Okay, enough talk about studying math. I want to talk about math. I've been studying real analysis recently, and so far the discussion has been on the properties of real numbers.
I mentioned that the course notes started with the natural numbers and addition, and proceeded to derive multiplication and exponents. The notes also develop zero and negative numbers before getting to rational numbers. The textbook skips straight to rational numbers and I will too.
A rational number is any number written as a fraction where both the numerator and the denominator are integers. In this conceptualization, the natural number 2 is a different concept than the rational number 2/1, even though they have all the same properties and can be used interchangeably in most contexts. Additionally, 1/2 is a different number than 2/4, even though they are equivalent.
One sometimes interesting, sometimes frustrating thing about the project to start over and redefine everything from scratch is that you have to revisit lots of basic math. In the early stages, this involves things which are totally internalized. I know that 1/2 = 2/4, but what does that mean in the context of this definition?When trying to develop ideas from the ground up, I end up second guessing myself frequently. Just because I "know" something is true doesn't mean I've proved it. Sometimes I embrace the challenge of proving everything, and other times I want to say, "I know this is true. Why can't I just say so and move on?"
Anyway, once the properties of rational numbers are established, the next step is to prove that the square root of 2 is not a rational number. I've seen this proof many times, but it took multiple times before it really sank in, so I will restate it here. Assume that x is a rational number such that x2 = 2. Then x can be written as a fraction, p/q, where both p and q are integers. Many different values for p and q can be chosen to give the same equivalent fraction, so pick p and q so that at most one is an even number. (If both are even, just divide both by 2 as many times as necessary until one is odd.) Then (p/q)2 = 2, so p2 = 2q2. Therefore p2 is even, so p is also even. But that means we can choose the integer m so 2m = p. Substituting back into the earlier equation, (2m)2 = 2q2, and by the same reasoning, q is also even. But we chose p and q so that at most one of the two numbers is even, and this contradiction means p and q do not exist, so the square root of 2 is not rational.
Now that we've shown that the square root of 2 is not a rational number, it's reasonable to ask what kind of number the square root of 2 is. I will go into the problems that the square root of 2 causes for the rational numbers and why the real numbers are the solution next time.