The Rational and Real Numbers
- The set of rational numbers is the set of numbers of the form p/q ,
where p and q are integers.
- Many times, it is convenient to work with numbers that cannot be expressed
in the form of p/q. For example, consider the concept of the square root
of a number, say the square root of 2. We may consider the set of rational
numbers r, whose square, r X r, is less than or equal to 2. The square root
of 2 is formally defined as the least upper bound to this set of numbers.
It can be shown that this least upper bound is not a rational number --
it is thus called an irrational number. Another example of an irrational
number is pi, defined as the ratio of the circumference of a circle to the
diameter of the circle.
- Thus, the set of rational numbers is not complete, in some sense, --
there are conceptually numbers "in between."
- The set of rational numbers together their limits are called the set
of real numbers . An irrational number is a real number that
is not rational. The set of real numbers is probably the set of numbers
you are most used to dealing with and thinking about.
- Contrary to what you might have thought, the set of real numbers is
still not complete, in a sense.
- Recall that we extended the set of rational numbers to the real numbers
so that we had a well defined square root of 2.
- You may have learned in high school that there is "no such thing"
as the square root of a negative number. It is true that there are no real
numbers whose square gives you a negative number. However, if we are willing
to extend our concept of real numbers a bit, we can define square roots
of negative numbers in a way consistent with the mathematical operations
we already have defined. This leads us to extend the set of real numbers
to the set of complex numbers.
The Complex Numbers <-- Next Slide