of a quadratic equation, students may have to take the square root of both sides of the equation which will give you a plus or minus sign in front of the square root. Which makes the connection on why there are two solutions to a quadratic equation and the quadratic formula, because a parabola has two roots.

Works Cited:

Brahambhatt, Rupendra. “Quadratic Formula: What, Why, and How It Changed Mathematics.” Interesting Engineering, Interesting Engineering, 16 July 2021, interestingengineering.com/quadratic-formula-what-why-and-how-it-changed-mathematics.

# A New Derivation of Snell’s Law without Calculus

Last week, I posted that my latest paper, “A New Derivation of Snell’s Law without Calculus,” has now been published in College Mathematics Journal. In that previous post, I didn’t provide the complete exposition because of my understanding of copyright restrictions at that time.

I’ve since received requests for copies of my paper, which prompted me to carefully read the publisher’s copyright restrictions. In a nutshell, I was wrong: I am allowed to widely distribute preprints that did not go through peer review and, with extra restrictions, the accepted manuscript after peer review.

So, anyway, here it is.

The above PDF file is an Accepted Manuscript of an article published by Taylor & Francis in College Mathematics Journal on January 28, 2022, available online: Full article: A New Derivation of Snell’s Law Without Calculus (tandfonline.com).

# A New Derivation of Snell’s Law without Calculus

I’m pleased to say that my latest paper, “A New Derivation of Snell’s Law without Calculus,” has now been published in College Mathematics Journal. The article is now available for online access to anyone who has access to the journal — usually, that means members of the Mathematical Association of America or anyone whose employer (say, a university) has institutional access. I expect that it will be in the printed edition of the journal later this year; however, I’ve not been told yet the issue in which it will appear.

Because of copyright issues, I can’t reproduce my new derivation of Snell’s Law here on the blog, so let me instead summarize the main idea. Snell’s Law (see Wikipedia) dictates the angle at which light is refracted when it passes from one medium (say, air) into another (say, water). If the velocity of light through air is $v_1$ while its velocity in water is $v_2$, then Snell’s Law says that

$\displaystyle \frac{\sin \theta_1}{v_1} = \displaystyle \frac{\sin \theta_2}{v_2}$

I was asked by a bright student who was learning physics if there was a way to prove Snell’s Law without using calculus. At the time, I was blissfully unaware of Huygens’s Principle (see OpenStax) and I didn’t have a good answer. I had only seen derivations of Snell’s Law using the first-derivative test, which is a standard optimization problem found in most calculus books (again, see Wikipedia) based on Fermat’s Principle that light travels along a path that minimizes time.

Anyway, after a couple of days, I found an elementary proof that does not require proof. I should warn that the word “elementary” can be a loaded word when used by mathematicians. The proof uses only concepts found in Precalculus, especially rotating a certain hyperbola and careful examining the domain of two functions. So while the proof does not use calculus, I can’t say that the proof is particularly easy — especially compared to the classical proof using Huygens’s Principle.

That said, I’m pretty sure that my proof is original, and I’m pretty proud of it.

format: plain


text:
## Analytic rank of an elliptic curve with algebraic rank 0

Let $E$ be an elliptic curve over $\mathbb{Q}$ with algebraic rank 0. Is there any way, one can argue that the analytic rank must be even? Of course this would follow from the standard conjectures, such as BSD, Sha is finite, and the parity conjectures, but are there any unconditional results?

-
No . – wccanard Mar 15 at 23:16
as long as it's known to be at most two! – Will Sawin Mar 16 at 3:22
mathoverflow.net/questions/123813/… – Srilakshmi Mar 16 at 6:23
In general no. If you know that the p-Selmer rank is 0 for some p, then yes. – Tim Dokchitser Mar 16 at 14:55
@Will Sawin and Tim Dokchitser: "as long as you have a condition, there is an unconditional result" ;-) Tim: are you not tempted to give a reference? ;-) – wccanard Mar 16 at 17:51
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Symmetries for a class of explicitly space- and time-dependent (1+1)-dimensional wave equations
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
1997 (English)In: Proceedings of the second international conference: Memorial Prof. W. Fushchych conference, July 7 - 13, 1997, Kyiv, Ukraine / [ed] Mykola Shkil, Kyev: Institute of Mathematics of the National Academy of Sciences of Ukraine , 1997, Vol. 1, 70-78 p.Conference paper, Published paper (Refereed)
##### Abstract [en]

