Numbers

A VERY INACCURATE HISTORY OF THE EVOLUTION OF MATHEMATICS

When men were savages living in jungles, they gradually acquired the ability to communicate by speaking. They gave names to things - trees, lakes, spiders, elephants - and to each other - Joe, Mike, Dave - and through this new-found ability they were able to improve the quality of their lives. They eventually got around to naming numbers - two antelopes, four geese, etc. As the number-names filled in, they became the FIELD of positive integers.

After a few decades - or millenia - it doesn't really matter - they learned to add. Three hunters here and four there - seven! And eventually they learned that no matter what two numbers they added, they always found a number as the sum. A modern mathematician would say the field of positive integers is closed under the operation of addition.

Then someone planned dinner for twelve and found she had only seven steaks. Quick, Oswald, more steaks. How many? Tricky, but at that point subtraction had to be invented. And Oswald, who liked to dig into things, tried to determine whether any number could be subtracted from any other. Of course he got stuck when the number being subtracted was at least as big as the original number. He complained to the brilliant Esmeralda about the limitation, and she solved the problem - by inventing more numbers! If zero and negative whole numbers - integers - are added to the field of positive numbers, any number can be subtracted from any other number. The field of integers is closed under the operations of addition AND subtraction. And Oswald and Esmeralda lived happily ever after.

Things went along idyllically (new? - look it up!) until Jedediah got tired of adding three and three and three and three. He thought about four threes and invented multiplication. And, like Esmeralda, he explored the possibilities, and found that indeed any two numbers could be multiplied to produce a number. The field of integers is closed under multiplication.

Idyllic times resumed untl one day Hepzibah wondered how many threes she needed to make twelve. Her multiplication table provided the answer with no trouble. But then she tried to find out how many threes were needed to make thirteen. No answer was avalable. That's when Simeon stepped up to the plate and invented fractions, such as 13/3. The analogy to subtraction as a reversal of addition led to the understanding that Simeon had invented a new operation, which soon became known as division. It took several centuries for the budding profession of mathematician to determine that the field of fractions (or "rational" numbers, since they are always the ratios of pairs of integers) is closed under all four operations: addition, subtraction, multiplication and division.

While all the above inventions were being made, there were of course many others: fire was discovered, clothing and shelter were found, adapted or made, and many new words kept making themselves. A and B became A plus B or A added to B. C take-away A became C minus A. Records were invented: piles of stones or scratches on trees were given specific meanings. Gradually, more and more information could be saved that way. People learned to remove pieces of bark from trees, and to make marks on them.. As the markings grew more complex, symbols were devised, among other things, for numbers and for operations that combined numbers.

Along with the records came the understanding that operations on numbers could follow one another: A could be added to B and the result could be multiplied by C.

They learned other things about numbers: A plus B and B plus A are equal. That becomes obvious if we look at a few dots. Turning the paper upside down reverses the order of the parts, but doesn't change the total number of dots.

2 + 3 = 3 + 2

Similarly, A x B is equal to B x A. Two rows of three dots each can be turned sideways to show three rows of two dots each.

2 x 3 = 3 x 2

We now say that addition and multiplication are commutative: we can commute, or interchange, numbers without affecting results. That's untrue for the reverse operations of subtraction and division:

A - B does not equal B - A, and A/B does not equal B/A.

Someone discovered that if a multiplier is applied to a sum, the result is the same as that for applying the multipler to each part of the sum and adding the results. The multiplier can be distributed among the parts of the sum: hence the "distributive" law. Restating the rule,

A x (B + C) = AxB + AxC.

Again, dots make the rule obvious.

2 x (3 + 4) = 2 x 3 + 2 x 4.

Another genius discovered another important rule:

A + (B + C) = (A + B) + C.

That is, if three numbers are to be added, it doesn't matter whether the first two are added and the result is added to the third or the if first is added to the sum of the second and third.

2 + (3 + 4) = (2 + 3) + 4

It doesn't matter which pair is associated first: hence the associative law.

