Featured Links: Math with Bad Drawings

Welcome to Featured Links! This is a blog category about interesting math-y stuff offsite.

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Card tricks, disappearing pizza, awkward poetry, and headlines from a parallel universe: what’s not to love? Math with Bad Drawings is a blog about teaching and math, sprinkled with the author’s signature whiteboard doodles.

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The blog has lots of silly posts, like A Math Professor Consults on a Hollywood Movie. Get ready to cringe.

Executive: Let’s say you wanted to get every car in the city to stop, by hacking into the traffic system and turning the lights to red. What equation would a mathematician use?

Professor: That’s not what mathematicians do.

ExecutiveBut if you did, what would you use?

Professor: Traffic cones.

Executive: No, I mean with computers.

And Don’t Let a Mathematician Write Your Political Slogans:

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Some posts are more serious and thought-provoking, like Fistfuls of Sand (or, Why It Pays to Be a Stubborn Teacher). This post made me appreciate my high school teachers in a totally different way.

Teaching often feels like throwing fistfuls of sand into the ocean. You watch the particles spread and sink, a pointless cloud, and you feel like a fool for ever imagining it would go better. You try again the next day, to the same result: nothing. You keep at it, bucket after bucket of sand, and it never seems to make any difference, until one day, you toss in one more cup of sand, and… there it sits on the surface, a tiny newborn island.

Other great teaching posts are Wrong But Not Stupid and The Hard-to-Tell Story of the High School Classroom.

Another interesting post, also published in Slate, is applicable to both the mathematician to the math-phobic. What It Feels Like to Be Bad at Math:

Thanks to a childhood of absurd privilege, I entered college well-prepared. As a sophomore in the weed-out class for Yale math majors, I earned the high score on the final exam. After that, it seemed plausible to me that I’d never fail at anything mathematical.

But senior spring, I ran into Topology. A little like a bicycle running into a tree.

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And of course, there are posts about math itself: check out 0.999…. and the Debate that Repeats Forever.

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Want to see more? Check out the Author’s Choice page or the main site.

Eye-Catching Math: Fibonacci Spirals

Eye-Catching Math is a series about beautiful mathematical images. Read more about it here.

Some patterns ought to never show up in the real world. The sequence of Fibonacci numbers is one such example. Fibonacci numbers periodically appear in popular culture: for example, they are featured in the bestselling novel and film The Da Vinci Code. But it would seem to have nothing to do with the natural world.

The Fibonacci series goes like this:

The first Fibonacci number is 1.

The second Fibonacci number is also 1.

The third number is equal to the first number plus the second number. So the third Fibonacci number is 1+1. In other words, the third Fibonacci number is 2.

It’s not hard. The fourth number is the second plus the third, or 1+2. So the fourth number is 3.

And so on. Each number is the sum of the previous two numbers. So if the eighth number is 13 and the ninth number is 21, the tenth number will be 13+21, or 34.

The beginning of the list of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

You might think this in an interesting sequence to study. Or you might not. Certainly, though, it seems these numbers would never show up in the natural world.

After all, this sequence is just a list of numbers that follow a pattern. The pattern isn’t “number of volts of electricity” or “number of oranges on a tree.” It is generated from pure arithmetic. So it seems it should have no application outside of number theory.

Right?

Fibonacci spiral 34

Nope. Fibonacci numbers are everywhere.

For example, the Fibonacci numbers tell you how to draw a particular type of spiral, called the Golden Spiral. (Each successive box above, starting from the center and working outwards, has sides of length of a Fibonacci number. The box lengths are 1, 1, 2, 3, 5, 8… One can “smooth out” this spiral with a number called the Golden Ratio. This is the limit of the ratio between successive Fibonacci numbers. If you want to learn more about this, look up the Golden Spiral.)

Mysteriously, natural spirals often resemble the Golden Spiral.

Spirally plants often contain a Fibonacci number of spirals, like the 5 spirals in this aloe plant:

Each of these pineapples almost certainly has a Fibonacci number of diagonal spirals.

This pinecone, too, has a Fibonacci number of spirals.

It’s almost impossible to count, but this flower has a Fibonacci pattern too.

For a mind-blowing series on Fibonacci numbers, check out Vi Hart‘s videos on YouTube. It’ll also explain most of the objects in this post.

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Fermat’s Last Theorem Part II: Why Bother?

In the last post we discussed Fermat’s infamous note,

It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers.

I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

Now let’s look at what he was actually saying.

Fermat claimed to have proven something fundamental about numbers. He wrote that for any integer* n greater than or equal to 3, there are no positive integers x, y, and z such that

xn+yn=zn .

*A positive integer is a whole number bigger than 0. The positive integers are 1, 2, 3, … and so on. Some examples of positive integers are 45 and 8. Some examples of numbers that are not positive integers are ½, -3, 0, and the square root of 7.

