This is what Theoni Pappas writes on page 110 of her Magic of Mathematics:
"Some ancient scribe had drawn circles with different sized radii. The scribe took the diameter (by doubling the radius) of each circle. And just for kicks, decided to wrap each circle's diameter around it."
Here pi is in the middle of telling the story of his discovery. Pappas writes that several cultures around the world independently discovered the concept of pi.
Then Pappas tells us of the discovery that pi isn't merely irrational, but transcendental. This means that pi isn't algebraic -- there exists no polynomial equation f, with integer (or rational) coefficients, such that f (pi) = 0.
This actually completes the indirect proof that pi is not constructible. Proof: assume that pi is constructible (via ruler and compass). But all constructible numbers are algebraic -- a contradiction since pi is transcendental. Therefore pi is not constructible -- it's impossible to square the circle. QED
Note: This spring break post has been edited so that I can write about Eugenia Cheng's new book, Beyond Infinity. The "Eugenia Cheng" label has been added to this post. Today I selected a random chapter to discuss.
Chapter 16 of Eugenia Cheng's Beyond Infinity is called "Weirdness." Here's how she begins:
"Before we end, here are some weird facts that come out of our new understanding of the infinitely small. Infinite things and finite things start getting mixed up in strange ways."
Back at my old school, a few students from time to time would call me "weird." Usually, this was when I tried to enforce an unpopular rule such as "no eating in the classroom" or "no phones." The implication was that only weird teachers enforce unpopular rules -- "non-weird" teachers let their students do whatever they want.
My response was, "If enforcing rules makes me weird, then fine, I'm weird!" And because I showed the students that I had no problem with being called "weird," they stopped using that word in their arguments with me after a few times.
After all, what's wrong with being "weird"? Notice that a person can rightfully call me "weird" for singing songs in class, for what sort of math teacher does that? But then again, no student ever called me "weird" for singing in class, or for doing anything abnormal yet enjoyable. The word "weird" only turned up when doing something normal teachers do -- enforce rules. It's as if "You're weird!" was actually code for "I don't want to!" (Over the next few months, I'll discuss this idea in more detail here on the blog, since it relates to why I ended up leaving the classroom this year.)
Cheng is telling us that infinity is weird, too -- but again, what's wrong with being weird? And so she starts out:
"First of all, here's how to make an infinite number of cookies with a finite amount of cookie dough."
In addition to being a mathematician, Cheng is a cook. She could have said, "here's how infinitely many numbers can add up to a finite number." But ever since her first book How to Bake Pi, she likes to refer to food and cooking as often as possible.
In her first example, the first cookie has a mass of 80 g, the second one 40 g, the third 20 g, the fourth 10 g, the fifth 5 g, and so on. Each cookie has half the mass of the previous cookie. Then there are infinitely many cookies, yet the grand total is only 160 grams.
In the second example, the first two cookies are 80 g and 40 g again. But this time, the third cookie has a mass of 26 2/3, or one-third the original cookie. The fourth cookie is 20 g, or one-fourth the original cookie. The fifth cookie is 16 g, or one-fifth the original cookie, and so on. This time, the grand total of all the cookies really is infinite.
In Cheng's final example, the numbers 80, 40, 26 2/3, 20, 16, and so on aren't the masses of the cookies -- now they're the radii. She tells us that now the grand total of the amount of dough used is finite -- yet if you line all these cookies up in a long line, they will stretch an infinite distance. Indeed, she explains:
"The progression of numbers that we're thinking about is 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...and is called the harmonic series. It is related to harmonics in music -- these are the wavelengths that give the harmonics of a note, expressed as ratios of the main note."
Pappas writes about harmonics and music in her own book as well -- and we'll catch up to those Pappas pages during the summer. I'll only write a little about harmonics and music from Cheng's book now, and then save the rest for Pappas. Cheng writes:
"On a violin if you play the G string without putting any fingers down, you get the note G. If you touch the string halfway up, the harmonic you get is the G an octave higher than the main G. If you touch it a third of the way up you get the D an octave and a fifth above the original G string. You get the next harmonics by touching the string a quarter of the way up, and then a fifth, and then a sixth."
She adds that it's possible to do the same on a cello, and I know that it's possible to play harmonics on a guitar as well. But harmonics aren't just for string instruments -- brass instruments play them too:
"The main note of a brass instrument is more likely to be a B-flat, but then the harmonics are at the same intervals above the main note: an octave, then a fifth above that, and so on."
The important idea here is that the sequence 1/2, 1/3, 1/4, and so on is the harmonic series. And the harmonic series actually diverges -- that is, its sum is infinite. Cheng demonstrates this by adding up the first few terms:
-- Let's see if we can make it past 1, unlike the first cookie example where we eat half the leftovers each time and never quite make it to eating all the dough until "forever," when we about get there:
1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12
which is bigger than 1.
-- Let's see if we can make it past 2:
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 > 2.
-- Now that Cheng has resorted to using a spreadsheet, let's see how long it takes to get past 5. It gets really slow but does eventually happen at 1/227.
But she admits that it takes too long, even with a spreadsheet, for the sum to exceed 10. So instead, she employs a method that she calls both sneaky and clever:
1. The first clump of fractions has just 1/2 in it, the first thing in the harmonic series.
2. The second clump has the next two fractions, 1/3 and 1/4.
3. The third clump has twice as many fractions as the previous clump: the next four fractions, that is,
1/5, 1/6, 1/7, 1/8.
4. The fourth clump has yet twice as many fractions:
1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16.
So to sum up, we have clumps like this:
1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16)...