In this paper the nonlinear wave equation $\partial^2 u/\partial x_0^2-\partial^2 u/\partial x_1^2+f(x_0,x_1,u)=0$, where $f$ is an arbitrary smooth function of its arguments, is considered from the symmetry standpoint. The form of the most general Lie point symmetry generator of this equation is obtained. The classes of functions $f$, for which the equation in question admits a one-parameter Lie point symmetry group, are constructed. Then, the authors investigate the possible form of generators of conformal transformations, assuming the usual form of generators of Lorentz and scaling transformations, and study the wave equations invariant under such operators. The symmetry groups of obtained equations are used for the construction of ansätze and reductions of these equations to ordinary differential equations. $Q$-conditional (nonclassical) symmetries of the wave equation are also considered. Namely, the determining equations for the coefficients of a $Q$-conditional symmetry operator are found and their compatibility is investigated.

##### Place, publisher, year, edition, pages
Kyev: Institute of Mathematics of the National Academy of Sciences of Ukraine , 1997. Vol. 1, 70-78 p.
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
Local ID: c5b49920-9bfd-11db-8975-000ea68e967bISBN: 966-02-0343-8 (print)OAI: oai:DiVA.org:ltu-38089DiVA: diva2:1011588
##### Conference
International Conference Symmetry in Nonlinear Mathematical Physics : 07/07/1997 - 13/07/1997
##### Note
Godkänd; 1997; 20061221 (kani)Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2017-11-25Bibliographically approved

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v. 2.29.1
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# Refinement of nested cases / aligned equations

For the following two approaches or sets of equations, I would like to

1- (in the first set) make the first for align with the two fors below

2- (in the second set) keep the font and interspaces the same as the first set

\documentclass{article}
\usepackage{mathtools,nccmath}
\begin{document}
\begin{fleqn}
\begin{gather}
t_k = \left\{
\begin{alignedat}{2}
&{\frac{k}{N}}^\nu & \quad &\text{for } \nu > 1\\
&\left\{\begin{alignedat}{2}
&t_0 \left( \frac{k}{N_0} \right)^\nu & \quad &\text{for } 0 \leq k \leq N_0\\
%
&t_0 + \frac{k-N_0}{N-N_0} \left(t - t_0\right) & &\text{for} \enspace N_0 < k \leq N, t_0 < t
\end{alignedat}\right.
\end{alignedat}\right.\\[\baselineskip]
%%
% https://tex.stackexchange.com/a/84072/2288
t_k =
{\frac{k}{N}}^\nu & \text{for } \nu > 1 \\
\left\{\begin{array}{@{}l@{}}
t_0 \left( \frac{k}{N_0} \right)^\nu \\
t_0 + \frac{k-N_0}{N-N_0} \left(t - t_0\right)
\end{array}\right.\kern-\nulldelimiterspace
& \begin{array}{@{}l@{}}
\text{for } 0 \leq k \leq N_0 \\
\text{for } N_0 < k \leq N, t_0 < t
\end{array}
\end{array}\right.}
\end{gather}
\end{fleqn}
\end{document}

• I tried understanding the definition of t_k, but I can't. In the second set there is no n, for instance. You're overcomplicating things, in my opinion. – egreg Jul 28 at 13:02
• @egreg This question is more of a practice to understand the nesting than a real math problem. Therefore, I summed up all my thoughts in a single question instead of asking many questions on the site. – Diaa Jul 28 at 13:21
• I'm not sure that's sensible notation in any case. – egreg Jul 28 at 13:31
• You wrote in a comment, "I summed up all my thoughts in a single question". That's actually not a clever strategy; if the issues are not self-evidently related to each other, it's almost certainly better to post separate queries. – Mico Jul 28 at 13:41
• @Mico I understand what you both mean, so I happily edit the question to make it mathematically consistent :) – Diaa Jul 28 at 13:42

• I added a \left( ... \right) "wrapper" around the first fraction term, as otherwise it's not obvious whether the \nu exponent term pertains to the entire fraction or to just the numerator.

• The solution also inserts typographic struts in the second "inner" array to assure that the rows' heights are the same as in the first "inner" array.

