The Integers
- 0 minus m, or simply -m, is defined in terms of subtraction above:
m + (-m) = m - m = 0
- The number line is illustrated above. It gives us a graphical way of
thinking about numbers, addition, and subtraction, which is quite useful,
e.g. :
- We can thus think of non-zero integers as having a "direction"
: right or left.
- Multiplication Revisited:
- 0 times anything equals 0, i.e. 0 X n = n X 0 = 0 for any n. Why?:
- 0 X n = (1 - 1) X n = 1 X n + (-1) X n = n - n = 0
- Minus one times anything "changes the direction" of the
number. i.e. (-1) X n = -n. Why? : (-1) X n = (0 - 1) X n = (0 X n) - (1
X n) = 0 - n = -n
- The change in direction is 180 degrees, i.e. the new direction is
opposite.
- This concept will be the key intuition leading to the extension
to the set of complex numbers!
- Division: The inverse of multiplication. If q is not zero, p
divided by q, written p/q , is defined as the number r such that r X q =
p.
- Often, there will be no such integer r so satisfying this condition
(i.e. p is not a multiple of q), so we again invent a set
of new numbers to get around the problem.
- This causes us again to extend the set of numbers we deal with to
the set of rational numbers.
The Rational and Real Numbers <-- Next Slide