My Favorite One-Liners: Part 18

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert  x = 0.\overline{432} = 0.432432432\dots into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number x.

1000x = 432.432432\dots

x = 0.432432\dots

Notice that the decimal parts of both x and 1000x are the same. Subtracting, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time.

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you will believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

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See also my fuller post on this topic as well as the index for the entire series.

 

My Favorite One-Liners: Part 14

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of \sec x, the first step is far from obvious:

\displaystyle \int \sec x \, dx = \displaystyle \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \, dx

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

x = 0.\overline{432} = 0.432432432\dots

into a decimal, the first step is to multiply x by 1000:

1000x = 432.432432\dots

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and poof! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting x from 1000x, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}.

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

Engaging students: Rational and Irrational Numbers

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 Daniel Adkins. His topic, from Pre-Algebra: rational and irrational numbers.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

The largest hurdle to overcome in mathematics, is the introduction to foreign, and new concepts. Quite often, individuals are stuck with their “old math”. When a new object appears before them, they won’t play with it, or recognize that even though it’s new, it still works with what we already know. This is especially true when it comes to introducing concepts of new sets of numbers, such as the imaginary numbers, or irrational numbers. More often than not, when it comes to irrational numbers, students freeze up. I believe that the best way to prevent this, is to show that students already know a lot about this set. By taking it step by step and reminding students what they already know about rational numbers, you can show them they have known about irrational numbers in some form or fashion for quite some time.

A simple two step project would be to first introduce the concept of an irrational number, then the instructor can draw a circle with a marked radius, and say, this is my pizza pie. Now I want a piece of pizza pie, but when it comes to pieces of pizza pie, I’m particular. I want to proficiently partake of my pizza pie by partitioning it perfectly, to where each piece is equally cut. If all I know is the radius though, how can I know where to cut it? Eventually students will point out that by finding the circumference, and then dividing the circumference by how many pieces you want, you can make sure they’re all equal. At this point, point out that you have a ratio of pieces to circumference, but how did the students get to the circumference in the first place? 2*pi*r so that means the radius of a circle is in a ratio to its circumference right? So we can right pi as some sort of fraction correct?  If the students are aware that this isn’t possible, then the digging isn’t necessary, but if they aren’t ask them to try and write it as a fraction.

The second part of this exercise would be to emphasize nested sets. Divide the students up into 2-4 groups, and have a several Natural, whole, integer, rational, and irrational numbers written on pieces of colored paper (with each team having 1 color). Students will line up in front of “nestable” baskets spread out in front of them labeled by the different sets of numbers as listed above, and will one at a time aim for the smallest set that their number can fall in. After all the papers have been thrown, the papers will be collected and compared as a class, and each paper made in the correct basket will count as a point for that team. At the end of it all, put the numbers back in their baskets and show how the baskets can all fit inside of each other, except that the irrational and rational baskets are the same size, and so they can’t nest inside of each other. This can be emphasized by drawing it on the board. This exercise reminds students of what it means for sets to share qualities, and that irrational numbers don’t have the same qualities that rational numbers do.

 

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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?

 

Throughout my high school career, it was never brought to my attention that there were conflicts within the history of math, or in fact that there even was a history of math. In fact, it wasn’t till my collegiate years that my classmates and I came to learn such things were at one point a problem, that math could have different viewpoints.

The individual who is credited with discovering irrational numbers is Hipassus of Metapontum. He was a philosopher who studied Pythagorean based concepts, and while trying to use the Pythagorean Theorem to solve for a ratio between a unit square’s side length and its diagonal, he learned that there wasn’t such a thing. At the time, the other Pythagorean philosophers believed that only positive rational numbers existed. So when Hipassus introduced his discovery to them, they weren’t exactly happy. The story varies, and no one may ever know what truly happened to him, but some of the more versed stories range from the other Pythagoreans simply killing him, to Pythagoras himself ostracized him, and upon the Gods discovering the abomination he discovered, they had him drowned by the sea to hide it away.

Regardless of the validity of these stories, it shows how discoveries like these can often cause turmoil in time periods.

 

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How was this topic adopted by the mathematical community?

