# 2.1: Axioms and Basic Definitions

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Real numbers can be constructed step by step: first the integers, then the rationals, and finally the irrationals. Here, however, we shall assume the set of all real numbers, denoted \(E^{1},\) as *already* *given*, without attempting to reduce this notion to simpler concepts. We shall also accept without definition (as *primitive* concepts) the notions of the *sum* \((a+b)\) and the *product*, \((a \cdot b)\) or \((a b),\) of two real numbers, as well as the *inequality relation* \(<\) (read "*less than*" ). Note that \(x \in E^{1}\) means "x *is in* \(E^{1,},\) i.e., "*x is a real number.*"

It is an important fact that all arithmetic properties of reals can be deduced from several simple axioms, listed (and named) below.

## Axioms of Addition and Multiplication

Definition

1. (closure laws) The *sum* \(x+y,\) *and the product* \(x y,\) *any real numbers are real numbers themselves.* In symbols,

\[\left(\forall x, y \in E^{1}\right) \quad(x+y) \in E^{1} \text{ and } (x y) \in E^{1}\]

2. (commutative laws)

\[\left(\forall x, y \in E^{1}\right) \quad x+y=y+x \text{ and } x y=y x\]

3. (associative laws)

\[\left(\forall x, y, z \in E^{1}\right) \quad(x+y)+z=x+(y+z) \text{ and } (x y) z=x(y z)\]

4. (existence of neutral elements)

(a) *There is a* (*unique*) *real number*, *called zero* (0), *such that, for all real* \(x, x+0=x\).

(b) *There is a* (*unique*) *real number, called one* (1), *such that* 1 \(1 \neq 0\) *and, for all real* \(x, x \cdot 1=x .\)

In symbols,

(a) \[\left(\exists ! 0 \in E^{1}\right)\left(\forall x \in E^{1}\right) \quad x+0=x;\]

(b) \[\left(\exists 1 \in E^{1}\right)\left(\forall x \in E^{1}\right) \quad x \cdot 1=x, 1 \neq 0.\]

(The real numbers 0 and 1 are called the *neutral elements* of addition and multiplication, respectively.)

5. (existence of inverse elements)

(a) *For every real* \(x,\) *there is a* (*unique*) *real, denoted* \(-x,\) *such that* \(x+(-x)=0\).

(b) *For every real* \(x\) *other than* \(0,\) *there is a* (*unique*) *real, denoted* \(x^{-1}\), *such that* \(x \cdot x^{-1}=1\).

In symbols,

(a) \[\left(\forall x \in E^{1}\right)\left(\exists !-x \in E^{1}\right) \quad x+(-x)=0;\]

(b) \[\left(\forall x \in E^{1} | x \neq 0\right)\left(\exists ! x^{-1} \in E^{1}\right) \quad x x^{-1}=1.\]

(The real numbers \(-x\) and \(x^{-1}\) are called, respectively, the additive inverse (or the symmetric) and the multiplicative inverse (or the reciprocal) of \(x . )\)

6. (distributive law)

\[\left(\forall x, y, z \in E^{1}\right) \quad(x+y) z=x z+y z\]

## Axioms of Order

Definition

7. (trichotomy) *For any real* \(x\) *and* \(y,\) *we have*

*\[\text{either} x<y \text{ or } y<x \text{ or } x=y\]*

*but never two of these relations together.*

8. (transitivity)

\[\left(\forall x, y, z \in E^{1}\right) \quad x<y \text{ and } y<z \text{ implies } x<z\]

9. (monotonicity of addition and multiplication)* For any* \(x, y, z \in E^{1}\), *we* *have*

*(a) \[x<y \text{ implies } x+z<y+z;\]*

(b) \[x<y \text{ and } z>0 \text{ implies ] x z<y z.\]

**Note 1:** The uniqueness assertions in Axioms 4 and 5 are actually redundant since they can be deduced from other axioms. We shall not dwell on this.

**Note 2:** Zero has no reciprocal; i.e., for no \(x\) is \(0 x=1 .\) In fact, \(0 x=0 .\) For, by Axioms VI and IV,

\[0 x+0 x=(0+0) x=0 x=0 x+0.\]

Cancelling \(0 x(\) i.e., adding \(-0 x\) on both sides \(),\) we obtain \(0 x=0,\) by Axioms 3 and 5 (a).

**Note 3:** Due to Axioms 7 and 8, real numbers may be regarded as given in a certain *order* under which smaller numbers precede the larger ones. (This is why we speak of "axioms of *order*.") The ordering of real numbers can be visualized by "plotting" them as points on a directed line ("the real axis") in a well-known manner. Therefore, \(E^{1}\) is also often called "*the real axis*," and real numbers are called "*points*"; we say "the *point* *x* instead of "*the number x*.

