Number Systems

The oldest and most elementary number system is the set of natural numbers (also known as counting numbers)

N = {1, 2, 3, …}

The natural numbers are constructed from the first natural number 1 by successively adding 1 each time:

1

2 = 1 + 1

3 = 2 + 1

.

.

.

The next natural number after N is then N + 1. . There is no largest natural number. Natural numbers are closed under addition and multiplication, so that

In words, if m and n are natural numbers, so are their sum and product. The natural numbers are (in general, there are exceptions) not closed under the operations of subtraction, division nor exponentiation.

For any given natural number m, there is a natural number that is larger than m, thus the natural numbers are unbounded above. The natural numbers are ordered, so that for any two natural numbers m and n, only one of m < n and n < m is true.

Functions of the natural numbers are called sequences. A sequence assigns a value (natural, integer, real, complex or otherwise) for each natural number (or each element of a subset of the natural numbers). For example,

assigns the square of the number to each natural number. A sequence can be understood as a set, so the present example could be written

{1, 4, 9, …}

More on sequences elsewhere.

The natural numbers have several limitations as mentioned above. If we allow zero and the negatives of natural numbers into a new set, we get the integers

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

Allowing these new numbers closes the set under subtraction, and allows every integer to have an additive inverse

The integers form a superset of the natural numbers. There is neither a largest nor a smallest integer, and so the set of all integers is both unbounded above and unbounded below and so is unbounded. Like the natural numbers, the integers are ordered and closed under addition and multiplication. Sequences defined on the integers are possible, but of less interest than those defined on the natural numbers.

The rational numbers are all numbers of the form

in other words, the set of all ratios of integers. We take ratios that are equivalent (like 1/2 and 2/4) to be the same rational number. For any two distinct rational numbers p and q, only one of p < q and q < p is true, so the rationals are ordered. The rational numbers are closed under addition, multiplication, subtraction and division (provided 0 is not in the denominator). For any two distinct rational numbers p and q, there is a rational number strictly between them (in fact, there are infinitely many).

There is one big problem with rational numbers. There are "holes" in the set of rational numbers, so that equations like

have no rational solution. There are rational numbers arbitrarily close to a solution, but no exact answers. The set of all numbers that form the solution of polynomials is called the algebraic numbers.  Even adding the algebraic numbers to the rationals is not enough to make a continuum, in other words, an unbroken, complete number line. That continuum is called the real number system.  We are going to avoid a precise construction of the real numbers, since that is normally a subject of advanced math courses (and does very little to get either good grades or solve practical math problems).  Suffice to say that filling in all the holes in the rational numbers (with the irrational numbers) completes the continuum.  Once complete, many very nice things happen to the number system.  For one thing, every bounded infinite sequence of numbers has at least one limit point (in ordinary language, when a sequence is defined so that infinitely many sequence numbers fall into a finite section of the real line, there is at least one real number which is the limit of an infinite number of them).   Many polynomials have real solutions but not rational ones.

Some polynomials don't have even real solutions. For example,

has no real solutions. If we try to solve it anyway,

The square root of -1 is not a real number. It is given a special name, the imaginary unit (i). Numbers that include the imaginary unit are called complex numbers. A generic complex number has the form

where a is called the real part of the complex number and b is called the imaginary part. With the complex numbers, we finally have an algebraically complete set. The fundamental theorem of algebra states that every nth degree polynomial (with whole number powers only) has n complex solutions. In symbols, the equation

has a representation of the form

with the numbers

the zeros or roots of the polynomial. This guarantees that all algebra problems with complex numbers have a meaningful solution (i.e. that some complex number is the answer).

The reader might wonder why we should worry about all these different number systems. As it turns out, knowing something about complex numbers sheds a little light on real numbers (particularly when it comes to sines and cosines). Knowing something about real numbers sheds a little light about rational numbers, and so on. It would be very hard to do engineering without understanding an irrational number, pi:

Likewise, it would be hard to calculate continuously compounded interest, design atomic bombs, operate nuclear power plans, design electric circuits, and a host of other, very practical, everyday problems without understanding another irrational number, e:

Modern chemistry and physics depend very heavily on complex number theory, and so does modern technology, such as the computer with which you are now reading this document. The language of science is fundamentally mathematical, and requires familiarity with all the number systems we have discussed here. Most technical subjects are technical because of an underlying dependence on mathematics. The real world assumes a basic understanding of math to do science, engineering, computers, medicine, biology and business. Even gambling requires math if one wishes to win, since figuring probabilities is precisely how lotteries and casinos stay in business.

The only skill more fundamental than math is the ability to read.

Why bother? Because nearly every career you can do absolutely depends on your math skills. How good you should get at a particular number system depends on what you do. For example, a businessperson rarely deals with numbers that are not money, so the rational numbers should do for most of his everyday needs, unless he has to borrow or lend money. Then he really needs to know a little about real numbers (particularly e). A gambler should be good at probability, and so needs to know the rational numbers well. Engineers and scientists deal with real numbers most of the time, and occasionally have to deal with complex numbers. A practicing mathematician has to know about all common number systems (many of which we have not mentioned).