Bernabé 1
Alejandra Bernabé
COSMOS Cluster 6
Irrational Numbers
Throughout the fifth century, people used whole numbers and fractions and neglected the
use of negative and irrational numbers. An irrational number is any real number that cannot be
expressed as a ratio a/b in which a and b are both integers and
0; therefore an irrational
number cannot be expressed as a simple fraction. Modern theories in subjects such as Physics,
Engineering, Chemistry, Astronomy, etc. acquire the utilization of irrational numbers, more
importantly, Mathematics. Irrational numbers have become integral pieces in such subjects and
are commonly used to create explanations of theories of the natural world. Although sometimes
difficult to explain, irrational numbers such as √2, π, and e are utilized in the industrialization of
the world.
The first proof of the existence of irrational numbers was created by the Greeks,
particularly a Pythagorean by the name of Hippasus of Metapontum, while identifying the side of
a pentagram. The Pythagorean method then claimed that there must be an indivisible unit which
fit evenly into a length than did the other. However, in the fifth century BC, Hippasus created an
assertion of contradiction to the Pythagorean claim. Hippasus created the first proof of an
irrational number, √2, by demonstrating the measurements of an isosceles right triangle. His
reasoning is as follows:
As we know, the square of an even integer is even and the square of an odd integer is
odd. In order to prove the irrationality of √2, suppose that √2 were a rational number, and say
√2 = ,
Bernabé 2
In which both a and b are integers. Suppose that the rational fraction, a/b, is in its lowest terms,
for this reason, we know that neither a nor b are even. Then square the equation above and
simplify to get:
2 =
2
2 ,
2 2.
2
The term 2b2 denotes an even integer; therefore a2 is an even integer, and hence a is an even
integer. Say a = 2c, in which c is also an integer and replace a by 2c in the equation a2 = 2b2 to
get
2
2
,
4
2
,
2
.
Now, the term 2c2 denotes an even integer; therefore, b2 is also an even integer, and hence b is an
even integer. The conclusion now states that both a and b are even integers, whereas a/b was
presumed to be simplified in lowest terms. This contradiction indicates that it is not possible to
express √2 in the rational form a/b, and therefore helps us conclude that √2 is an irrational
number (Numbers: Rational and Irrational, 42).
The number pi, also an irrational number, serves as the most important irrational number
in the world because it is utilized to find the circumference of a circle, and find the area and
volume of a sphere or square. Pi, along with the square root of two, are often used in the
construction of buildings in our society. Pi was originally founded as an irrational number by
many mathematicians. One particular mathematician that provided a proof with the use of
continued fractions was Johann Heinrich Lambert. Lambert proved pi to be irrational by the
following continued fraction:
Bernabé 3
tan
3
5
7 ⋱
Johann Heinrich Lambert proved that if x were non-zero and rational, x must be an
1, and so classifies
irrational number. Lambert then demonstrated that
irrational and therefore
to be
is an irrational number (Numbers: Rational and Irrational, 52).
The irrational number, e, is frequently utilized in the advanced mathematical classes for
representing the natural logarithm. The series of representation of e classifies the number e to be
irrational. As Joseph Fourier stated and proved by contradiction, the number e is an irrational
number as follows:
Initially, suppose that the number e is a rational number. This proof replies on the fact that e can
be expressed in an infinite series
1
Assume
1
1!
1
2!
1
3!
1
…
4!
, in which p and q are positive integers; therefore e is a positive integer.
Then, the terms of the equation must be multiplied by q! on both sides as follows:
!
Since
!
!
1!
!
2!
!
3!
!
4!
!
!
⋯
, ! is a positive integer as is the sum of
!
!
1!
!
2!
!
3!
!
4!
⋯
!
!
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Thus, the other terms are equal to the sum of two integers. Therefore they sum to an integer and
in which we will call the remaining terms R.
1
1
1
!
1
2 !
1 !
1
1
1
1
2
1
1
1
2
1
3
1
1
1
1
1
⋯
1
1
1
1
1
1
⋯
3 !
1
1
1
1
1
1
…
⋯
1
1
1
Reflecting to the original definition of R, we can now see that since q is positive, so is R. Thus,
1
stating R is an integer between 0 and , contradicts our original assumption of e being a rational
number. Thus, e is an irrational number (Irrational Numbers, 57).
Although irrational numbers are difficult to comprehend, they serve in a major portion of
our modern mathematical industry. Irrational numbers are often used in the daily lives of our
society whether it is in the creation of buildings, chemical equations, or basic physics. Irrational
numbers effectively help modern society to understand the workings of the natural and future of
our world.
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Works Cited
Dedekind, Richard. Essays on the Theory of Numbers: I. Continuity and Irrational Numbers. II.
The Nature and Meaning of Numbers. New York: Dover Publications, 1963. Print.
Niven, Ivan. Irrational Numbers. Washington: Mathematical Association of America, 1956.
Print.
Niven, Ivan. Numbers: Rational and Irrational. [New York]: Random House, 1961. Print.