Irrational Number

Pronunciation: /ɪˈræ ʃə nl ˈnʌm bər/ Explain

An irrational number is a real number that can not be expressed as the ratio of two integers. Since a rational number is a real number that can be expressed as the ratio of two integers, this means that any real number is either rational or irrational, but can not be both.

In math, one can represent numbers in ways that mask their true identity. For example, the square root of Square root of 4 can be reduced to 2, and so Square root of 4 not an irrational number, even if it is represented by using a square root.

Examples of irrational numbers include π, square root of 2, and .

Here are some examples of numbers that are not irrational: 5.2, 12/7, 3, and Square root of 4.

check mark Understanding Check

Decide if each number is irrational. Remember to reduce and simplify the number as much as possible. Then click 'Rational' or 'Irrational' to see if you are right.

NumberIs irrational?
Square root of 5 Rational
Irrational
Square root of 16 Rational
Irrational
18/9 Rational
Irrational
Table 1: Rational or irrational

Properties of irrational numbers

PropertyDescription
ContinuityThe set of irrational numbers is not a continuous set.
AdditionAddition of irrational numbers is the same as addition of real numbers.
Closure with respect to additionThe set of irrational numbers is not closed with respect to addition since square root of 2+ -square root of 2 = 0. 0 is not an irrational number.
Associative property of additionThe set of irrational numbers is associative with respect to addition. Since all irrational numbers are also real numbers, and the set of real numbers is associative with respect to addition, the associative property of addition applies to irrational numbers.
Additive identityThe additive identity for real numbers is zero. Zero is not in the set of irrational numbers. The set of irrational numbers does not have the property of identity with respect to addition.
Additive inverseThe additive inverse of an irrational number a is -a since a+(-a) = 0. The set of irrational numbers is invertible with respect to addition.
Group with respect to additionSince the additive identity of irrational numbers is not itself an irrational number, the set of irrational numbers does not form a group with respect to addition.
MultiplicationMultiplication of irrational numbers is the same as multiplication of real numbers.
Closure with respect to multiplicationThe set of irrational numbers is not closed with respect to multiplication since square root of 2·square root of 2 = 2. 2 is not an irrational number.
Associative property of multiplicationThe set of irrational numbers is associative with respect to multiplication. Since all irrational numbers are also real numbers, and the set of real numbers is associative with respect to multiplication, the associative property of multiplication applies to irrational numbers.
Multiplicative identityThe multiplicative identity for real numbers is one. One is not in the set of irrational numbers. The set of irrational numbers does not have the property of identity with respect to multiplication.
Multiplicative inverseThe multiplicative inverse of an irrational number a is -a since a+(-a) = 0. The set of irrational numbers is invertible with respect to multiplication.
Group with respect to multiplicationSince the multiplicative identity of irrational numbers is not itself an irrational number, the set of irrational numbers does not form a group with respect to multiplication.
CardinalityThe cardinality of the irrational numbers is 1 = 2ℵ0. The set of irrational numbers is uncountable.
Table 1: Properties of irrational numbers.

Proof of the existence of irrational numbers

The first proof of the existence of irrational numbers is attributed to a Pythagorean, a member of an ancient Greek religion. It proceeds as follows:

StepEquations/DiagramsDiscussion
1 Claim Assume that the length of the hypotenuse and the length of the legs of an isosceles right triangle are both rational numbers. Since ratio of two rational numbers can be reduced to another rational number, there is a ratio of two integers c:b that can represent the ratio of any two rational numbers.
2 An isosceles right triangle. The hypotenuse is labeled 'a' and the legs are labeled 'b' Let c:b be the ratio of lengths of the hypotenuse to a leg of a isosceles right triangle expressed in smallest terms where c and b are both integers. Since c:b is expressed in the smallest terms, c and b have no factors in common.
3
A2 + B2 = C2Pythagorean theorem
b2 + b2 = c2Substitute values from the diagram
2b2 = c2Combine like terms
Start with the Pythagorean theorem: A2 + B2 = C2. Then substitute the variables from the diagram in step 1. Substitute b in for A and B. Substitute c in for C. Then combine the like terms to get the equation 2b2 = c2.
4 2b2 = c2 The definition of an even number is a number that has 2 as a factor. Since c2 = 2b2, 2 is a factor of c2. This means that c2 is an even number.
5 c is even. Since c2 has a factor of 2 and c2 = c·c, c must also have a factor of 2. So c is even.
6 b is odd. Since c:b is in the lowest terms, b has no factors in common with c. So b can not have a factor of 2. This means that b must be odd.
7 c = 2y Since c is even, there is an integer y such that c = 2y.
8 4y2 = 2b2 Since c2 = 2b2, substitute 2y in for c. This gives the equation (2y)2 = 2b2. Expand the exponent on the left side of the equation to get 4y2 = 2b2.
9 2y2 = b2 Take the equation 4y2 = 2b2 and simplify. This gives the equation 2y2 = b2.
10 b is even. By a similar argument to steps 3 and 4, b must be even.
11 Contradiction Since b can not be both even and odd, a contradiction exists. This means that the measures of both the leg and the hypotenuse of a isosceles right triangle can not be rational numbers. So a number that is not a rational number must exist.
Table 2: Proof of the existence of irrational numbers.

Cite this article as:

McAdams, David E. Irrational Number. 8/28/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/irrational.html.

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Revision History

8/28/2018: Corrected spelling. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
3/3/2010: Added "References", added section on the proof of existence of irrational numbers. (McAdams, David E.)
8/13/2008: Changed some math constructs to images. Added understanding check (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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