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are discussed later in this topic, we end up with a negative and a positive
zero.
Rational and irrational numbers
Anumbersystemcanalsobeusedtorepresentpartsofawhole.Forexam-
ple,whenacarpentercutsoneboardintotwoboardsofequallengthwecan
represent the result with the fraction 1/2. The verbalization of this opera-
tionstatesthatthefraction1/2indicatesoneofthetwopartswhichconsti-
tute an object. Rational numbers are those expressed as a ratio of two
integers,forinstance,1/2,2/3,7/248.Notethatthisuseoftheword rational
is related to the mathematical concept of a ratio, not to reason.
The denominator of a rational number expresses the number of poten-
tialparts.Inthissense2/5indicatestwooffivepossibleparts.Thereisno
reason why the number 1 cannot be used to indicate the number of poten-
tial parts, for example 2/1, 128/1. In this case the ratio x /1 indicates x ele-
ments of an undivided part. Therefore, it follows that x /1 = x . The
implication is that the set of rational numbers includes the integers, since
an integer can be expressed as a ratio by using a unit denominator.
But not all non-integer numbers can be written as an exact ratio of two
integers. The discovery of the first irrational number is usually associ-
ated with the investigation of a right triangle by the Greek mathematician
Pythagoras (approximately 600 BC).
The Pythagorean theorem states that in any right triangle the square of
the longest side (hypotenuse) is equal to the sum of the squares of the
other two sides.
c
a=1
b =1
For this triangle, the Pythagorean theorem states that
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