Square roots and logarithms without a calculator (Part 12)

I recently came across the following computational trick: to estimate $\sqrt{b}$ , use

$\sqrt{b} \approx \displaystyle \frac{b+a}{2\sqrt{a}}$ ,

where $a$ is the closest perfect square to $b$ . For example,

$\sqrt{26} \approx \displaystyle \frac{26+25}{2\sqrt{25}} = 5.1$ .

I had not seen this trick before — at least stated in these terms — and I’m definitely not a fan of computational tricks without an explanation. In this case, the approximation is a straightforward consequence of a technique we teach in calculus. If $f(x) = (1+x)^n$ , then $f'(x) = n (1+x)^{n-1}$ , so that $f'(0) = n$ . Since $f(0) = 1$ , the equation of the tangent line to $f(x)$ at $x = 0$ is

$L(x) = f(0) + f'(0) \cdot (x-0) = 1 + nx$ .

The key observation is that, for $x \approx 0$ , the graph of $L(x)$ will be very close indeed to the graph of $f(x)$ . In Calculus I, this is sometimes called the linearization of $f$ at $x =a$ . In Calculus II, we observe that these are the first two terms in the Taylor series expansion of $f$ about $x = a$ .

For the problem at hand, if $n = 1/2$ , then

$\sqrt{1+x} \approx 1 + \displaystyle \frac{x}{2}$

if $x$ is close to zero. Therefore, if $a$ is a perfect square close to $b$ so that the relative difference $(b-a)/a$ is small, then

$\sqrt{b} = \sqrt{a + b - a}$

$= \sqrt{a} \sqrt{1 + \displaystyle \frac{b-a}{a}}$

$\approx \sqrt{a} \displaystyle \left(1 + \frac{b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{2a + b-a}{2a} \right)$

$= \sqrt{a} \displaystyle \left( \frac{b+a}{2a} \right)$

$= \displaystyle \frac{b+a}{2\sqrt{a}}$ .

One more thought: All of the above might be a bit much to swallow for a talented but young student who has not yet learned calculus. So here’s another heuristic explanation that does not require calculus: if $a \approx b$ , then the geometric mean $\sqrt{ab}$ will be approximately equal to the arithmetic mean $(a+b)/2$ . That is,

$\sqrt{ab} \approx \displaystyle \frac{a+b}{2}$ ,

so that

$\sqrt{b} \approx \displaystyle \frac{a+b}{2\sqrt{a}}$ .

Jordan Lyles Becomes First Brewer To Wear Irrational Number

From the Onion, and posted in honor of the imminent Opening Day of the 2020 Major League Baseball season.

Engaging students: Square roots

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 Julie Thompson. Her topic, from Algebra: square roots.

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

When I think of square roots my mind immediately takes me to the very popular movie ‘The Wizard of Oz’. In a scene near the end of the movie, the scarecrow incorrectly states the Pythagorean Theorem. He states it so fast that some people may not have time to process what he is saying is incorrect. The theorem he states is as follows: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” There are a couple things wrong with this statement. First of all, the Pythagorean Theorem is based on right triangles, not isosceles. Secondly, we take the square of two specific sides and set it equal to the square of the third side, not the square root of ‘any two sides’ equal to the square root of the remaining side.

As an engage, I think it would be very interesting to first show the clip of the movie to capture my students’ attention, and then have a discussion about why the theorem is wrong and what the correct theorem actually is!

Also, I found an awesome worksheet from Mathbits that is all about this scene from the movie and goes through a couple examples that shows why his theorem can’t work, and also allows students to prove why it is false!!

Click to access OzMath.pdf

How can technology be used to effectively engage students with this topic?

A very engaging website that was actually introduced to me in college: KAHOOT! I first played Kahoot in my TNTX 1200 class here at UNT. It was very exciting and fun for me, as a college student, to play, so I know middle and high school students will love it as well. Kahoot is an online quiz game where students use their own technology to join in to the game with a game pin provided by the teacher. Students get to give themselves a game nickname which makes it fun to be able to see their name pop up on the scoreboard. Then a variety of questions on the topic are asked, one at a time, with a time limit for the students to answer in (usually about 20-30 seconds). This is a quick game that can be used as an engage at the beginning of class to get students thinking and excited about the topic for the day. In this case…square roots! I found a great Kahoot created by ‘remangum’ that focuses on finding square roots of numbers (it throws in a couple cube roots). Once you get passed about 7 questions, they throw some variables into the mix. One of the question asks to find d:

Sqrt (d*d)=9, where * is multiplication. In this case, d=9 because sqrt(81)=9. I like this because it allows the students to think a little harder and problem solve.

