THE PASCAL TRIANGLE
In modern mathematics, more and more stress is placed on the context
in which statements are true. In elementary mathematics this generally
means an emphasis on a clear understanding of which number systems possess
certain properties. We begin, then, by describing briefly the number systems
with which we will be concerned. These number systems have developed through
successive enlargements of previous systems.
Sunday, February 8, 1998
At one time a "number" meant one of the natural numbers:
1,2,3,4,5,6,... . The next numbers to be introduced were the fractions:1/2,
1/3, 2/3, 1/4, 3/4, 1/5,..., and later the set of numbers was expanded
to include zero and the negative integers and fractions. The number system
consisting of zero and the positive and negative integers and fractions
is called the system of rational numbers, the word "rational"
being used to indicate that the numbers are ratios of integers. The integers
themselves can be thought of as ratios of integers since 1=1/1, -1= -1/1,
2=2/1, -2=-2/1, 3=3/1, etc.
The need to enlarge the rational number system became evident when
mathematicians proved that certain constructible lengths, such as the lengthof
a diagonal of a unit square, cannot be expressed as rational numbers.
The system of real numbers then came into use. The real numbers
include all the natural numbers; all the fractions; numbers such asand
which represent constructible lengths; numbers such asand
which do not represent lengths constructible from a given unit with compass
and straightedge; and the negatives of all these numbers. Modern technology
and science make great use of a still larger number system, called the
complex numbers, consisting of the numbers of the form a
+ bi with a and b real numbers and i2