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.
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