• Finally, \addlinespace directives (provided by the booktabs package) are inserted to increase the vertical separation between rows.

That said, I don't believe that this layout is all that clear. For instance, shouldn't there be an indication somewhere that rows 2 and 3 pertain to the case \$\nu \le 1?

\documentclass{article}
\usepackage{nccmath}   % for 'fleqn' env.
\usepackage{array}    % for \newcolumntype macro
\newcolumntype{L}{>{\displaystyle}l}
% Define two (typographical) struts (to be used in math mode only):
\newcommand\strutA{\vphantom{%
\displaystyle\left( \frac{k}{N_0} \right)^{\nu}}}
\newcommand\strutB{\vphantom{%
\displaystyle\frac{k-N_0}{N-N_0}}}

\begin{document}
\begin{fleqn}
$$t_k = \left\{ \begin{array}{@{} L @{\quad} L @{}} \left(\frac{k}{N}\right)^{\!\nu} & \text{for \nu > 1} \\ \addlinespace \left\{ \begin{array}{@{} L @{}} t_0 \left(\frac{k}{N_0}\right)^{\!\nu} \\ \addlinespace t_0 + \frac{k-N_0}{N-N_0} (t - t_0) \end{array} \right.\kern-\nulldelimiterspace & \begin{array}{@{} L @{}} \text{for 0 \leq k \leq N_0}\strutA \\ \addlinespace \text{for N_0 < k \leq N, t_0 < t}\strutB \end{array} \end{array} \right.$$
\end{fleqn}
\end{document}

• Thanks for your answer. Actually, \nu >1 is a general condition for every single case. So, if you could propose a better place for ot, I would be grateful. – Diaa Jul 28 at 14:24
• @Diaa - If \nu>1 is always true, then only the first of the three rows can ever be true, right? In that case, why bother with showing rows 2 and 3? – Mico Jul 28 at 14:42
• @Diaa - Please indicate whether \nu>1 and t_0<t are both always true. – Mico Jul 28 at 15:49
• Only \nu > 1 is the real conditions here. t0 is a mark over the timeline. So, t can be less than t0. Sorry for late reply, I was away from PC – Diaa Jul 28 at 16:18
• @Diaa - Honestly, I am still quite unsure about what it is you're trying to express. Hence, I am in no position to suggest anything one might label "the right way". But I can say confidently that the three-row cases-like expression you've presented is not the right way to go. For sure, if the information in row 1 is always true, you should not place that piece of information in a cases-like structure along with two other rows. Readers have every right to expect that rows in cases-like structure are mutually exclusive and exhaustive. – Mico Jul 28 at 17:50


text:
L2P - Free Software Directory

# L2P

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

### L2P

http://redsymbol.net/software/l2p/
'L2P' creates PNG images from LaTeX math expressions. It can work with either a fragment of LaTeX code (such as $\frac{x^2+1}{3-x}$) or with a full LaTeX document. It is designed to be fast, robust, and to offer the user a high degree of flexibility and control. It includes complete documentation.

### Documentation

User manpage available in HTML format from http://redsymbol.net/software/l2p/dist/l2p-doc.html

released on 6 December 2004

## Licensing

PublicDomainJanet Casey6 December 2004

Contact(s)Role
Aaron Maxwell Maintainer

## Resources and communication

Audience Resource type URI
Bug Tracking,Developer,Support E-mail mailto:amax@redsymbol.net

## Software prerequisites

Kind Description
Required to use LaTeX
Required to use ImageMagick

This entry (in part or in whole) was last reviewed on 29 May 2010.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the page “GNU Free Documentation License”.


text:
# Weight

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and subject to a disclaimer.

Weight is a property of objects in a gravitational field. The weight of an object is the force exerted on it by the gravitational field, usually one caused by a single very large (planet-sized or larger) object.

