Wednesday, November 15, 2006

Real numbers – the slippery slope to insanity

Let’s build real numbers from scratch using the Dedekind cut. Use the square root of 2 as an example. Does √2 exist? Specifically, is there an x that can satisfy the equation x^2 = 2 (x squared = 2)?

As the Ancient Greeks already knew, x is definitely not a rational number (fractions and whole numbers). With an understanding that extends only to the rational numbers, how can x be created using and building from existing knowledge?

To perform the Dedekind cut, group all the rational numbers into two sets a and b as follows:
a = {x : x^2}
b = {x : x^2 > 2}

The set a contains all the rational numbers that are less than two when squared; b is the opposite. A few of the elements in a and b are:
a = {1.03, 0.2432, -1.2, …}
b = {3.5, 2.72, -342.22, …}

I must stress this again: all elements in a and b are rational numbers. We currently trying to construct the concept of real numbers from rational numbers.

Interestingly, there is no largest element in a. Give my any element in a and I can find you another element (still in a) that is larger than yours, despite the constraint imposed upon by the rule that all elements in a must obey x^2 a is constructed (it only has elements that squares to less than two), all elements in a will never exceed a certain value. For example, the elements in a will definitely not exceed 19, because 19 squared is 361, definitely greater than 2. We say that 19 is an upper bound of a. No element in a exceeds 19.

But 19 is a very boring upper bound for a. It is like saying "the population of China is definitely less than 27 trillion." Duh.

We can use a smaller value for the upper bound of a, maybe 12. 12 squared is 144, which is greater than 2. For sure, no element in a will be greater than 12.

The upper bound of a can be made smaller and smaller, using 10, 7.4, 5.4, 3.5, 1.5…

And here comes the crucial step: the least upper bound of a is the square root of 2. From two groups of rational numbers, we have conjured a real number.

Recall that there is no largest element of a. Similarly, there is no smallest element of b (ignoring the signs). Like in the previous illustration, we can write a list of elements of b, all of them decreasing and there would never be a smallest element.
5.4
2.54
2.1
1.5
1.46
1.4149
1.4142135626

Interestingly, the least upper bound of a does not lie in a and does not lie in b (after all, a and b can only contain rational numbers). The definitions of sets a and b leave an infinitesimally small gap between them, a gap that can only fit one (real) number- √2.

And that’s all there is to creating real numbers from rationals. It probably feels a little inadequate, a little unsatisfying- as if the real number was not properly created. But bear in mind that the Dedekind cut is merely creating the concept of real numbers. It tells us that the square root of two is not some dodgy product of black magic, but is in fact a very real number.

What the Dedekind cut does not do is tell us the value of √2. It cannot do that. Nothing can- our only understanding of √2 is that it satisfies the equation x^2=2.

We can make approximations (x ≈ 1.4142135624 ), we can generate algorithms to tell us the 96 millionth digit in √2, but we can never know the exact value of √2- it has infinitely many digits.



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