This means that, for example, no whole numbers x, y and z will make true x3+y3=z3, or x5+y5=z5, or x21+y21=z21.

You can think of the equation xn+yn=zn as a statement that is always false, as long as x, y, z and n are the types of numbers specified by Fermat’s theorem. An analogous English sentence would be “this watermelon is blue.” No matter which watermelon I am talking about, the sentence is not going to be true.

Quick review: xn can be thought of as “x multiplied by itself n times.” For example, 15=1*1*1*1*1, which is equal to 1.  And 32=3*3, which is equal to 9.*

*This is a good way to understand as long as the exponents are positive integers: 1, 2, 3, 4, 5, 6, 7, … and so on. The idea gets a little more complicated if the exponent is a fraction like 1/3, an irrational number like the square root of 2, a negative number, or 0. But Fermat was specifically talking about positive integers greater than or equal to 3. That’s just 3, 4, 5, 6, 7… and so on.

Imagine a slot machine or old-fashioned odometer with slots for x, y, z and n. Spin the x wheel to 3, y wheel to 8, z wheel to 10, and n wheel to 6. That equation is 36+86=106. This is false. It’s easy to check with a calculator. Spin again, and you might get 145+15=95. This is false, too. Try it yourself: pick three positive integers for x, y, and z, and one integer greater than or equal to 3 for n. Plug them all in to Fermat’s theorem.

Is the equation true?

(I’ll bet you a donut it isn’t.)

You may be convinced, after trying enough examples, that there really are no numbers that satisfy the equation.

You could even get a computer to try millions of combinations and check them all. But even with computers, no person has ever found a set of numbers that make true the equation in Fermat’s theorem.

Isn’t that enough? Why did Fermat need proof, if no one can make the equation work?

The need for proof becomes clearer when you consider another possible mathematical claim. In a class with 23 students, there is a fifty-fifty chance that two of them have the same birthday.

Do you think this is true? Take a guess based on your intuition.

Most people think this is false when they first see it. It seems that there should be a much lower chance of two students having the same birthday. Fifty-fifty seems way too likely.

But it is true: the chance that two students have the same birthday is really fifty percent. You can prove it mathematically.

Math is full of weird surprises like this one.  Many numerical patterns that seem to be true end up failing. One such pattern (#3 on this list) works for every number up to 8424432925592889329288197322308900672459420460792433.

One simply cannot guess and check such an example.

This brings us back to our question about Fermat. How do we know for sure that there is no example out there that contradicts Fermat’s theorem? What if, when we try 330, 555 and 9834825 to the 90th power, the equation will suddenly turn out true?

There are infinitely many possibilities. We can’t try all of them.

This is where proof comes in. With a proof, we can say, without a doubt, whether Fermat’s Last Theorem is true or false.

But this brings us to an even more basic, fundamental question about math. It is a simple question. A question asked by many. The question is: Why bother?

It’s easy to use something without understanding it. A person can drive a car without knowing how to build an internal combustion engine. A three-year-old can turn on a TV before he even knows the word “electricity.” An engineer can use calculus to design an airplane without knowing how or why calculus is true.

Most of the time, this is fine. But sometimes, a person wants to understand something. It may be something concrete, like why ice melts. Or it may be more abstract, like Van Gogh’s motivation to paint Starry Night. These questions don’t have practical use: it is totally possible to enjoy a cold drink or a painting without deep understanding. But knowledge itself is something people enjoy.

Mathematics is a rich landscape, full of intricate and beautiful ideas. In the last post, we compared a mathematical topic to a cavern of crystals. A proof is like light shed on a crystal. We may have guessed what the crystal would look like, but only once we illuminate it – that is, once we prove a statement once and for all – can we start to appreciate its beauty.

For many people, this is THE reason to do math. It is an incredible experience to “get” something. It is amazing to suddenly see harmony in a complicated structure. People who love math, for the most part, aren’t problem solvers or number crunchers. They are motivated by the desire to know. And math is full of beautiful things to know.

Fermat’s Last Theorem Part I: A Dead Man’s Margin and the Nature of Proof

In the late seventeenth century, a peculiar mystery began. It is a bit of a favorite, as great math legends go. It’s a classic story math professors love to tell.

A man named Pierre de Fermat jotted something down in the margin of a mathematics book. Translated from the original Latin, it said,

It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers.

I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

This dense-looking paragraph can be written neatly as a mathematical equation. But we’ll discuss the equation in the next post. The point here is about Fermat’s promise.

In short, Fermat thought he could prove the truth of that mathematical statement. Nobody had proven it before. His argument was “truly marvelous”—but, alas, the margin was too narrow to hold it.

Here begins the mystery. Fermat died without ever writing or speaking about this proof again. His son Samuel found the cryptic scribble after his father’s death.