Newton's Law of gravitation gives the force f exerted by gravity as:

${\displaystyle f=G\,{\frac {M_{\mathrm {1} }M_{\mathrm {2} }}{R_{\mathrm {E} }^{2}}}}$

Force is equal to mass multiplied by acceleration, and thus the weight of an object is equal to its mass multiplied by the acceleration it experiences due to the gravitational attraction of the nearby larger object. This acceleration is given as:

${\displaystyle a=G\,{\frac {M}{R^{2}}}}$ (where R is the distance from the center of mass of the larger object)

On the surface of Earth, the weight of an object is approximately equal to its mass multiplied by 9.8 m s−2, with minor local variations due to the irregularity of the earth's surface. On other planets, the weight of an object of a given mass will be different than its weight on earth, and thus moving the object will require different amounts of force. Most sources of force are not dependent on their weight, and so most actions will have different results in different gravitational fields. The classic example of this is to consider a person jumping on different planets. Due to the different gravitational fields of the different planets, the person's weight will be different, but the force produced by his muscles will be essentially the same, as the force is produced by a chemical reaction. Therefore, a person can jump significantly higher in a weaker gravitational field, such as occurs on earth's moon, but will be unable to jump as high in a stronger gravitational field, such as occurs on Jupiter.

## Units

In SI units, weight is measured in units of force, the newton and its derivatives. In U.S. customary units, weight is measured in pounds-force and other units which are fractions or multiples of the pound. One pound-force is the weight of 0.453 592 37 kg subject to a standard gravity of 9.80665 m s−2, or approximately 4.448 N.

Note that the pound, without the "-force" qualifier, is considered a unit of mass.

## Apparent Weight

Objects that are either in a downward vertical freefall or in an orbit around a planet such as Earth are often referred to as being "weightless," or as "experiencing weightlessness." However, this is a misnomer. As mentioned earlier, weight is the force exerted on an object by a gravitational field, and this force is certainly present in order for objects to either pick up speed as they fall vertically, or maintain an orbit rather than travel in a straight line away from the planet. However, for a person falling or orbiting, these situations feel no different than floating in space far away from any planet or other source of gravity.

To characterize these situations more accurately, we can instead say that a person or object's apparent weight is zero. In this context, apparent weight is the reading one would get from a scale located under the person or object. Apparent weight can be zero, equal to the true weight, less than (but nonzero) the true weight, or greater than the true weight. The following examples illustrate each of these possibilities.

• As mentioned already, a person or object in vertical freefall or in orbit around a planet has zero apparent weight. A scale under the person or object would have a reading of zero.
• A person or object standing at rest on the ground has an apparent weight equal to their true weight. A scale under the person would indicate the person's weight, and is in fact how weight is typically measured.
• A person on an elevator that has a downward acceleration has an apparent weight that is less than their true weight. Note that a downward acceleration is not the same as a downward motion, and occurs for either (1) an object moving upward while slowing down, or (2) an object moving downward while speeding up. In either case, the person would feel lighter than normal, and a scale under the person would have a reading that is less than their true weight.
• A person on an elevator that has an upward acceleration has an apparent weight that is more than their true weight. Note that an upward acceleration is not the same as an upward motion, and occurs for either (1) an object moving upward while speeding up, or (2) an object moving downward while slowing down. In either case, the person would feel heavier than normal, and a scale under the person would have a reading that is more than their true weight.


text:
### Deep Learning Based Recommendation Engines using Keras

One thing that I have been thinking a lot about since I wrote my chapter on matrix factorization methods, and since I am currently writing a chapter on graph theory is on the idea of a recommendation engine. (If you want to get the book free as I write it click here) They have become super relevant, and probably one of the most important pieces of technology in retail marketing.

Recommendation engines keep you glued to your streaming service, clicking on articles on your favorite news outlet, and buying from your favorite retailers. Recommendation engines have taken the place of sales associates in the online space. They are literally everywhere you go online. The problem is the level of sophistication varies greatly. I myself have a recommendation engine on this blog. It is not using a very fancy algorithm at all. In fact, all it does is prioritize content that has similar tagging that I manually add to the articles. Nothing really sophisticated here, mainly because I don’t want to spend valuable time maintaining it.

So as I was thinking about graph theory, my mind wandered over to the connections between matrix factorization, and the basic idea of recommendation engines. It turns out that you can think of recommendation engines as learning embeddings on a bipartite graph. I won’t go into the gory math details here. At any rate, I got to thinking about how deep learning algorithms have gotten very good at learning embeddings, and I realized that I could get my deep learning algorithms to generate recommendations for a given user.

So, I am not going to productionalize this model. I don’t think that I have enough data from my logs to train such a model effectively, and again I don’t want to maintain a model.