Here is the link: https://play.kahoot.it/#/?quizId=8ed37283-e3fc-4389-a8ed-ff7200993731

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

Many students who enter middle school/ early high school wonder why they have to learn all these pointless concepts such as square roots and the order of operations. They might even think to themselves, “When will I ever need to know this in the future when I have a job?” According to homeschoolmath.net, “The answer is that you need algebra in any occupation that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade, commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You also need algebra to take your SAT test or GED.” This is very important to let students know, but they may not believe you or care. For instance, they may say that’s true for math and science professions, but they are planning to major in something totally different and they won’t need math. Math actually can be useful in other fields, but for the sake of this question, I will stick to math and science.

In their future classes, such as Algebra II, they will be using things such as the quadratic formula. This will involve plugging in and simplifying things under a radical, as well as dealing with square roots in whole equations rather than just on their own. Also, understanding the nature of square roots will help them in future courses such as PreCalculus when they must study all the characteristics of the square root function. As an engaging aspect to all of this, I may mention that, “Studying algebra also has a benefit of developing logical thinking and problem solving skills. Algebra can increase your intelligence! (Actually, studying any math topic — even elementary math — can do that, if it is presented and taught in such a manner as to develop a person’s thinking.)”

Quotations from: https://www.homeschoolmath.net/teaching/why_need_square_roots.php

References

A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card). (2012, January 15). Retrieved September 09, 2016, from https://reflectionsinthewhy.wordpress.com/2012/01/15/a-visual-approach-to-simplifying-radicals-a-get-out-of-jail-free-card/

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Buncombe, A. (16, April 4). Square Root Day: There are only nine days this century like this. Retrieved September 09, 2016, from http://www.independent.co.uk/news/world/americas/square-root-day-there-are-only-nine-days-this-century-like-this-a6967991.html

Fowler, D., & Robson, E. (n.d.). Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Historical Mathematica, 366-378. Retrieved September 9, 2016, from https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf.

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/

Engaging students: Graphing Square Root Functions

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 student Alexandria Johnson. Her topic, from Algebra II: graphing square root functions.

An interesting word problem that students should be able to solve after completing a lesson on graphing square root functions would be: “The Chandra satellite detected x-rays coming from the region of the sky containing the galaxy cluster JKS041. The electrons in the gas are emitting the X-rays, and colliding at high speed with the protons in the gas. The energy of the x-rays at the time they were emitted by the hot gas was 21,400 electron Volts (eV). This energy is shared equally between the electrons and protons. The speed of a proton is related to its kinetic energy by E = 1/2mV^2 where E is the energy in Joules, V is the proton speed in meters/sec, and m is the mass of a proton (m = 1.7 x 10-27 kg). About how fast are the protons moving? (Note: 1 eV = 1.6 x 10^-19 Joules)”. Students can arrange the problem into a square root function to solve for velocity: V=sqrt(2E/m). Using the information provided students can convert eV to E and solve for m. Once this information is found, students can plug in the numbers to solve for V. Note: this question is difficult and some students may struggle with the calculations. A simpler question about the relationship between kinetic energy and velocity could be used in place of this one. Question provided by https://spacemath.gsfc.nasa.gov/weekly/6Page70.pdf.

In Physics, students will be able to use square root functions to describe the relationship between different variables. Having the knowledge of graphing square root functions will allow students to represent these relationships graphically. For example, to find kinetic energy, students use the formula E=(1/2)*m*v^2, where m=mass and v=velocity. Students can manipulate the equation to find v which would be v=sqrt(2E/m). Given m, students should be able to graph the relationship between v and E. When solving for volume, students can rearrange the equation into the form y=a*sqrt(x-h)+k, where h=0, k=0 y=v, x=E, and a=sqrt(2/m). knowing how to graph a square root function, students can graph this equation.

