Real Number System

This is a guide on the real number system.

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The integers and rational numbers are part of a larger system of numbers. As we
know, the integers and rational numbers correspond to some points on the line.
The real numbers are those numbers which correspond to all points on the line.
Another way of describing them is to say that they consist of all numbers which
have a decimal expansion, possibly infinite.

It is a rather long and tedious process to develop the theory of the real
numbers systematically. We shall first summarize some of the algebraic
properties which they satisfy. Unless otherwise specified, number means real
number.

The sum and product of two numbers are numbers, and they satisfy the following
properties, similar to those of rational numbers.

Addition is commutative and associative, meaning that for all real numbers,
a,b,c, we have:

\(a + b = b + a\)
and
\(a + (b + c) = (a + b) + c\)

We also have:

\(0 + a = a\)

To each real number a there is associated a number denoted by -a such that :

\(a + (-a) = a\)

As with integers, or rational numbers, we represent a and -a on opposite sides
of 0 on the line.

The number -a is called the additive inverse of a. We also read it as minus a.
If b is a number such that \(a + b = 0\), then we must have \(b = -a\), as we
see by adding -a to both sides of the equations \(a + b = 0\)

Multiplication is commutative and associative, meaning that for all real numbers
a,b, and c, we have:

\(ab = ba\) 
and
\(a(bc) = (ab)c\)

We also have:
\(1a = a\)
and 
\(0a = 0\)

Multiplication is distributive with respect to addition, meaning that:

\(a(b + c) = ab + ac\)
and
\((b + c)a = ba + ca\)

So far, these properties are the same as those satisfied by the integers and
rational numbers. In particular, further properties which were proved using only
these basic ones are now valid for the real numbers. For instance we recall the
important formulas:

\((a + b)^2 = a^2 + 2ab + b^2\)

\((a - b)^2 = a^2 - 2ab + b^2\)

\((a + b)(a - b) = a^2 - b^2\)

These are true if a,b, are real numbers because their proofs used only
commutativity and associativity.

If a is a real number that does not equal 0, then there exists a real number
denoted by \(a^{-1}\) such that:

\(a^{-1}a = aa^{-1} = 1\)

As with rational numbers, this number \(a^{-1}\) is called the multiplicative
inverse of a. Instead of writing \(a^{-1}\), we write \(\frac{1}{a}\) and we
write: a/b instead of \(b^{-1}a\) or \(ab^{-1}\).

The proofs of properties concerning inverses before depended only on the basic
we have mentioned so far, and are applicable to the real numbers. We have
cross-multiplication, cancellation rules and so on. We have the uniqueness of
the multiplicative inverse. Namely, if multiplying both sides by \(a^{-1}\)
shows that \(b = a^{-1}\).