. The notion of a set is basic in math. A set is a collection of objects. It could be any kind of objects: The set of presidents with IQs lower than 100. The set of people in my household, and so on. A set can be empty, that is, have no members (e.g., the set of people named Bush who I have voted for). It can also be infinite (we´ll get into a little more detail on that later).Sets can be specified by list all the members (e.g. {Tom, Jill, Dave}), or by describing the characteristics they have in a way that is clear.Sets are usually written enclosed in curly braces {}, but I´m not going to be very formal about that stuff here. While sets can be anything, in this diary we´ll be concerned about sets of numbers of one sort or another.

Once you have a set of numbers, you can do things to them. These things are called operations, and, since we´re concerned with math, we´ll be talking about mathematical operations. Mostly, if not entirely, we´ll be talking about mathematical operations that take two objects and return a third.Examples of mathematical operations are addition, subtraction, and so on. When a mathematical operation on two elements of a set always returns another object on that set, that operation is said to be closed on that set.This will be clearer with an example or two, given below.

. The operation of addition
is closed on the set of natural numbers. If you add two natural
numbers, you always get another natural number.3 + 2 = 5. You can
be sure of this, even without doing the actual computation: 1,223,
871 + 2,102,876? I don´t know the answer, but it will be
another natural number. So, as long as we stick with counting and
addition, we´re fine with just the natural numbers.
Subtraction, though, is a different story. Sometimes the result is
another natural number: 7 –– 5 = 2. But sometimes it
isn´t.What is 5 – 7? Hmmmm. What about 5 –
5?Neither answer is part of the natural numbers.For a very long
time, mathematicians thought these were nonsense problems. The first
person to deal with them sensibly at all appears to have been
Brahmagupta, in India, in the 7^{th} century CE; who may
also have been the first to use 0 as a solution to an equation. The
Greeks, amazing though they were, didn´t get it. So, we have
now expanded the number system from the natural numbers to the
integers. But we aren´t done.

The next operation to consider is probably multiplication. But that doesn´t pose any threat to the integers; the integers are closed under multiplication. For instance -5*-2 = +10. But once you have the idea of multiplication as repeated addition, it´s natural to think of division, or repeated subtraction, and the integers are not closed under division. For example 5 divided 3 is not an integer. 1 is too little, 2 is too big. For this, we need fractions, also known as rational numbers, because they are ratios. Now we are OK with division. But those mathematicians never leave well enough alone. Multiplication is repeated addition. What, they supposed, would repeated multiplication be like? No problem. They invented exponents. These are written as superscripts, where the superscript indicates how many times to multiply something by itself.

Still, we are okay, as long as we stick to
exponents that are positive integers. But, of course, we don't stop
there. Since we know that we can take any number and raise it to a
power, it´s natural to wonder what number raised to some
power, would equal a number. For example, we know that
3^{2} = 9, which means that, if we want to know what number,
squared, equals 9, the answer is 3. These are called square roots
(root is a synonym, more or less, for solution). They are written
as fractional exponents.So

So far so good. But what about, say, the square root of 2? Well, it´s not an integer, because 1 is too small, and 2 is too big. But it´s not a fraction, either, and the proof of that is pretty neat. It´s another reductio ad absurdum proof.That is, we start off by assuming that what we want to show is wrong, then we deduce something absurd from that assumption that :

1. Start with the assumption , we can also assume that p and q have no common factor, because, if they do, we can simply divide it out (e.g. 6/12 = ½ by dividing by 6).

2. Square both sides to get

3. Multiply both sides by q^{2} to get

4.So, p^{2} is even.(It is double some other
number, so it must be even)

5. So p is even (if you square an odd number, you get an odd number, if you square an even number, you get an even number).

6.q is odd. (p is even, so if q were also even, they would have a common factor of 2, but we assumed no common factor)

7. Since p is even we can set it equal to 2r

8. Substitute p = 2r into the equation in step 3 to get

9. Multiply out to get

10. Divide by 2 to get

11. But that means q is even (see step 5) and we already showed it was odd.

QED

This REALLY bothered the Greeks when they figured it out. They swore everyone who knew it to secrecy, and there´s even supposition that, when one guy blabbed, he was killed. (They took math seriously in those days).

So far, all the numbers we´ve dealt with have been the solution to algebraic equations. For instance, the square root of 2 is the solution to But there are still MORE numbers. There are, in fact, lots of other numbers. There are the transcendental numbers, like π; there are the imaginary and complex numbers; and there are the transfinite numbers. But this is long enough.I´ll talk about those in other diaries.

Sources:I used material from many sources, but the two biggest were Calvin C. Clawson´s Mathematical Mysteries and John Allen Paulos´ Beyond Numeracy. I recommend both.

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