A useful resource when creating a lesson about graphing square root functions is https://teacher.desmos.com/. This website provides teachers with existing activities that the students can complete. Also, it allows the teacher to create activities for the student. An activity that is already created for teacher use is called Polygraph: Square root functions. In this activity, students play a game similar to the board game Guess Who. Students pair up and are given a set of graphs of square root functions. Partner 1 chooses a graph. Then, Partner 2 asks questions about the graphs to try to find the graph that Partner 1 chose. Students compare various graphs and communicate these differences. Though the website doesn’t offer any other premade activities at this time, teachers can use the activity type “marble slides” to create an activity that shows how a, h and k affect the parent function of square roots.
Work cited

“Chandra Spies the Most Distant Cluster in the Universe.” Space Math, NASA, Chandra Spies the Most Distant Cluster in the Universe. Accessed 15 Sept. 2017.
“Square Root Functions.” Desmos Classroom Activities, teacher.desmos.com/polygraph/custom/560ad29158fd074d156300b6. Accessed 15 Sept. 2017

Engaging students: Square roots

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 Jessica Martinez. Her topic, from Algebra: square roots.

How has this topic appeared in the news?

There is a (sort of) holiday for square root days; sort of because square root days only come 9 time every century and this year we celebrated 4/4/16. Since it’s not as frequent as Pi Day, it’s a lesser known “holiday”, but even then, it still pops up in the news. I found this online article for a UK news site that described other square root-related fun facts in history. It also included a post from Good Morning America with the hashtag #squarerootday, which gave me this idea: I would like to encourage my students to participate in all of the fun square root-related activities that celebrate this day (if there was one that school year). The founder of square root day has suggestions that include but are not limited to: square dancing, drinking root beer out of square glasses, or even taking a drive on route 66. In the days leading up to this fantastic math-related day, I would consider giving my kids an extra credit point for posting a picture of themselves doing something square root related on the class twitter with the tag #squarerootday (or a post on some other class social media). If there wasn’t a square root day during that academic year, I still think it would be fun to tell my students about this holiday.

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

My students should have already learned about perfect squares and their multiplication tables up to 12 or 13, at least. For a simple refresher, I could have my students color/highlight perfect squares on multiplication tables. Then taking the square root of something is the inverse of creating perfect squares, unless what’s under the square root sign isn’t a perfect square. Then what’s under the radical is something that they need to divide into its prime factors so that they can simplify. My students should have also at least learned about prime numbers, if not prime factoring. A way to solve square roots would be pairing up the prime factors under the square root so that you can “take it out” from under the radical; for my students, I could have them think of the square root sign as a jail cell, and the only way that the numbers could “get out” of the cell is if they had a “prime partner” to escape with (i.e. a pair of 2s, a pair of 3s etc.).

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

So one of the oldest records of square roots in history would be The Old Babylonian tablet YBC 7289, which dates back anywhere from 2000-1600 BC. It depicts a square with two diagonals drawn and on the diagonals are numbers; when they are calculated, you get a very close approximation of the square root of 2 for the diagonal. Their value for the square root of two was about 1.41421297; I could have my students quickly calculate the square root of two (about 1.41421356) and mention to my students that this is pretty impressive for a civilization without modern day technology. The fact that they used clay tablets for math calculations shows how little they had to work with. Yet Babylon was also one of the most famous ancient cities in Mesopotamia; it’s mentioned multiple times in the bible and they were pretty advanced in mathematics for their area, despite the lack of resources we have today. They used a sexagesimal number system, which is base 60; they could solve algebra problems and work with what we now call Pythagorean triples; they could also solve equations with cubes.

References

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/

Lessons from teaching gifted elementary students (Part 8g)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received, in the students’ original handwriting. They wanted me to add adjacent numbers on the bottom row to produce the number on the next row, building upward until I reached the apex of the triangle. Then, after I reached the top number, they wanted me to take the square root of that number. (Originally, they wanted me to first multiply by 80 before taking the square root, but evidently they decided to take it easy on me.)

And, just to see if I could do it, they wanted me to do all of this without using a calculator. But they were nice and allowed me to use pencil and paper.

So far, I’ve used Pascal’s triangle to obtain

$y = \displaystyle \sum_{k=0}^{11} (k+1)^2 {11 \choose k}$

$= \displaystyle \sum_{k=2}^{11} k(k-1) {11 \choose k} + \sum_{k=1}^{11} 3k {11 \choose k} + \sum_{k=0}^{11} {11 \choose k}$ .

$= \displaystyle \sum_{k=2}^{11} k(k-1) \left( \frac{11!}{k!(11-k)!} \right) + \sum_{k=1}^{11} 3k \left( \frac{11!}{k!(11-k)!} \right) + \sum_{k=0}^{11} \left( \frac{11!}{k!(11-k)!} \right)$ .

