**Mathematics** presents “logical tools” for learning.

**Arithmetic** forms the first part of mathematics that presents the “number skill”. Arithmetic starts with counting.

**Counting** is a tool for learning how many things are there. Counting starts with one. The next count is one more.

A **unit** is the thing being counted one at a time. If one is counting houses, then each *house* is a unit. If one is counting inches of a length, then each *inch* is a unit.

The **digits** are the ten symbols – 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 – that are used to write numbers.

**Numbers** provide a system to represent the counts. A number is made up of one or more digits, just like words are made up of one or more letters.

**Addition** is counting together of numbers. **Subtraction** is opposite of addition.

**Multiplication** is repeated addition of a number. **Division** is opposite of multiplication.

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## THE FIELD AXIOMS

Real numbers are undefined objects that satisfy certain properties.

### AXIOM #1: CLOSURE PROPERTY OF ADDITION

*If x and y are real numbers, then x+y is a unique real number.*

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.

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### AXIOM #2: CLOSURE PROPERTY OF MULTIPLICATION

*If x and y are real numbers, then xy is a unique real number.*

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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### AXIOM # 3: COMMUTATIVE PROPERTIES OF ADDITION & MULTIPLICATION

*If x and y are real numbers, then x+y = y+x.*

*If x and y are real numbers, then x+y = y+x.*

*If x and y are real numbers, then xy = yx.*

*If x and y are real numbers, then xy = yx.*

**ADDITION:** Two numbers may be added in any order. For example,

**5 + 8 = 8 + 5**

One may visualize the numbers as items of a one-dimensional array. For example,

**$ $ $ $ $ $ $ $ $ $ $ $ $**

One may count the items in this array as “5 first and 8 next”; or “8 first and 5 next”. The result is the same.

**$ $ $ $ $ $ $ $ $ $ $ $ $**

**$ $ $ $ $ $ $ $ $ $ $ $ $**

“Subtraction” is accounted by this law by treating the number being added as a negative integer. The sign moves with the following number. The first unsigned number is treated as having a positive sign. For example,

**[8 – 5] is +8 –5 = –5 +8**

**MULTIPLICATION:** Two numbers may be multiplied in any order. For example,

**5 x 8 = 8 x 5**

One may visualize the numbers as items of a two-dimensional array. For example,

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

### $ $ $ $ $ $ $ $

One may count the items in this array, as “5 rows of 8 each”, or “8 columns of 5 each”. The result is the same.

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

“Division” is accounted by this law by using the reciprocal (multiplicative inverse) of the divisor as the multiplicand.

**[8 ÷ 2] is 8 x ½ = ½ x 8**

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### AXIOM # 4: ASSOCIATIVE PROPERTIES OF ADDITION & MULTIPLICATION

*If x, y and z are real numbers, then (x+y)+z = x+(y+z).*

*If x, y and z are real numbers, then (xy)z = x(yz).*

In addition, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

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

Similarly, in multiplication, three things, arranged in the same order, may be associated in two different ways. The result is the same. For example,

**3 x (5 x 8) = (3 x 5) x 8**

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### AXIOM 5: DISTRIBUTIVE PROPERTY

*If x, y and z are real numbers, then x(y+z) = xy + xz.*

Multiplication distributes over addition. For example, a factor may be multiplied by the other factor as a sum of two parts with the same outcome as follows.

**5 x 17 = 5 x (10 + 7) = 5 x 10 + 5 x 7 = 50 + 35 = 85**

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### AXIOM 6: IDENTITY ELEMENTS

*The identity element for addition is 0, i.e., for any real number x, *

*x+0 = x.*

*The identity element for multiplication is 1, i.e., for any real number x, *

*x.1 or 1x = x.*

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 7: INVERSES

*A unique ADDITIVE INVERSE exists for every real number, i.e., for every x the additive inverse is -x such that *

**x + (-x) = 0.**

**x + (-x) = 0.**

*A unique MULTIPLICATIVE INVERSE exists for every real number, i.e., for every non-zero x the multiplicative inverse is 1/x such that*

**x . (1/x) = 1.**

**x . (1/x) = 1.**

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)

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.

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