Hipassus’s discovery caused such a drastic response because of two reasons; first off, it contradicted the core belief of Pythagoreans that Mathematics and geometry were indefinitely correlated, as in they were completely inseparable. But it also raised another problem that would eventually be brought up by another philosopher named Zeno. The problem was in the discrete vs. continuous argument, and how geometry couldn’t solve it. All in all, when Hipassus introduced this concept, it was met with malice. Many individuals would write this off as simply how things were “back then”, but a closer examination at something like imaginary numbers will reveal a similar pattern. It wasn’t until the Middle Ages when Middle Eastern mathematicians introduced concepts of algebra that irrational numbers became fully accepted within the mathematical community.

All in all, the stories behind things like irrational and imaginary numbers should be shared within schools much more often. Not only is it extremely interesting, and can convince students to do their own research, but it also shows that people were afraid to learn new thing, that these foreign concepts that are terrifying now, were terrifying to the people who discovered them too. It teaches students that instead of ostracizing others for bizarre concept, but instead to analyze them themselves. Because those bizarre concepts, may become commonplace. It shows students that Hipassus was on the right side of history, even though he was alone for quite a while.

 

References:

https://brilliant.org/wiki/history-of-irrational-numbers/

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html

https://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-with-consecutive-odd-even-integers.faq.question.580533.html

http://tulyn.com/8th-grade-math/irrational-numbers/wordproblems

 

Irrational / Everything’s relative

One popular (though maybe apocryphal) story from the history of mathematics involves the discovery of irrational numbers by Pythagoras and his disciples. The following quote is from the book Fermat’s Last Theorem by Simon Singh:

One story claims that a young student by the name of Hippasus was idly toying with the number \sqrt{2}, attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e. that \sqrt{2} is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

When I was a boy, the story was told that Pythagoras could not accept irrational (i.e.., cannot be written as the ratio of two integers) numbers because their existence would mean that we live in an irrational (i.e., insane, crazy) world, and so he had the unfortunate discoverer silenced.

When I present this story to my own students, they’re usually incredulous about the story, doubting that someone so smart could act so stupidly (or irrationally). Then I’ll tell them a much more recent story, from less than 100 years ago, about how a scientific principle morphed into a statement of ethics. Einstein’s theories of special relativity and general relativity were developed in the early 1900s; his theory of general relativity explained precession in the orbit of Mercury and predicted the deflection of starlight by the Sun’s gravity, which were both unexplained by Newtonian mechanics.

Writing to a popular audience, Einstein summarized his theory as follows:

The ‘Principle of Relativity’ in its widest sense is contained in the statement: The totality of physical phenomena is of such a character that it gives no basis for the introduction to the concept of “absolute motion”; or, shorter but less precise: There is no absolute motion.

The following sentences from Paul Johnson’s Modern Times summarize the popular reaction to Einstein’s work:

But for most people, to whom Newtonian physics, with their straight lines and right angles, were perfectly comprehensible, relativity never became more than a vague source of unease. It was grasped that absolute time and absolute length had been dethroned; that motion was curvilinear… At the beginning of the 1920s the belief began to circulate, for the first time at a popular level, that there were no longer any absolutes: of time and space, of good and evil, of knowledge, above all of value. Mistakenly, but perhaps inevitably, relativity became confused with relativism.

Indeed, the modern catchphrase “everything’s relative” was spawned shortly after the discovery of special and general relativity, a moral principle that Einstein himself abhorred.

So, after telling the story about Pythagoras and \sqrt{2}, I’ll use this story to hold up a mirror to ourselves, demonstrating that the passage of time has not made us immune from translating mathematical or scientific principles into statements of ethics.

The antiderivative of 1/(x^4+1): Part 3

This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:

\displaystyle \int \frac{1}{x^4 + 1} dx

To compute this integral, I will use the technique of partial fractions. In yesterday’s post, I used De Moivre’s Theorem to factor the denominator over the complex plane, which then led to the factorization of the denominator over the real numbers.

In today’s post, I present an alternative way of factoring the denominator by completing the square. However, unlike the ordinary method of completing the square, I’ll do this by adding and subtracting the middle term and not the final term:

x^4 + 1= x^4 + 2x^2 + 1 - 2x^2

= (x^2 + 1)^2 - (x \sqrt{2})^2

= (x^2 + 1 + x\sqrt{2})(x^2 + 1 - x \sqrt{2}).

The quadratic formula can then be used to confirm that both of these quadratics have complex roots and hence are irreducible over the real numbers, and so I have thus factored the denominator over the real numbers:

\displaystyle \int \frac{dx}{x^4 + 1} = \displaystyle \int \frac{dx}{\left(x^2 - x \sqrt{2} + 1 \right) \left(x^2 + x \sqrt{2} + 1\right)} .

and the technique of partial fractions can be applied.