$= \displaystyle \sum_{k=2}^{11} \frac{11!}{(k-2)!(11-k)!} + 3 \sum_{k=1}^{11} \frac{11!}{(k-1)!(11-k)!} + \sum_{k=0}^{11} \frac{11!}{k!(11-k)!}$ .

$= \displaystyle \sum_{i=0}^{9} \frac{11!}{i!(9-i)!} + 3 \sum_{j=0}^{10} \frac{11!}{j!(10-j)!} + \sum_{k=0}^{11} \frac{11!}{k!(11-k)!}$ .

$= \displaystyle \sum_{i=0}^{9} 11 \times 10 \frac{9!}{i!(9-i)!} + 3 \sum_{j=0}^{10} 11 \frac{10!}{j!(10-j)!} + \sum_{k=0}^{11} \frac{11!}{k!(11-k)!}$

$= \displaystyle 110 \sum_{i=0}^{9} {9 \choose i} + 33 \sum_{j=0}^{10} {10 \choose j} + \sum_{k=0}^{11} {11 \choose k}$

$= 110 \times 2^9 + 33 \times 2^{10} + 2^{11}$

$= 92,160$ .

I’m almost done… except my students wanted me to find the square root of this number without using a calculator.

There are a couple ways to do this; the method I chose was directly extracting the square root by hand… a skill that was taught to children in previous generations but has fallen out of pedagogical disfavor with the advent of handheld calculators. I lost my original work, but it would have looked something like this (see the above website for details on why this works):

And so I gave my students their answer: $x \approx 303.578\dots$

Engaging students: Square roots

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 Tiffany Jones. Her topic, from Algebra: square roots.

B.1 How can this topic be used in your students’ future courses in mathematics or science?

One area of mathematics I wish I had more practice with in grade school is numerical reasoning. I feel that, as a student, I was allowed to use my calculator too much and am struggling to remove my calculator crutch. I hope to encourage my students to sharpen their numerical reasoning skills and to not rely on their calculator. Does this number make sense? Is it too high, too low? Is a negative result valid given the scenario of the problem? The following video introduces a method to estimate the square root of non-perfect squares to the nearest tenth by hand:

“Estimating Square Roots To the Nearest Tenth by Hand” by Fort Bend Tutoring

It gives the students another tool for their toolbox of numerical reasoning, practice using formulas, reviews long division by hand, and strongly encourages students to remember the perfect squares.

I think that introducing this idea as an engage could intrigue student to wonder why the formula works and to wonder what else they are able to do quickly by hand.

Fort Bend Tutoring’s YouTube channel offers videos on a wide verity of high school mathematics topics and courses. The videos cover several examples. They are engaging, not dry and there is also a “theme song” to the videos. I feel that these videos can sever as a great addition to lessons as extra help to the students.

D.1 What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The following story was first told to me in a calculus one course. While the telling of the story was to serve as amusement and did not directly relate to the topic of the day, it stuck with me. It comes to mind frequently when working with the Pythagorean theorem and with irrational squares. And when given this assignment, I saw square roots as an option, this story again came to mind. I think having an interesting story cross my mind makes a problem overall more fun. I would want to give that to my students. The article “The Dangerous Ratio” by Brain Clegg does a wonderful job of telling the story, its implications, and gives a mock dialogue so reads can work through the logic. At the end of the article, there is a link to an activity about the proof that the square root of 2 is irrational.

E.1

How can technology be used to effectively engage students with this topic?

I really like the idea of a flipped classroom and hope to be able to practice it in my classroom. While a completely flipped classroom will take some time to implement, videos such as Math Antics’ “Exponents & Square Roots” will be a great place to start.

This particular video address a previously learned topic, namely exponents and relates it to the new topic. It provides definitions and visuals to remember how the terms relate to each other and how to read the symbols. It goes through several examples of varying level and shows the viewer how to use technology such as calculators to solve hard problems. In addition, the video addresses some common misconceptions such as mistaking the root sign and the division sign. Moreover, it ties everything together with a quick review at the end.

One of my favorite aspects so of flipped classrooms, is that the student can review the video over and over. Math Antics does an excellent job of talking the math out to the viewer. The animations are amusing yet helpful. While a lot of information is covered, the video is not dry.

Resources:

“Estimating Square Roots To the Nearest Tenth by Hand” by Fort Bend Tutoring – https://www.youtube.com/watch?v=bUh7Hj-3dkw

“The Dangerous Ratio” by Brian Clegg – http://nrich.maths.org/2671

“Exponents & Square Roots” by Math Antics –https://www.youtube.com/watch?v=C_iKTTI1E34

How I Impressed My Wife: Part 5h

Earlier in this series, I gave three different methods of showing that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Using the fact that $Q$ is independent of $a$ , I’ll now give a fourth method.

