*[Note: These axioms are numbered in the order provided in the 1961 edition of CALCULUS Volume 1 by Tom M. Apostol.]*

## FUNDAMENTALS OF REAL NUMBER SYSTEM

Real numbers are undefined objects that satisfy certain properties.

Addition is an operation such that for every pair of real numbers x and y we can form the sum of x and y, which is another real number denoted by x+y. The sum x+y is uniquely determined by x and y.

Multiplication is an operation such that for every pair of real numbers x and y we can form the product of x and y, which is another real number denoted by xy or by x.y. The product xy is uniquely determined by x and y.

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**THE FIELD AXIOMS**

**AXIOM 1:** *Commutative laws, x+y = y+x; xy = yx*

Addition is counting together. Two numbers may be added in any order to come up with the same result. A number must include its sign or vector. In that sense, addition is combining two things, with their attributes, in any order. For example,

5 + 8 = 8 + 5

**($$$$$) + ($$$$$$$$) = ($$$$$$$$) + ($$$$$)**

Multiplication is repeated addition. One may visualize a two-dimensional arrangement in terms of rows and columns. To come up with the total number of items in that two-dimensional matrix, one may count by columns or by rows. The result would be the same. For example,

5 . 8 = 8 . 5

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

(adding 5 by columns 8 times) = (adding 8 by rows 5 times)

5+5+5+5+5+5+5+5 = 8+8+8+8+8

**.**

**AXIOM 2:** *Associative laws, x+(y+z) = (x+y)+z; x(yz) = (xy)z*

Three things, arranged in the same order, may be associated in two different ways, as shown above. The result is the same when they are associated in a single dimension of addition, or in the two dimensions of multiplication. For example,

3 + (5 + 8) = (3 + 5) + 8

3 . (5 . 8) = (3 . 5) . 8

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**AXIOM 3:** *Distributive law, x(y+z) = xy + xz*

A factor in a multipliation may be broken into two parts. Each part may then be multiplied by the second factor. When the two partial products are combined, we get the product of the original two factors. For example,

5 . 17 = 5 . (10+7) = 5 . 10 + 5 . 7 = 50 + 35 = 85

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**AXIOM 4:*** Existence of identity elements*

There exist two real numbers, which we denote by 0 and 1, such that for every real x we have

0 + x = x + 0 = x (the idea of adding nothing)

1 . x = x . 1 = x (the idea of a single occurrence)

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**AXIOM 5:** *Existence of negatives*

For every real number x there is a real number y such that

x + y = y + x = 0 (the idea of negating something into nothing)

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**AXIOM 6:** *Existence of reciprocals*

For every real number x (except 0) there is a real number y such that

x . y = y . x = 1 (the idea of reducing something to its unit)

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**Note 1:** 0 is an exception because it denotes nothing, whereas all other numbers denote something.

**Note 2:** 1 is unique because it denotes a unit, whereas all other non-zero numbers denote multiple units.

**Note 3:** The numbers 0 and 1 in Axioms 5 and 6 are those of Axiom 4.

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**THE ORDER AXIOMS**

**AXIOM 7:** *If x and y are positive, so are x+y and xy.*

**AXIOM 8:** *For every real x (except 0), either x is positive or -x is positive, but not both.*

**AXIOM 9:** *The number 0 is not positive.*

*.*

**These 9 axioms, however, do not account for the existence of irrational numbers. **