The Positive Integers 
- These are probably the first type of numbers you encountered as a child.
- Addition: 1+1 = 2, 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 = 4, etc.
- Thus, 2 + 3 = (1 + 1 )+ (1 + 1 + 1) = 1+1+1+1+1 = 5, etc. (Associative
law)
- Furthermore, 3 + 2 = 1+1+1+1+1 = 2 + 3, etc. (Commutative law)
- Multiplication: 1 X 1 = 1. The rest follows from the distributive
law:
- 1 X anything gives you the same thing back:
- 1 X 2 = 1 X (1+1) = 1 X 1 + 1 X 1 = 1+1 = 2
- 1 X3 = 1 X (1+1+1) = 1+1+1 = 3, etc.
- 2 X 3 = (1+1) X 3 = 1 X 3 + 1 X 3 = 3 + 3 = 6 , etc.
- Zero: n + 0 = 0+ n = n for any number n
- Here is our first extension of the number system!
- Subtraction: The inverse of addition: m "minus" m =
m - n is defined as
the number k such that k + n = m
This is well defined, so far, as long as m is greater than or equal to n.
- What if m is less than n? We can take the approach that "there
is no such thing" as m-n if m < n. However, as you know by now,
it is convenient to define a new set of numbers, the negative integers,
-1, -2, -3, etc. You could argue that a negative number is "imaginary,"
but clearly it is no more imaginary than a positive number.
- Thus, we can extend the set of positive integers to
the set of all integers, positive, negative, or zero -- or simply the set
of integers. More extensions of this sort will lead us naturally to the
set of complex numbers. In this sense, a complex number is no more "imaginary"
than a negative number!
The Integers -- Next Slide