Since $Q$ is independent of $a$ , I can substitute any convenient value of $a$ that I want without changing the value of $Q$ . As shown in previous posts, substituting $a =0$ yields the following simplification:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_{0}^{2\pi} \frac{dx}{\cos^2 x + 2 \cdot 0 \cdot \sin x \cos x + (0^2 + b^2) \sin^2 x}$

$= \displaystyle \int_{0}^{2\pi} \frac{dx}{\cos^2 x + b^2 \sin^2 x}$

$= \displaystyle \int_{-\pi}^{\pi} \frac{dx}{\cos^2 x + b^2 \sin^2 x}$

$= \displaystyle \int_{-\infty}^{\infty} \frac{ 2(1+u^2) du}{u^4 + (4 b^2 - 2) u^2 + 1}$

The four roots of the denominator satisfy

$u^2 = \displaystyle 1 - 2b^2 \pm 2|b| \sqrt{b^2 - 1}$

So far, I’ve handled the cases $|b| = 1$ and $|b| > 1$ . In today’s post, I’ll start considering the case $|b| < 1$ .

Factoring the denominator is a bit more complicated if $|b| < 1$ . Using the quadratic equation, we obtain

$u^2 = \displaystyle 1 - 2b^2 \pm 2|b| i \sqrt{1-b^2}$

However, unlike the cases $|b| \ge 1$ , the right-hand side is now a complex number. So, To solve for $u$ , I’ll use DeMoivre’s Theorem and some surprisingly convenient trig identities. Notice that

$(1-2b^2)^2 + (2|b| \sqrt{1-b^2})^2 = 1 - 4b^2 + b^4 + 4b^2 (1 - b^2) = 1 - 4b^2 + b^4 + 4b^2 - b^4 = 1$ .

Therefore, the four complex roots of the denominator satisfy $|u^2| = 1$ , or $|u| = 1$ . This means that all four roots can be written in trigonometric form so that

$u^2 = \cos 2\phi + i \sin 2\phi$ ,

where $2\phi$ is some angle. (I chose the angle to be $2\phi$ instead of $\phi$ for reasons that will become clear shortly.)

I’ll begin with solving

$u^2 = \displaystyle 1 - 2b^2 + 2|b| i \sqrt{1-b^2}$ .

Matching the real and imaginary parts, we see that

$\cos 2\phi = 1-2b^2$ ,

$\sin 2\phi = 2|b| \sqrt{1-b^2}$

This completely matches the form of the double-angle trig identities

$\cos 2\phi = 1 - 2\sin^2 \phi$ ,

$\sin 2 \phi = 2 \sin\phi \cos \phi$ ,

and so the problem reduces to solving

$u^2 = \cos 2\phi + i \sin 2\phi$ ,

where $\sin \phi = |b|$ and $\cos \phi = \sqrt{1-b^2}$. By De Moivre’s Theorem, I can conclude that the two solutions of this equation are

$u = \pm(\cos \phi + i \sin \phi)$ ,

$u = \pm( \sqrt{1-b^2} + i |b|)$ .

I could re-run this argument to solve $u^2 = \displaystyle 1 - 2b^2 - 2|b| i \sqrt{1-b^2}$ and get the other two complex roots. However, by the Conjugate Root Theorem, I know that the four complex roots of the denominator $u^4 + (4 b^2 - 2) u^2 + 1$ must come in conjugate pairs. Therefore, the four complex roots are

$u = \pm \sqrt{1-b^2} \pm i |b|$ .

Therefore, I can factor the denominator as follows:

$u^4 + (4 b^2 - 2) u^2 + 1 = (u - [\sqrt{1-b^2} + i|b|])(u - [\sqrt{1-b^2} - i|b|])$

$\qquad \times (u - [-\sqrt{1-b^2} + i|b|])(u - [-\sqrt{1-b^2} + i|b|])$

$= (u - \sqrt{1-b^2} - i|b|)(u - \sqrt{1-b^2} + i|b])$

$\qquad \times (u +\sqrt{1-b^2} + i|b|)(u +\sqrt{1-b^2} + i|b|)$

$= ([u - \sqrt{1-b^2}]^2 +b^2)([u + \sqrt{1-b^2}]^2 +b^2)$

To double-check my work, I can directly multiply this product:

