## The Basics of Math

##### 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.

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.

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

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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### 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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### 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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### 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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### 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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