A universe of numbers, positive and negative fractions and integers, four kinds of operations, and three "laws" makes an awful lot on of inventing for our jungle inhabitants. I'd guess they waited a couple of generations before continuing. But where is there to continue to?

Well, consider that multiplication started with repeated addition - three plus three plus three plus three was simplified to four threes, or four times three. What happens with repeated multiplication? The answer is whole new worlds!

Integers multiplied a whole number of times pose no problems: 3 x 3 x 3 x 3 can be given a simple name - 3 to the 4th power - and symbol - 3 ^ 4, where 3 can be called a base and 4 is an exponent. Further, negative or rational numbers in the base works just fine. The results are still numbers within our previously defined field of rational numbers, positive and negative.

But as soon as we try anything other than a positive integer for the exponent we're in trouble. What does it mean to multiply three by itself 2.3 times, or minus 2 times? The answers come by extending some simple observations.

Using only positive integer exponents, we see, for example that

2^3 x 2^4 = 2x2x2 x 2x2x2x2 = 2^7

and that in general when two exponentials with the same base are multiplied, the exponents add. If we define exponents other than positive integers as being such that the addition rule always applies, then 2^1/2 x 2^1/2 must be 2^1 or just 2. 2^1/2 must therefore be the square root of 2! Simlarly, 2^1/3 must be the cube root of 2.

Note that 2^4 is twice 2^3, so 2^3 is 1/2 of 2^4. Further, 2^2 is 1/2 of 2^3, 2^1 is 1/2 of 2^2, and by extension 2^0 is 1/2 of 2^1, or 1. Similar argument lead to the conclusion that x^0 = 1independently of the value of x! and x^-n must always be 1/x^n. Note that the square root of 2 (or 3 or 5 or 7 . . .) is not the ratio of any two whole numbers - is not a rational number. Numbers of that class are called, rationally enough, irrational.

But we're not yet out of the woods. We've introduced square roots, and they can readily be found for positive numbers, but what of the square roots of negative numbers? As we've seen, each time an operation called for numbers that didn't exist in the field of numbers under discussion, the field was extended: from positive integers to signed integers to rational numbers to numbers that were rational or irrational. The square root of -1 is not a member of that last field, so again the field is extended. It is given the symbol i, and its definition is simply "the square root of -1". This bothered some people enough so that they insisted on calling it the imaginary unit. I personally feel that's unfortunate, because it's just a number, arrived at by the same kind of logical process that led to all our earlier kinds of numbers. Anyway, we now have a field of numbers that is closed under the operations of addition, subtraction, multiplication, division and exponentiation. Note that it takes two quantities to identify a number which has both "real" and "imaginary" parts and is therefore called complex.

We interrupt to try a few examples of operating on complex numbers.

Recall that previous extensions of the field of numbers weere introduced as a result of inverse operations: negative numbers arose with subtraction, and rational numbers came from division. Here we have both irrational and complex numbers arising from exponentiation, and we haven't even mentioned trying to reverse exponentiation. First we note that addition is commutative, and therefore has only one inverse - subtraction. It makes no difference whether we ask a plus what makes b, or what plus a makes b. Similarly, because multiplication is commutative, it has only one inverse. One can ask what times a gives b or a times what gives b.

But exponentiation is not commutative. a^b is not equal to b^a. 2^3 is 8 and 3^2 is 9. It matters whether we ask what base to the power a gives b or a to what power gives b. For example, what number to the third power gives 8. An answer is 2, because 2^3 = 8 (note that I was careful to say AN answer, not THE answer). We say that 2 is a third (or cube) root of 8 The alternative question is to what power must 2 be raised to get 8. The answer, 3, is called the logarithm of 8, base 2. I believe that roots of rational numbers are called algebraic numbers, which form a limited class. Non-algebraic numbers are called transcendental, and I believe that logarithms soon lead to them.

If those poor savages had only known what they were getting into!