$([u - \sqrt{1-b^2}]^2 +b^2)([u + \sqrt{1-b^2}]^2 +b^2)$

$= (u^2 - 2u \sqrt{1-b^2} + 1 - b^2 + b^2) (u^2 + 2u \sqrt{1-b^2} + 1 - b^2 + b^2)$

$= ([u^2 +1] - 2u\sqrt{1-b^2})([u^2+1] + 2u\sqrt{1-b^2})$

$= [u^2+1]^2 - [2u\sqrt{1-b^2}]^2$

$= u^4 + 2u^2 + 1 - 4u^2 (1-b^2)$

$= u^4 + u^2 (2 - 4[1-b^2]) + 1$

$= u^4 + u^2 (4b^2 - 2) + 1$ .

So, at last, I can rewrite the integral $Q$ as

$Q = \displaystyle \int_{-\infty}^{\infty} \frac{ 2(1+u^2) du}{ ([u - \sqrt{1-b^2}]^2 +b^2)([u + \sqrt{1-b^2}]^2 +b^2)}$

I’ll continue with this fourth evaluation of the integral, continuing the case $|b| < 1$ , in tomorrow’s post.

How I Impressed My Wife: Part 4f

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Originally, my wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by $\sec^2 x$ and then employing the substitution $u = \tan x$ (after using trig identities to adjust the limits of integration).

But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Here’s my progress so far:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_0^{2\pi} \frac{2 \, dx}{1+\cos 2x + 2 a \sin 2x + (a^2 + b^2)(1-\cos 2x)}$

$= 2 \displaystyle \int_0^{2\pi} \frac{d\theta}{(1+a^2+b^2) + 2 a \sin \theta + (1 - a^2 - b^2) \cos \theta}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\theta}{S + R \cos (\theta - \alpha)}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\phi}{S + R \cos \phi}$

$= \displaystyle -\frac{4i}{R} \oint_C \frac{dz}{z^2 + 2\frac{S}{R}z + 1}$ ,

where this last integral is taken over the complex plane on the unit circle, a closed contour oriented counterclockwise. In these formulas, $R = \sqrt{(2a)^2 + (1-a^2-b^2)^2}$ and $S = 1 + a^2 + b^2$ . (Also, $\alpha$ is a certain angle that is now irrelevant at this point in the calculation).

This contour integral looks complicated; however, it’s an amazing fact that integrals over closed contours can be easily evaluated by only looking at the poles of the integrand. In recent posts, I established that there was only one pole inside the contour, and the residue at this pole was equal to $\displaystyle \frac{R}{ 2 \sqrt{S^2-R^2} }$ .

This residue can be used to evaluate the contour integral. Ordinarily, integrals are computed by subtracting the values of the antiderivative at the endpoints. However, there is an alternate way of computing a contour integral using residues. It turns out that the value of the contour integral is $2\pi i$ times the sum of the residues within the contour; see Wikipedia and Mathworld for more information.

Therefore,

$Q = \displaystyle -\frac{4i}{R} \oint_C \frac{dz}{(z - r_1)(z- r_2)}$

$= \displaystyle -\frac{4i}{R} \cdot 2\pi i \cdot \frac{R}{ 2 \sqrt{S^2-R^2} }$

$= \displaystyle \frac{4\pi}{\sqrt{S^2-R^2}}$

Next, I use some algebra to simplify the denominator:

$S^2 - R^2 = (1+a^2+b^2)^2 - (1-a^2-b^2)^2 - (2a)^2$

$S^2 - R^2 = [(1 + a^2 + b^2) + (1-a^2-b^2)][(1 + a^2 + b^2) - (1 - a^2 -b^2)] - 4a^2$

$S^2 - R^2 = 2[2 a^2 + 2b^2] - 4a^2$

$S^2 - R^2 = 4b^2$

Therefore,

$Q = \displaystyle \frac{4\pi}{\sqrt{4b^2}} = \displaystyle \frac{4\pi}{2|b|} = \frac{2\pi}{|b|}$

Once again, this matches the solution found with the previous methods… and I was careful to avoid a common algebraic mistake.

In tomorrow’s post, I’ll discuss an alternative way of computing the residue.

Inverse Functions: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions on inverse functions that appear in Precalculus and Calculus.

Square Roots, nth Roots, and Rational Exponents

Part 1: Simplifying $\sqrt{x^2}$