To begin with, consider the real numbers, denoted by ℝ, as a line extending infinitely far in both directions. In this book, the notation, ≡ indicates something is being defined. Thus the integers are defined as

the natural numbers,

and the rational numbers, defined as the numbers which are the quotient of two integers.

are each subsets of ℝ as indicated in the following picture.
As shown in the picture,
Axiom 2.1.2 x + 0 = x, (additive identity).
Axiom 2.1.3 For each x ∈ ℝ, there exists −x ∈ ℝ such that x +
Axiom 2.1.4
Axiom 2.1.5 xy = yx,(commutative law for multiplication).
Axiom 2.1.6
Axiom 2.1.7 1x = x,(multiplicative identity).
Axiom 2.1.8 For each x≠0, there exists x^{−1} such that xx^{−1} = 1.(existence of multiplicative inverse).
Axiom 2.1.9 x
These axioms are known as the field axioms and any set (there are many others besides ℝ) which has two such operations satisfying the above axioms is called a field. Division and subtraction are defined in the usual way by x − y ≡ x +
In the first part of the following theorem, the claim is made that the additive inverse and the multiplicative inverse are unique. This means that for a given number, only one number has the property that it is an additive inverse and that, given a nonzero number, only one number has the property that it is a multiplicative inverse. The significance of this is that if you are wondering if a given number is the additive inverse of a given number, all you have to do is to check and see if it acts like one.
Theorem 2.1.10 The above axioms imply the following.
Proof:Suppose then that x is a real number and that x + y = 0 = x + z. It is necessary to verify y = z. From the above axioms, there exists an additive inverse, −x for x. Therefore,

and so by the associative law for addition,

which implies

Now by the definition of the additive identity, this implies y = z. You should prove the multiplicative inverse is unique.
Consider 2. It is desired to verify 0x = 0. From the definition of the additive identity and the distributive law it follows that

From the existence of the additive inverse and the associative law it follows
0  = + 0
x = +

= + 0
x = 0 + 0x = 0x 
To verify the second claim in 2., it suffices to show x acts like the additive inverse of −x in order to conclude that −

and so x = −
To demonstrate 3.,

and so using the definition of the multiplicative identity, and the distributive law,

It follows from 1. and 2. that 1 = −

Therefore, by the uniqueness of the additive inverse proved in 1., it follows
To verify 4., suppose x≠0. Then x^{−1} exists by the axiom about the existence of multiplicative inverses. Therefore, by 2. and the associative law for multiplication,

This proves 4. ■
Recall the notion of something raised to an integer power. Thus y^{2} = y ×y and b^{−3} =
Also, there are a few conventions related to the order in which operations are performed. Exponents are always done before multiplication. Thus xy^{2} = x
Also recall summation notation.
Definition 2.1.11 Let x_{1},x_{2},

Thus this symbol, ∑ _{j=1}^{m}x_{j} means to take all the numbers, x_{1},x_{2},
As an example of the use of this notation, you should verify the following.
Example 2.1.12 ∑ _{k=1}^{6}
Be sure you understand why

As a slight generalization of this notation,

It is also possible to change the variable of summation.

while if r is an integer, the notation requires

and so ∑ _{j=1}^{m}x_{j} = ∑ _{j=1+r}^{m+r}x_{j−r}.
Summation notation will be used throughout the book whenever it is convenient to do so.
Example 2.1.13 Add the fractions,
You add these just like they were numbers. Write the first expression as