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Number Systems

The Natural Numbers · Integers · Rational Numbers · Real Numbers · Complex Numbers · Why Bother? · Recommended Books

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).

College Algebra (Schaum's Outlines)

The classic algebra problem book - very light on theory, plenty of problems with full solutions, more problems with answers

Schaum's Easy Outline: College Algebra

A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students