## The Data

Obviously, to train a deep learning model, we need some data. In general, we only need one type of data to do this an edge list in a bipartite network where we have two node types, a user type and a product type notice the italics here indicate that there is a lot of wiggle room in what we consider a user and a product. In general a user is what we are making a recommendation to and a product is what we are going to make a recommendation about.

We actually, only need those columns because we can be tricky about how we implement the last column, i.e. we can generate negative cases synthetically, if we only have positive cases in our data. But from a machine learning model, we need something like a rating for the user and product pair. Fortunately for us, the data that we will use has a rating.

We are going to be using the movielens small data set. You can download it from here. It has the following format:

userId movieId rating timestamp 1 1 4.0 964982703 1 3 4.0 964981247 1 6 4.0 964982224 1 47 5.0 964983815 1 50 5.0 964982931

There isn’t a whole bunch of feature engineering that we need to do here. I do want to change my ratings around just a little bit. I can do that with the following code:

df['rating'] = df['rating'].map({1:-1,2:-1,3:0,4:1,5:1,1.5:-1,2.5:-1,3.5:1,4.5:1})

Other than that, our data was pretty clean. So we’ll leave it at that.

## The Model

So at this point what you probably are most interested in is the model for handing out recommendations. The best thing to do in this case is to just take a play out of the collaborative filtering playbook and come up with some sort of matrix factorization method.

Probably, the simplest thing to do is to just create a latent space for users and a latent space for movies and then take the dot product of those two spaces. We can then spit out a rating for the user/movie pair. With that in place all we need to do is back propagate the error between the prediction and the true rating the user gave.

user_input = Input(shape=[1], name="User-Input")
user_embedding = Embedding(len(df.userId.unique())+1, 5, name="User-Embedding")(user_input)
user_vec = Flatten(name="Flatten-users")(user_embedding)

movie_input = Input(shape=[1], name="Movie-Input")
movie_embedding = Embedding(len(df.movieId.unique())+1, 5, name="Movie-Embedding")(movie_input)
movie_vec = Flatten(name="Flatten-movies")(movie_embedding)

prod = Dot(name="Dot-Product", axes=1)([user_vec, movie_vec])
model = Model([user_input, movie_input], prod)

That’s the entire code for our model. Let’s go through it in three chunks. The first chunk says take a user id as input, and then embed the user into a 5-dimensional space. Flatten it out so that we have a vector.

The next chunk is very similar in that we take a movie id as input and spit out a 5 dimensional vector for the movie.

The last chunk takes the dot product between these two vectors and produces a single number. We then define the model by saying that we want to take the inputs and output the dot product between their latent embedding space. The last line says how to do the back propagation.

That’s it. It really is just collaborative filtering. I don’t think that we have done any real deep learning. All of this could have been defined and done without using a deep learning library like Keras, but Keras sure made it convenient.

## Getting Predictions

There are a few ways that you can approach the prediction problem. You can do everything at run-time, and serve up predictions dynamically on the fly. There is something to be said about that.

In my experience, however, I just don’t have the horsepower that would require. So what you can do is you can pre-compute the recommendations and store them in a database somewhere. It is really easy to do since all that you have to do is to put in a user id and a movie id. You can then just store the score and ids in a database table and retrieve them later when your user logs in the next time.

So how do you actually get the scores? Here is an example.

import numpy as np
model.predict([np.array([610]),np.array([1])])

This will give us a recommendation score for user 610 and movie 1. Assuming that you have a database, you can just look these up to make sense of them. I won’t do that here.

Here is the output you get:

array([[1.0163633]], dtype=float32)

It looks like this movie should be very much recommended to this user. I only trained for 3 iterations, because time. I would suspect that after more iterations you wouldn’t see values above 1 like this in the recommendations since our recommendation scores run between -1 and 1.

## Let’s go deeper

So when I say let’s go deeper, I mean let’s take advantage of back propagation and add layers to our architecture. Here, I just add two layers to each of the “towers” in my model. Essentially, I create a deep neural network on top of each of my embedding vectors and then I use the same trick. You can also add layers to the bottom of the network if you want to. I’ll leave that to you if you are interested, but the idea here is that you can mess around with the architecture all day and night to try to squeeze out all the performance that you can from this algorithm.