Now you are likely waiting for the hook. What was so sensational here? If you are unfamiliar with mathematical language, Fermat’s note might look no more interesting than, say, a microwave construction manual.

But in terms of the study of abstract patterns, this idea was HUGE.

When the note was revealed, a race began to find Fermat’s “marvelous” proof.  Brilliant mathematicians tried and failed to prove his statement. The problem became known as Fermat’s Last Theorem.

Fermat’s Last Theorem may sound familiar to you. It was in the news relatively recently – just twenty years ago. Fermat died in 1665, but it wasn’t until 1994 that his theorem was finally proved, by British mathematician Andrew Wiles. It was considered a monumental mathematical achievement.

Why?

Or more specifically, what was it that Fermat failed to accomplish?

The key word here is “proof.”

In everyday life, the word “proof” is somewhat ambiguous. Usually it means “enough evidence to convince people of something.” A ten-year-old can “prove” she is athletic by shooting a basketball from mid-court. A defendant in the courtroom can “prove” his innocence by providing a valid alibi. A scientist or historian can present a theory and “prove” it with experiments or artifacts.

In mathematics, on the other hand, there is only one kind of proof—and it is of a completely different nature. A mathematical proof is an argument that follows the rules of logic perfectly. And because it follows the rules of logic perfectly, it is literally impossible to refute. If I prove something mathematically today, then in a thousand years, my argument will still be unbreakable.

This kind of proof is unique to mathematics. In the scientific kind of proof, theories are always at risk, because new information can come along and contradict the old. But math is built from the bottom up. If done properly, it is unbreakable, no matter what future scientists discover.

Its strength comes from its style of reasoning. Math begins with basic assumptions and follows through with the logical consequences of those assumptions. The assumptions are called “axioms.”

Axioms and logic together are the foundation of every mathematical proof. No test tubes, or field samples, or even apples dropping from a tree—just axioms and logic. With just axioms and logic, any true mathematical idea can be formulated and proven.

The name “axiom” sounds formidable, but the ideas are simple. Axioms are un-splittable kernel ideas. They are agreed upon by everyone. One axiom might be, “between any two points, a line can be drawn.”* This axiom may seem pretty obvious. And it is. Axioms are meant to be as simple as possible. This is how they form the framework of a mathematical field.

After axioms are chosen, the framework is set. Then there is only one way forward: logical reasoning.

Mathematicians who are building proofs start by make deductions from the axioms (or, in practice, by work that was already built up from the axioms). Usually, the logical deductions that come straight from the axioms are nearly as simple as the axioms themselves. Those deductions, in turn, can lead to slightly more complex conclusions. In some ways it is simple, like hammering nails into wood to build a house. But in other ways, it is wide open for creativity: the mathematician is creating the blueprint as he builds. Step by step, logical reasoning gives rise to an entire mathematical theory. By the time we get to full-fledged concepts like spheres or integration, axioms are usually long out of sight.

* As a side note, there is one type of concept that is arguably more basic than an axiom: the objects that the axiom talks about. One must assume the existence of points and lines to even begin discussing, for example, geometry. To do arithmetic, one needs to assume the existence of addition and multiplication. Whereas a triangle can be defined in terms of points, lines and angles, a point cannot be defined in simpler mathematical terms. And whereas exponents can be defined in terms of addition and multiplication, addition and multiplication cannot be defined in simpler mathematical terms.

The beauty of this logical structure is that it lies outside subjective experience. Once the axioms are in place, the rules of logic determine the conclusions. One might imagine each mathematical field as a cavern of undiscovered crystals. The cavern is too dark to see. But the crystals are there. Once we have chosen the cavern we wish to explore, we cannot change what the crystals inside it will look like. We can only shed light on the crystals that are already there.

Logic, of course, warrants its own discussion. But here, the important thing is that any two people applying logic to the same idea will never contradict each other, even if they take different particular steps.

Math can be difficult. It can be complex. It can lead to surprising conclusions or even be completely counterintuitive.

But because every step in each theory was taken carefully and logically, we know that no flaw that will undermine the steps the mathematician has taken. No contradictions will be deduced later. No input from the physical world will change it. All of the discoveries within a mathematical field are consistent with each other– and sometimes consistent with entirely different fields as well. The crystals that form in the caverns all sparkle with the same facets, because they all grew through the same process.

When a person sees the consistent ideas grow through pure logic, the effect is, for many, indescribable. It is a pleasure and a joy. It is one of the main reasons people study mathematics.

And proofs are the way to get there.

Fermat promised an idea tantalizing to those who appreciate proofs—a “marvelous” bit of reasoning that would prove the conjecture he’d written. We’ll discuss Fermat’s Last Theorem more in the next post.

Read Part II >> 

Resources:
Wolfram MathWorld
MacTutor