= 1.4142135 ˇˇˇDuring the Determination of the square root, we have encountered calculations, which we could continue for ever such as the calculation of the
= 1.4142135 ˇˇˇ
You might think
that this action will finish or lead to a periodic decimal, so
that it could be finished. You would then have found a finite
decimal fraction or a periodic infinite decimal fraction, which you could convert into a simple
fraction.
Hence you might ask whether is equal to a simple fraction.
Let = p/q,
where p and q are integers without common
factors and q > 1. On squaring, we find
2 = (p/q)˛ = p˛/q˛ = pˇp/qˇq.
Since p, q have no common factors and q > 1, you have on the right hand side again a fraction. But the integer 2 cannot be equal to a fraction, whence the assumption is wrong.
The calculation of
continues and has no end. Therefore it is a new type of number.
So far, we have encountered only integers, common fractions and decimal fractions
with finite numbers of decimals or periodic decimals. All these numbers, which can be
positive or negative, share the property of being representable
as common fractions and are called
rational
numbers. You
might ask whether we should accept the existence of such new
numbers. Since mathematics aims to remove as many restrictions as
possible, it has accepted these numbers and called them irrational
numbers. These
numbers cannot be expressed in terms of rational numbers - as
common fractions or decimal fractions with finite or periodic
decimal digits. You can easily construct such numbers, for
example:
2.14 1144 111444 11114444 ˇˇˇ,
where the two digits 1 and 4 occur once, then twice, then three times, etc.This decimal fraction does not have a period nor will have the number
2.1456 11456 111456 ˇˇˇ
Using such laws of formation, you can generate as many irrational numbers as you like. You cannot specify exactly an irrational number as a decimal fraction! At some stage or other, you must stop and round them off. Strictly speaking, for such a rounded-off number, for example,
= 1.4142,
the use of an equality sign is inaccurate, yet it is used!
is rational whenever a
is the nth power of a rational number, otherwise it is
irrational.
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The extension to positive and negative irrationals is obvious. Rational and irrational numbers are called real numbers.
Historical remarks
If the lengths of two sides in a right-angled
triangle are rational, most often the length of the third side will be irrational, since 
. It
became a hobby of the Pythagorean
School to look for right-angled triangles
in which a, b, c are rational. It is sufficient
to solve the equation 
for integer values, since division of the equation by
a squared number will generate rational solutions. Integers which
satisfy the equation a˛ + b˛ = c˛
are called Pythagorean numbers. They are given by: a = m˛ - n˛,
b = 2mn, c = m˛ + n˛, m, n integers,
for example:
| m = 2 | n = 1 | 3˛ + 4˛ = 5˛ | ||
| m = 3 | n = 2 | 5˛ + 12˛ = 13˛ |
This question relating to integer solutions can
be generalized to every equation or system of equations. It has
gained special significance for the equation x˛+y˛=z˛
with the extension of the exponent 2 to arbitrary n.
Fermat has 
stated
that the equation xn + yn
= zn cannot be solved by
integers except for n = 1 and 2; this is called today Fermat's
Theorem. All efforts to prove
it have been unsuccessful. The German mathematician Erich Kummer (1809 - 1877) succeeded in proving it
for the numbers n<100. Among the larger numbers are
exceptions for which his study was not sufficient. Wolfskehl, an industrialist at Darmstadt in Germany, studied this
problem all his life and bequeathed at his death in 1906 100,000
German Marks to the Göttingen Society of Sciences for whoever
finds a general proof of the theorem or a counter example.