So here is the code for my new model:

user_input = Input(shape=[1], name="User-Input")
user_embedding = Embedding(len(df.userId.unique())+1, 5, name="User-Embedding")(user_input)
user_vec = Flatten(name="Flatten-users")(user_embedding)
user1 = Dense(100, name='uDense1')(user_vec)
user2 = Dense(100, name='uDense2')(user1)

movie_input = Input(shape=[1], name="Movie-Input")
movie_embedding = Embedding(len(df.movieId.unique())+1, 5, name="Movie-Embedding")(movie_input)
movie_vec = Flatten(name="Flatten-movies")(movie_embedding)
movie1 = Dense(100, name='mDense1')(movie_vec)
movie2 = Dense(100, name='mDense2')(movie1)

prod = Dot(name="Dot-Product", axes=1)([user2, movie2])
model = Model([user_input, movie_input], prod)


text:
# Generating Pascal's triangle

I am trying to generate Pascal's triangle table in Python 3. I started to like using this following pattern when dealing with 2D arrays.

1. Generate a list of valid indices
2. Use a function that works with those indices in a list comprehension. (may produce side effects)

I wonder, is this pattern pythonic and how common is it in practice?

def pascal_triangle(level):
def compute(table, row, col):
if row == 0:
table.append([])
if col == 0 or row == col:
table[row].append(1)
elif col <= row:
table[row].append(table[row-1][col] + table[row-1][col-1])
table = []
valid_idxs = [(r,c) for r in range(level) for c in range(level)]
[compute(table, r, c) for r, c in valid_idxs]
print(table)

• Hi, I appreciate the fact that you considered my answer as satisfying. However, I reckon it would be best for you not to accept it too quickly as it can prevent other people to want to come and bring new ideas to the table. – SylvainD Jul 25 '14 at 16:18

In :

table = []
valid_idxs = [(r,c) for r in range(level) for c in range(level)]
[compute(table, r, c) for r, c in valid_idxs]
print(table)

the line [compute(table, r, c) for r, c in valid_idxs] definitly shouldn't be a list comprehension as we are building a list that we do not use. If it is side-effects we are interested in, we might as well use a for.

table = []
valid_idxs = [(r,c) for r in range(level) for c in range(level)]
for r, c in valid_idxs:
compute(table, r, c)
print(table)

Once, this is done, it seems quite clear that the first list comprehension is not really required neither as it could be :

table = []
for r in range(level):
for c in range(level):
compute(table, r, c)
print(table)

This being said, even though it looked like a good idea to use a function to extract common code, the responsabilities are not clearly defined making things a bit hard to track. Let's put the body of the function back in the loop and try to see what can be improved :

def pascal_triangle(level):
table = []
for r in range(level):
for c in range(level):
if r == 0:
table.append([])
if c == 0 or r == c:
table[r].append(1)
elif c <= r:
table[r].append(table[r-1][c] + table[r-1][c-1])
print(table)

First thing to improve : print the content of table before or after the table.append([]) : isn't it a bit weird to build all lists as we go through the c variable ? Also, the first list created is non-empty but the other one are. Let's make the process clearer and add an empty list at each r iteration.

def pascal_triangle(level):
table = []
for r in range(level):
print(table) # look how the tables gets populated now : isn't it cool ?
table.append([])
for c in range(level):
if c == 0 or r == c:
table[r].append(1)
elif c <= r:
table[r].append(table[r-1][c] + table[r-1][c-1])
print(table)

Let's go one step further, please note that the deeper loop will not do anything if c > r (this wasn't true before previous change but it is now correct). We can easily remove this pointless iterations (and the pointless check) :

def pascal_triangle(level):
table = []
for r in range(level):
print(table)
table.append([])
for c in range(r+1):
if c == 0 or r == c:
table[r].append(1)
else:
assert(c<=r)
table[r].append(table[r-1][c] + table[r-1][c-1])
print(table)

Now, it's time for the special trick : we could remove the test if c == 0 by performing this before the loop. Also, we could't remove the if c == r by performing this after the loop (but we need to check that this would have happened which is when r>0).

def pascal_triangle(level):
table = []
for r in range(level):
print(table)
table.append([])
table[r].append(1)
for c in range(1, r):
table[r].append(table[r-1][c] + table[r-1][c-1])
if r:
table[r].append(1)
print(table)

