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If we also want to include division, the integers are still too limited. We need to
include fractions or
rational numbers
:
0, 1, -1, 1/2, -1/2, 2, -2, 3/2, -3/2, 1/3, -1/3, . . .
The rational numbers are nice and neat but they leave out important quan-
tities such as π or 2 . These are called
irrational numbers
because they
cannot be expressed as integers or fractions of integers. Both π and 2
can only be expressed as infinite series - as a sum of an infinite num-
ber of terms. In practice, for a useful approximate value of π or 2 ,
we need only to sum up a few terms of the series. For example, for π, we could
use the so-called Gregory-Leibniz expansion:
π = 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + . . .)
and for 2 , we could use the Taylor expansion
( )
( ) ( ) ( ) (
)
2
=+ −
11212413 246135 2 468
/
/
× +× ×× −××
/
/
× ×× +
...
There are many other methods for calculating π and for taking square roots. All
these methods lead to the well-known decimal approximations for π and 2
π = 3.14159265 . . .
and
2141421356
= .
...
In the struggle to understand what could and could not be proved, the ques-
tion arose of what numbers could be calculated. This led to the concept of
an “effective procedure” - a set of rules telling you, step-by-step, what to do
to complete a calculation. In other words, if there is an effective procedure
for some computational problem it means that there is an algorithm that
can be executed to solve the problem. These methods for calculating π and
2 are examples of effective procedures. They may not be the most efficient
way to calculate π or 2 but these algorithms will work and will produce an
answer.
The number system that includes irrational numbers like these is the sys-
tem of
real numbers.
In everyday life, we use approximations to real numbers
and do our calculations accurate to a specific number of decimal places.
How many real numbers are there? Georg Cantor (
B.6.6
), who developed
the theory of infinite numbers in the late 1800s, showed that the number of
integers is the same as the number of natural numbers. He did this by setting
up a one-to-one correspondence as follows:
B.6.6. The name of Georg Cantor
(1845-1918) is associated with
set theory and with tackling the
problem of infinity in a mathemat-
ically rigorous way. Cantor came
to the conclusion that the infinite
set of real numbers is larger than
the infinite set of natural numbers.
Furthermore, he was able to show
that there is an infinite number of
infinities. Cantor's ideas met with
considerable resistance from fellow
mathematicians. The great German
mathematician, David Hilbert, was
an exception and was early to rec-
ognize the significance of Cantor's
work. Hilbert later said: “No one
shall expel us from the Paradise that
Cantor has created.”
B3
Integers
0
-1
1
-2
2
-3
3
-4
. . .
Natural numbers
0
1
2
3
4
5
6
7
. . .
Although it may seem that there are more integers than natural numbers,
Cantor showed that the integers could in principle be counted off against the
natural numbers in this way. Although both were infinite, the existence of such
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