If you ask a MITS (man in the street, not MIT grad)
what math is about, chances are he´´ll mention numbers.
Not all math is, in fact, about numbers, but some of it is. But what numbers! Mathematicians aren´t content with the numbers everyone knows, nor with what everyone knows about them. They´ve been inventing new kinds of numbers, and finding out new things about them, for thousands of years. Natural numbers, negative numbers, rational and irrational numbers. Imaginary numbers, complex numbers, transfinite numbers and more. If you want
to read more, join me below the fold.
The first numbers to
be discovered, both by people historically and by individuals in
childhood, are called the natural numbers: 1, 2, 3, and so on.People
have known about these for a very long time, well back into
prehistory; and little kids learn to count before they start
school. There´s even some evidence that some animals have some
notion of (at least some) numbers. If any part of math is
‘natural´ it´s counting. One of the ways counting
can be used is to determine how many of something you have. That
leads, naturally enough, to addition. If you have some of something, and you get more, how many have you got? You have to add. Well enough. But what about if you have some of something and lose some?
That´s subtraction, and that leads to our first extension of
the number system, but first, two short digressions about sets and
mathematical operations.
. 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 7th 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
32 = 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 q2 to get
4.So, p2 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.