Now Cheng proceeds to estimate the sum of each clump. To do this, she takes the smallest fraction in each clump and multiplies it by the number of fractions in the clump. Since she always takes the smallest fraction, this is guaranteed to be an underestimate.
The first clump, of course, is exactly 1/2. For the second clump, we have:
1/3 + 1/4 > 1/4 + 1/4 = 1/2
Here she points out that this is a over-the-top way to show that the sum exceeds 1/2. But she explains that the purpose of this is to generalize all the way to infinity, while adding the fractions with a common denominator or using a spreadsheet don't generalize. Yes, "Techniques that generalize well often look silly when you apply them to basic examples. It's one of the reasons that learning math is hard, because often all the basic examples make the math look silly or pointless or contrived." These words from Cheng are something I'll keep in mind if I ever return to being a math teacher.
Let's keep going and take the third clump:
1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 4/8 = 1/2
Now look at the fourth clump:
1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 > 8/16 = 1/2
We can go on like this forever. Actually, Cheng explains: "A better way of saying 'keep going forever' or 'and so on' is to say what happens for the nth clump, where n could be anything. The nth clump has 2^(n - 1) terms and ends at 1/2^n. Each term is greater than or equal to 1/2^n and so the total is greater than 2^(n - 1)/2^n = 1/2, as we claimed."
And as there are infinitely many clumps and each clump exceeds 1/2, the sum is infinite. Therefore the harmonic series diverges. In our second cookie example, the amount of dough used follows the harmonic series and so is infinite.
"We can draw the harmonic series as a sort of bar chart. Here each bar has width 1, so the area of each rectangle is the width times the height, which is 1/n for each n."
At this point she is trying to link the sum of the harmonic series to the area under the curve y = 1/x, which between x = 1 and x = b is ln b. In fact, she adds:
"This way of fitting a 'bar chart' snugly under a graph is at the heart of finding the area under a curve. This is otherwise known as 'integration.'"
Yes, this is exactly the integral that we teach students in AP Calculus classes. In fact, returning to the previous example, Cheng points out that some texts define ln b to be the area under the curve -- and I recall that my old Calculus text did exactly that.
She explains that using bars to estimate an area under a curve is the general idea behind the Riemann integral -- namely the integral that AP Calculus students are expected to learn. She writes:
"When you learn integration in school, you usually learn it as some kind of nifty formula so that if you're asked to integrate a particular function, you manipulate the function and know what the answer is. But the reason that works is this process of chopping the graph into infinitesimally small bars."
I don't wish to spend too much time discussing the rest of this chapter, since so much of it depends on the many diagrams Cheng includes here, but which I'm not posting. But I do quote the following:
"We'll now go back to that last, most weird cookie conundrum, where we had an infinite number of cookies made from a finite amount of dough, and yet the cookies stretched out to infinity when we lined them up. How is that possible?"
Here she's trying to explain that while the harmonic series diverges, the series of squares 1/4, 1/9, 1/16, 1/25, and so on converges to a finite amount. She actually shows this by squeezing squares of side 1/2, 1/3, 1/4, 1/5, and so on into a square of side 1. Not only do infinitely many small squares fit in the large square, but we can leave out the upper right 1/4 of the square and they still fit. (Again, this is better seen in a diagram than read.) The total sum works out to be pi^2/6 - 1 -- which, as we've seen above in Pappas, is transcendental, not merely irrational. (Note: Usually the sum is written starting with 1, and so that sum is pi^2/6, without the "-1" term.)
Cheng's final example in this chapter is:
"The weird situation with the cookies is related to a very strange and slightly more mathematical example: the fact that you can take an infinite area, rotate it through the air, and still only have swept out a finite volume."
Of course, I won't explain it fully here. But let's skip to the end of this chapter:
"It's the joy of infinity. The real joy, to me, is understanding the logic behind it, and seeing how to deal with these things with mathematical rigor."
And this applies not just to infinity, but to Geometry as well. In Geometry, we rigorously demonstrate how the inequalities in triangles are related, such as the Triangle Exterior Angle Inequality and the Unequal Sides Theorem:
Unequal Sides Theorem (Triangle Side-Angle Inequality, TSAI):
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.
Given: Triangle ABC with BA > BC
Prove: angle C > angle A
1. Triangle ABC with BA > BC 1. Given
2. Identify point C' on ray BA 2. On a ray, there is exactly one point at a given distance from
with BC' = BC an endpoint.
3. angle 1 = angle 2 3. Isosceles Triangle Theorem
4. angle 2 > angle A 4. Exterior Angle Inequality (with triangle CC'A)
5. angle 1 > angle A 5. Substitution (step 3 into step 4)
6. angle 1 + angle 3 = angle BCA 6. Angle Addition Postulate
7. angle BCA > angle 1 7. Equation to Inequality Property
8. angle BCA > angle A 8. Transitive Property of Inequality (steps 5 and 7)
The next theorem is proved only informally in the U of Chicago. The informal discussion leads to an indirect proof.
Unequal Angles Theorem (Triangle Angle-Side Inequality, TASI):
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
The contrapositive of the Isosceles Triangle Theorem is: If two angles in a triangle are not congruent, then sides opposite them are not congruent. But which side is opposite the larger angle? Because of the Unequal Sides Theorem, the larger side cannot be opposite the smaller angle. All possibilities but one have been ruled out. The larger side must be opposite the larger angle. QED
Eugenia Cheng's Beyond Infinity was a highly enjoyable read. I definitely recommend it to anyone who is interested in learning about infinity -- as well as how mathematics works in general.