Zero - Charles Seife [83]
Appendix B
The Golden Ratio
Divide a line in two, such that the ratio of the small part to the large part is equal to the ratio of the large part to the whole line. For the sake of simplicity, let’s say that the small part is 1 foot long.
If the small part is 1 foot long, and the large part is x feet long, then the length of the whole line is obviously 1 + x feet long. Putting our relationship into algebra, we find that the ratio of the small part to the large part is
1/x
while the ratio of the large part to the whole thing is
x/(1 + x)
Since the ratio of the small to the large is equal to the ratio of the large to the whole, we can set the two ratios equal to each other, yielding the equation
x/(1 + x) = 1/x
We wish to solve the equation for x, which is the golden ratio. The first step is to multiply both sides by x, which yields
x2/(1 + x) = 1
We then multiply by (1 + x), which gives us the equation
x2 = 1 + x
Subtract 1 + x from both sides, yielding
x2 - x - 1 = 0
Now we can solve the quadratic equation. This gives us two solutions:
Only the first one, with a value of about 1.618, is positive, thus it is the only one that made sense to the Greeks. Thus the golden ratio is approximately 1.618.
Appendix C
The Modern Definition of a Derivative
Nowadays, the derivative is on firm logical grounds, because we define it in terms of limits. The formal definition of a derivative of a function f(x) [which we denote as f'(x)] is
To see how this gets rid of Newton’s dirty trick, let’s look at the same function we used for demonstrating Newton’s fluxions: f(x) = x2 + x + 1. The derivative of this function is equal to
Multiplying out, we get
Now the x2 cancels out the - x2, the x cancels out the - x, and the 1 cancels the - 1, leaving us with
Dividing through by ?—remember, ? is always nonzero because we haven’t taken the limit yet—we get
f'(x) = limit
(as ? goes to 0) of 2x + 1 + ?
Now we take the limit and let ? approach zero. We get
f'(x) = 2x + 1 + 0 = 2x + 1
which is the answer we desired.
It’s a small shift in thinking, but it makes all the difference in the world.
Appendix D
Cantor Enumerates the Rational Numbers
To show that the rational numbers are the same size as the natural numbers, all Cantor had to do was come up with a clever seating pattern. This is precisely what he did.
As you may recall, the rationals are the set of numbers that can be expressed as a/b for some integers a and b (with b nonzero, of course). To start with, let us consider the positive rational numbers.
Imagine a grid of numbers—two number lines crossed at zero, just like the Cartesian coordinate system. Let’s put zero at the origin, and at every other point on that grid, let’s associate with the rational number x/y where x is the point’s x-coordinate and y is the point’s y-coordinate. Since the number lines go off to infinity, every positive combination of xs and ys has a spot on that grid (Figure 58).
Now let’s create a seating chart for the positive rational numbers. For seat one, let’s start at 0 on the grid. Then let’s move to 1/1: that’s seat two. Next, wander over to ½: seat three. Then to 2/1 (which, of course, is the same thing as 2): seat four. Then to 3/1: seat five. We can wander back and forth along the grid, counting off numbers as we go. This yields a seating chart:
Figure 58: An enumeration of the rational numbers
Eventually, all the numbers have a seat; some, in fact, have two seats. It’s easy to remove the duplicates—just skip over them when making the seating chart. The next step is to double the list, adding the negative rationals after the corresponding positive rational. This gives us a seating chart:
Now all the rational numbers—positives, negatives, and zero—have a seat. Since nobody is left standing and no seats are open, this means that the rationals are the same size as the counting numbers.
Appendix E
Make Your Own Wormhole Time Machine
It’s easy—just follow these four simple steps.
Step 1: Build a small wormhole. Both ends will be in