Download Stage 5 WID FUNCTIONS PROJECT The Real Numbers

Stage 5 WID FUNCTIONS PROJECT
The Real Numbers
MAT095 EDUCO
October 13, 2006
In out last discussion, we defined the idea of a function, and when we
showed pairings and stick diagrams we used discrete sets of numbers. What
we wish to investigate now are functions that not only map discrete sets into
discrete sets, but that also map continuous sets into continuous sets.
One continuous set we wish to investigate is the set of real numbers. The
set of real numbers contains several subsets. The most elemental subset of
the set of real number is the set of counting numbers, C = {1, 2, 3, · · ·}.
The subset of the real numbers that contains the counting numbers is the
set of whole numbers, W = {0, 1, 2, 3, · · ·}. The next set in the hierarchy
is the set of integers, I = {· · · − 3, −2, −1, 0, 1, 2, 3 · · ·}. The set of integers
are contained in the the next set up the ladder which is the set of rational
numbers, Q = { ab |b = 0 a and b are integers}. An example of a rational
number is 12 , or −2
. The last subset of the real numbers is the set of irrational
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numbers. This set is separate from all the others and does not have any
subsets
like√the rational numbers. Examples of members of this set are π,
√
2, and − 5. In fact all the square roots prime numbers are irrational. In
fact, all numbers that cannot be represented and the ratio of two integers, ab
are irrational and there are many, many of them. A diagram of how these
sets relate to one another is shown below.
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The Real Numbers
Q
Z
I
W
C
The set of real numbers is a hierarchy with the set of irrationals added
on. What is important about the real numbers is that it is the set of all
numbers on the number line. That is, for every point on the line there is
a real number and for every real number there is a point on the line. We
say the real numbers are complete. That is, there are no numbers that are
unaccounted for.
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Exercises
1. Identify which set the following numbers belong to. Either C, W, I Q,
or Z.
(a) 4
(b)
1
2
(c) π
(d) -5
√
(e) − 5
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2. The idea of a subset was described above. When one subset is a subset
of another set it is completely contained inside it. For example, the set
of positive odd numbers is a subset of the set of integers. A real world
example of a subset might be: the set of all males is a subset of the set
of all humans. Give three real world examples of subsets.
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