Now, we can try to be more fancy : instead of calling table[r] every time : let's introduce a line variable and use it :

def pascal_triangle(level):
table = []
for r in range(level):
line = []
line.append(1)
for c in range(1, r):
line.append(table[r-1][c] + table[r-1][c-1])
if r:
line.append(1)
table.append(line)
print(table)

We can see that this looks a lot like a list comprehension scenario and indeed, we can use it :

def pascal_triangle(level):
table = []
for r in range(level):
table.append([1] + [table[r-1][c] + table[r-1][c-1] for c in range(1, r)] + ([1] if r else []))
print(table)


text:
## Differential and Integral Equations

### Ground state solutions for a semilinear problem with critical exponent

#### Abstract

This work is devoted to the existence and qualitative properties of ground state solutions of the Dirchlet problem for the semilinear equation $-\Delta u-\lambda u=\vert u\vert^{2^*-2}u$ in a bounded domain. Here, $2^*$ is the critical Sobolev exponent, and the term ground state refers to minimizers of the corresponding energy within the set of nontrivial solutions. We focus on the indefinite case where $\lambda$ is larger than the first Dirichlet eigenvalue of the Laplacian, and we present a particularly simple approach to the study of ground states.

#### Article information

Source
Differential Integral Equations, Volume 22, Number 9/10 (2009), 913-926.

Dates
First available in Project Euclid: 20 December 2012


text:
## Intermediate Algebra for College Students (7th Edition)

Published by Pearson

# Chapter 1 - Section 1.4 - Solving Linear Equations - Exercise Set: 38

#### Answer

$x=\dfrac{25}{7}$

#### Work Step by Step

For easier work, it is better to get rid of the fractions. This can be achieved by multiplying the LCD of $30$ on both sides of the equation to obtain: $30(\frac{3x}{5} - \frac{x-3}{2}) = 30(\frac{x+2}{3}) \\\frac{90x}{5} - \frac{30(x-3)}{2} =\frac{30(x+2)}{3} \\18x - 15(x-3) = 10(x+2) \\18x - 15x +45 = 10x+20 \\3x+45 =10x+20$ Subtract $3x$ and $20$ on both sides to obtain: $45-20=10x-3x \\25=7x \\\frac{25}{7} = \frac{7x}{7} \\\frac{25}{7}=x$ Check: $\begin{array}{ccc} &\dfrac{3(\frac{25}{7})}{5} - \dfrac{\frac{25}{7}-3}{2} &= &\dfrac{\frac{25}{7}+2}{3} \\&\dfrac{(\frac{75}{7})}{5} - \dfrac{\frac{25}{7}-\frac{21}{7}}{2} &= &\dfrac{\frac{25}{7}+\frac{14}{7}}{3} \\&\frac{75}{7(5)} - \dfrac{\frac{4}{7}}{2} &= &\dfrac{\frac{39}{7}}{3} \\&\frac{15}{7} - \frac{4}{7(2)}&= &\frac{39}{7(3)} \\&\frac{15}{7} - \frac{2}{7} &= &\frac{13}{7} \\&\frac{13}{7} &= &\frac{13}{7}\end{array}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
leo silva
2011-02-10
2013-05-08

• leo silva
2011-02-10

I am using Windows XP, R-2.12.2, notepad++ 5.8.7 ,NppToR 2.5.2 and i dont have admin privileges.

After the instalation, i used the NppToR -> settings to configure the paths and everything worked.
If i close NppToR and try to run it again i get an error about the notepad++ directory and i need do configure the path again to make it work.

I tried this using the NppToR installer and the portable version and i got the same problem.

Here is my NppToR settings
R=C:\Documents and Settings\xxxxxxxx\My Documents\R\R-2.12.1
Rcmdparms=

• Andrew Redd
2011-02-10

Disturbing.  Could you post screenshot of the error?  That would help me track it down.  It sounds like the ini file is not being saved.   Could you to check if there is an npptor.ini file in the directory of the portable version.  Let's stick with that since I think that we will be easier to get things working for you.

• Joe Ledvina
2011-04-24

I'm having the same problem with the "protable" version of NppToR - it's not portable (it reads and writes to an ini file in %APPDATA%), and it also doesn't remember the path to Notepad++ if I exit and restart NppToR. I've been troubleshooting it and I can confirm the following on 64-bit Windows 7:

1. The problem occurs whether or not I edit either npptor.ini file (the one in the app directory or the one in the "default data" directory). I did not rename either file.