There’s a theorem that says that any polynomial over the real numbers can be factored over the real numbers using linear terms and irreducible quadratic terms. However, as seen in this example, there’s no promise that the terms will have rational coefficients.

I’ll continue the calculation of this integral with tomorrow’s post.

Proofs that the square root of 2 is irrational

Here’s a resource with over 25 different proofs for demonstrating that \sqrt{2} is irrational: http://www.cut-the-knot.org/proofs/sq_root.shtml

Engaging students: Rational and Irrational Numbers

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 Allison Metzler. Her topic, from Pre-Algebra: rational and irrational numbers.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The video below is a scene from Star Trek. While most students will not have seen this version of Star Trek or perhaps any version at all, most are familiar with the franchise. Because the students will recognize the popular TV show, this video will immediately grab their attention and keep it for the whole video. The video clearly displays how it’s impossible for the computer to compute pi because it is a “transcendental” number. Thus, since pi is irrational, the computer will never be able to find the last digit of pi, causing it to focus on this insolvable problem forever. This video would provide the students with not only entertainment, but also a way to easily remember what an irrational number is. I would also point out that if Spock would have told the computer to compute a rational number such as any fraction or whole integer, it would have taken a matter of seconds.

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C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Rational and irrational numbers can be found in music theory which is incorporated in classical music. First, Pythagoras was credited for discovering that “consonant sounds arise from string lengths related by simple ratios: -Octave 1:2 –Fifth 2:3 –Fourth 3:4” which are all rational numbers. Rational numbers are also found in the sound frequency and the diatonic scale. In order to get an equal tempered scale, we must get from the note C to the note C’ in twelve equal multiplicative steps we must find x such that x12=2. This causes us to take the twelfth root of 2 which produces an irrational number. The benefit of tuning a piano to tempered scale is that (1) “Sharps and flats can be combined into a single note” and (2) “Performers can play equally well in any key.” Rational and irrational numbers can also be found in other areas of music as evidenced below.

”At least one composition, Conlon Nancarrow’s Studies for Player Piano, uses a time signature that is irrational in the mathematical sense. The piece contains a canon with a part augmented in the ratio square root of 42:1.”

Also, when you play a fretted instrument (i.e. guitar, banjo, balalaika, bandurria, etc.), you are playing irrational numbers. According to http://www.woodpecker.com/writing/essays/math+music.html, the reason guitars are so hard to tune is that “our ears don’t like the irrational numbers”. However, they are needed to make “complex chordal music.”

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D1. What interesting things can you say about the people who contributed to the discovery and the development of this topic?

The video below displays who discovered irrational numbers while also getting into why the square root of 2 is irrational. I would play the video until the 4 minute mark so that I can keep the attention of the students. I would then go further into who contributed to the discovery of irrational numbers.

The Pythagoreans were set on the idea that all numbers could be expressed as ratios of integers. However, Hippasus of Metapontum, a philosopher at the Pythagorean school of thought, discovered otherwise. He supposedly used the Pythagorean Theorem (a2 + b2 = c2) on an isosceles right triangle where the congruent sides were each 1 unit. Using the theorem, he found that the hypotenuse was the square root of 2 which proved to be incommensurable. The other Pythagoreans were so horrified with this discovery, that it’s said they had Hippasus drowned. They wanted to punish him while also keeping irrational numbers a secret. However, it’s hard to prove that this information is true because of the vague accounts of who discovered irrational numbers. Therefore, I would inform my students of this interesting story, but also tell them about the uncertainty of what actually happened.

References:

Discovery of Irrational Numbers (n.d.). In Brilliant. Retrieved February 7, 2014, from https://brilliant.org/assessment/techniques-trainer/discovery-of-irrational-numbers

Hippasus (2014, January 14). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Hippasus

Pre-Algebra 32-Irrational Numbers. YouTube, 2012. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=q_wstDWjnKQ&gt;.

Reid, H. (n.d.). On Mathematics and Music. In Woodpecker. Retrieved February 7, 2014, from http://www.woodpecker.com/writing/essays/math+music.html

Shatner, William, and Leonard Nimoy, perf. Star Trek. YouTube, 2009. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=H20cKjz-bjw&gt;.

