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 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.
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.
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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