2. The "portable" NppToR never writes to either of the ini files, only to the file in %APPDATA%. I always used the "exit" option in the tray menu to exit, in case NppToR is supposed to write to %APPDATA%, then move the files on exit.

## 3. If I enter a portable path to my portable Notepad++ (using %Drive%) in the settings dialog, NppToR can launch Notepad++, but if I exit and restart it, it pops up the error referenced above by leprech:

Error: Failed attempt to launch program or document:
Params: <>
Specifically: The system cannot find the file specified.
Line#
--> 312: Run,%Nppexe%

After I restart NppToR, the "Notepadd++ Home" box will be blank in the settings dialog, but the path ("%Drive%\path\to\Notepad++\") remains stored in the ini file in %APPDATA%.

4. If I set the FULL path to Notepad++, NppToR can never launch Notepad++ - it produces the above error message every time.

Cheers,
Joe

• Andrew Redd
2011-04-27

I've uploaded version 2.5.3 that should fix this issue among a few other small improvements.  Please let me know if the issue persists or not.  Thanks.

• Joe Ledvina
2011-04-27

Unfortunately, your fix did not change the behavior at all. It still doesn't read the ini in %A_ScriptDir% - it saves settings to %APPDATA%, (but it does not save "Notepad++ Home" - it only works until I close the app).

I have a tiny bit of experience with AHK, so I peeked at the source when I was trying to get it to work over the weekend. If I could compile it (it threw an error about iniGUI.ahk, which isn't included in the source), I would just remove these iniread and if(Global) sections to make my own portable-only executable:

iniRead, Global, %inifile%, install, global, 0 ;0=false
if(Global)

Could the problem be your iniread line assigns a value to "Global" whether it exists in the inifile or not, and then your if(Global) statement only checks to see that the variable named "Global" exists? According to the AHK help file, the last value of the iniread line is an optional default value, set if the key doesn't exist in the ini file.

Should it therefore be:

iniRead, Global, %inifile%, install, global, 1 ;0=false
if(Global=0)

?

• CHEN YU
2012-03-09

I am using both portable Notepad++ and portable NpptoR

which is experiencing the SAME problem.

I wonder when it would be fixed??

OS: Windows 7 SP1 x64
R: 2.14.2 x64

• Leandro Marino
2012-04-07

Hi,

I'm with the same problem. I have edited the npptor.ini but when i enter in the NppToR settings appears a blank in the Notepad ++ Home. I am attaching the ini file, the settings screen and the error.

1.  Inifile

2. Error message

3. Settings Screen

OS: Windows 7 x64 (i don't know the service pack)
R: R2.15.0 installed into the flash drive.

• Andrew Redd
2012-09-03

This should definitely be fixed in 2.6.2.  Please report is if anyone is still having problems.


text:
## Including formulas

Doxygen allows you to put formulas in the output (this works only for the HTML and formats, not for the man page output). To be able to include formulas (as images) in the HTML documentation, you will also need to have the following tools installed
• latex: the compiler, needed to parse the formulas. To test I have used the teTeX 0.9 distribution.
• dvips: a tool to convert dvi files to postscript files I have used version 5.86 from Radical Eye software for testing.
• gs: the ghostscript interpreter for converting postscript files to bitmaps. I have used Aladdin Ghostscript 5.10 for testing.

There are two ways to include formulas in the documentation.

1. Using in-text formulas that appear in the running text. These formulas should be put between a pair of \f$commands, so  The distance between \f$(x_1,y_1)\f$and \f$(x_2,y_2)\f$is \f$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\f\$.

results in:

The distance between and is .

2. Unnumbered displayed formulas that are centered on a separate line. These formulas should be put between \f$and \f$ commands. An example:
  \f[
|I_2|=\left| \int_{0}^T \psi(t)
\left\{
u(a,t)-
\int_{\gamma(t)}^a
\frac{d\theta}{k(\theta,t)}
\int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi
\right\} dt
\right|
\f]

results in:

Formulas should be valid commands in 's math-mode.

Warning:
Currently, doxygen is not very fault tolerant in recovering from typos in formulas. It may have to be necessary to remove the file formula.repository that is written in the html directory to a rid of an incorrect formula