Time Signature (2014, February 6). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Time_signature

Wassell, S. R. (2012, March 29). Rational and Irrational Numbers in Music Theory. In docstoc. Retrieved from http://www.docstoc.com/docs/117428973/Rational-and-Irrational-Numbers-in-Music-Theory

Thoughts on 1/7 and other rational numbers (Part 3)

In Part 2 of this series, I discussed the process of converting a fraction into its decimal representation. In this post, I consider the reverse: starting with a decimal representation, and ending with a fraction.

Let me say at the onset that the process I’m about to describe appears to be a dying art. When I show this to my math majors who want to be high school teachers, roughly half have either not seen it before or else have no memory of seeing it before. (As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.) I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators.

So let me describe how I describe this procedure to my students. To begin, suppose that we’re given a repeating decimal like 0.\overline{432} = 0.432432432\dots. How do we change this into a decimal? Let’s call this number x.

I’m now about to do something that, if you don’t know what’s coming next, appears to make no sense. I’m going to multiply x by 1000. Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How would I ever have thought to do that on my own?” To allay these concerns, I explain that this step comes from the patented Bag of Tricks. Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students. Multiplying by 1000 on the next step is absolutely not obvious, unless you happen to know via clairvoyance what’s going to come next.

Anyway, let’s write down x and also 1000x.

1000x = 432.432432\dots

x = 0.432432\dots

Notice that the decimal parts of both x and 1000x are the same. Subtracting, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. To make this more real and believable to them, I then ask them to pop out their calculators to confirm that this actually worked. (Indeed, many students need this confirmation to be psychologically sure that it really did work.)

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Then I ask my students, why did we multiply by 1000? They’ll usually give the correct answer: so that the decimal parts will cancel. My follow-up question is, what should we do if the decimal is 0.\overline{24}? They’ll usually respond that we should multiply by 100 or, in general, by 10^n, where n is the length of the repeating block.

This strategy, of course, works for $0.\overline{142857}$, eventually yielding

0.\overline{142587} = \displaystyle \frac{142857}{999999} = \displaystyle \frac{1}{7}

after cancellation.

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The same procedure works for decimal expansions with a delay, like x = 0.72\overline{3}. This time, I’ll ask them how we should go about changing this into a fraction. I usually get at least one of three responses. I love getting multiple responses, as this gives the students a chance to came the “different” answers, compare the different strategies, and

Answer #1. Multiply x by 1000 since the repeating pattern starts at the 3rd decimal place.

1000x = 723.333\dots

x = 0.7233\dots

\therefore 999x = 722.61

x =\displaystyle\frac{722.61}{999} = \displaystyle\frac{72261}{99900} = \displaystyle \frac{217}{300}

Answer #2. Multiply x by 10 since the repeating block has length 1.

10x = 7.23333\dots

x = 0.7233\dots

\therefore 9x = 6.51

x = \displaystyle \frac{6.51}{9} = \displaystyle\frac{651}{900} = \displaystyle\frac{217}{300}

Answer #3. First multiply x by 100 to get rid of the delay. Then multiply 100 x by an extra 10 since the repeating block has length 1.

1000x = 723.333\dots

100x = 72.33\dots

\therefore 900x = 651

x = \displaystyle\frac{651}{900} = \displaystyle\frac{217}{300}

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The above discussion concerned repeating decimals. For completeness, converting terminating decimals into a fraction is easy. For example,

0.124 = \displaystyle \frac{1}{10} + \frac{2}{100} + \frac{4}{1000} = \displaystyle \frac{124}{1000} = \displaystyle \frac{31}{250}

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One more thought. The concept behind Part 2 of this series shows that a rational number of the form k/n, where both k and n are integers, must have either a terminating decimal expansion or else a repeating decimal expansion (possibly with a delay). In this post, we went the other direction. Therefore, we have the basis for the following theorem.

Theorem. A number x is rational if and only if it has either a terminating decimal expansion or else a repeating decimal expansion.

The contrapositive of this theorem is perhaps intuitively obvious.

Theorem. A number x is irrational if and only if it has a non-terminating and non-repeating decimal expansion.

In my experience, most students absolutely believe both of these theorems. For example, most students believe that \sqrt{2} has a decimal expansion that neither terminates nor repeats. That said, most math majors are surprised to discover that it does take quite a bit of work — like a formal write-up of Parts 2 and 3 of this series — to actually prove this statement from middle-school mathematics.