Observe that the axioms only state certain properties of real numbers *without specifying what these numbers are*. Thus we may treat the reals as just *any* mathematical objects satisfying our axioms, but otherwise arbitrary. Indeed, our theory also applies to any other set of objects (numbers or not), provided they satisfy our axioms with respect to a certain relation of order \((<)\) and certain operations \((+)\) and \((\cdot),\) which may, but need not, be ordinary addition and multiplication. Such sets exist indeed. We now give them a name.

Definition 1

A *field* is any set \(F\) of objects, with two operations \((+)\) and \(( .)\) defined in it in such a manner that they satisfy Axioms 1-6 listed above (with \(E^{1}\) replaced by \(F,\) of course).

If \(F\) is also endowed with a relation \(<\) satisfying Axioms 7 to 9, we call \(F\) an ordered field.

In this connection, postulates 1 to 9 are called axioms of an (ordered) field.

By Definition \(1, E^{1}\) is an ordered field. Clearly, whatever follows from the axioms must hold not only in \(E^{1}\) but also in any other ordered field. Thus

we shall henceforth state our definitions and theorems in a more general way, speaking of ordered fields in general instead of \(E^{1}\) alone.

Definition 2

An element \(x\) of an ordered field is said to be *positive* if \(x>0\) or *negative* if \(x<0 .\)

Here and below, \(" x>y "\) means the same as \(" y<x . "\) We also write \(" x \leq y "\) for \(" x<y\) or \(x=y^{\prime \prime} ;\) similarly for \(" x \geq y. "\)

Definition 3

For any elements \(x, y\) of a field, we define their *difference*

\[x-y=x+(-y)\]

If \(y \neq 0,\) we also define the *quotient* of \(x\) by \(y\)

\[\frac{x}{y}=x y^{-1}\]

also denoted by \(x / y\).

**Note 4:** *Division by* 0 *remains undefined*.

Definition 4

For any element \(x\) of an ordered field, we define its *absolute value*,

\[|x|=\left\{\begin{array}{ll}{x} & {\text { if } x \geq 0 \text { and }} \\ {-x} & {\text { if } x<0}\end{array}\right.\]

It follows that \(|x| \geq 0\) *always*; for if \(x \geq 0,\) then

\[|x|=x \geq 0\]

and if \(x<0,\) then

\[|x|=-x>0 . \quad( \text{ Why? } )\]

Moreover,

\[-|x| \leq x \leq|x|,\]

for,

\[\text{if } x \geq 0, \text{ then } |x|=x;\]

and

\[\text{if } x<0, \text{ then } x<|x| \text{ since } |x|>0.\]

Thus, in all cases,

\[x \leq|x|.\]

Similarly one shows that

\[-|x| \leq x.\]

As we have noted, all rules of arithmetic (dealing with the four arithmetic operations and inequalities) can be deduced from Axioms 1 through 9 and thus apply to *all* ordered fields, along with \(E^{1}\). We shall not dwell on their deduction, limiting ourselves to a few simple corollaries as examples.

(i) \(a(-b)=(-a) b=-(a b)\);

\((\) ii) \(\quad(-a)(-b)=a b\).

**Proof**-
By Axiom 6,

\[a(-b)+a b=a[(-b)+b]=a \cdot 0=0.\]

Thus

\[a(-b)+a b=0.\]

By definition, then, \(a(-b)\) is the

*additive inverse*of \(a b,\) i.e.,\[a(-b)=-(a b).\]

Similarly, we show that

\((-a) b=-(a b)\)

and that

\[-(-a)=a.\]

Finally, (ii) is obtained from (i) when \(a\) is replaced by \(-a . \square\)

*In an ordered field*, \(a \neq 0\) *implies*

*\[a^{2}=(a \cdot a)>0\]*

*(Hence \(\hskip 4pt\) \(1 =1^{2}>0 . )\)*

**Proof**-
If \(a>0,\) we may multiply by \(a(\) Axiom 9(b) to obtain

\[a \cdot a>0 \cdot a=0, \text{ i.e., } a^{2}>0.\]

If \(a<0,\) then \(-a>0 ;\) so we may multiply the inequality \(a<0\) by \(-a\) and obtain

\[a(-a)<0(-a)=0;\]

i.e., by Corollary 1,

\[-a^{2}<0,\]

whence

\[a^{2}>0 \hskip 4pt \square\]