### Let’s commence, with something non-controversial, at the commencement.

#### by Gilbert Keith

Assume an ordered set of rational numbers.

They must satisfy the following

- Only one of the following can hold for a and b: a < b or a = b or a > b
- for all a, a = a
- a = b ==> b = a
- a = b and b = c ==> a = c
- a ≤ b and b < c ==> a < c
- a < b and b ≤ c ==> a < c
- Positive numbers are those greater than zero
- Negative numbers are those less than zero

In addition they obey the following laws of arithmetic:

Addition –

- a pair of numbers can be added via the operation a + b, which yields the sum of a and b. Addition always yields a real number
- If a = a’ and b = b’ then a + b = a’ + b’
- a + b = b + a
- (a + b) + c = a + (b + c) = (a + c) + b
- a < b ==> a + c < b + c

Subtraction –

- It is the inverse of addition. If a and b are any numbers, and x is a number such that a + x = b. Then by definition x = a – b.

Multiplication –

- Multiplication of a and b is represented as the operation a * b and gives the product, which is always a real number.
- If a = a’ and b = b’ then a * b = a’ * b’
- a * b = b * a
- (a * b) *c = a * (b * c) = (a * c) * b
- (a + b) * c = a * c + b * c
- a < b and c > 0 ==> a * c < b * c
- a * 0 = 0

Division –

- it is the inverse of multiplication, which is represented as the operation a ÷ b. If x is a number such that a * x = b then by definition x = b ÷ a, and it can only be performed when a is non-zero.