Finishing the number system...
Since somewhere around 500 BC, beginning with the counting numbers, 1,2,3...up to very many, mathematicians have expanded the concept of number to include negative numbers, zero, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers and the list goes on. What mathematicians do for a living is to continue defining larger and more complicated objects together with rules for manipulating them, similar to the rules for adding and multiplying counting numbers. Here we want to talk about real numbers, imaginary numbers and complex numbers.
The real numbers include the counting numbers (positive integers), fractions (rational numbers), irrational numbers, zero and the negative of all of these. The irrational numbers fill in all the gaps between the fractions so that the real numbers are continuous from minus as large as we can imagine to positive as large as we can imagine. We often represent this
continuum
of real numbers as a line called the real number line.
Every real number has its place on this line. Adding or multiplying real numbers gives us another real number but there are some peculiar rules about multiplying. The product of a negative and positive real is a negative real. The product of two negative reals is a positive real. It is not just some mathematician's whim that dictates these rules but the very nature of real numbers.
As a matter of convenience in writing down very large and very small numbers with lots of zeros in them like
1000000000000000000000000000000000000000000000000000000000000
we take advantage of the fact that each zero added represents multiplication by 10 and write the above monster as 1X10^{60}, where 10^{60} means 10 multiplied by itself 60 times. This trick is sometimes called scientific notation. To avoid the use of superscripts, which most calculators cannot handle, this number may be written 1e60. For very small numbers like
0.0000000000000000000000000000000000000000000000000000000000001
we take advantage of the definition of 10^{1} which is 1/10. Since each zero between the decimal point and the first nonzero digit represents division by 10 we write 1X10^{60} or 1e60.
There is another operation on real numbers called squaring, meaning to multiply a number by itself. The inverse of this operation, called taking the square root, means to find a number which when multiplied by itself gives back the number of which you are taking the square root. Now try to imagine a number which when multiplied by itself gives us 1. According to the rules for multiplication there is no real number which when multiplied by itself gives a negative real. Some folks were probably content to let half the real numbers go without square roots but some mathematicians felt they must have missed something in defining the real numbers. They called the thing they missed "i", the square root of 1, and defined another number line that was i multiplied by a real number line. The numbers inhabiting this new number line were called imaginary numbers. Multiplication by i has the effect of rotating the object i was multiplied by 90 degrees counterclockwise. The real number line, drawn horizontal, and the imaginary number line, drawn vertical, then are perpendicular and define a plane called the complex plane.
Every point on this plane can be identified with a number in the form a+b*i where a and b are real numbers. Numbers consisting of two parts like this, one part real and one part imaginary, are called complex numbers. Real numbers are those where b=0 and imaginary numbers are those where a=0 so all numbers can be considered complex. Normally complex variables, numbers used in mathematical expressions, are symbolized by the letter z, just as real variables are often symbolized by x. The discovery of complex numbers suddenly made many difficult mathematical problems easy. Problems are always harder when you are working with an incomplete tool kit.
