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 can be reduced to 2, and so not an irrational number, even if it is represented by using a square root.
Examples of irrational numbers include π, , and .
Here are some examples of numbers that are not irrational: 5.2, , 3, and .
Property  Description 

Continuity  The set of irrational numbers is not a continuous set. 
Addition  Addition of irrational numbers is the same as addition of real numbers. 
A, elements a, b of set A, and an operation * on set A, set A is closed with respect to * if and only if a * b is a member of set A. Click for more information.">Closure with respect to addition  The set of irrational numbers is not closed with respect to addition since +  = 0. 0 is not an irrational number. 
Associative property of addition  The 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 identity  The 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 inverse  The 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 addition  Since 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. 
Multiplication  Multiplication of irrational numbers is the same as multiplication of real numbers. 
Closure with respect to multiplication  The set of irrational numbers is not closed with respect to multiplication since · = 2. 2 is not an irrational number. 
Associative property of multiplication  The 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 identity  The 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 inverse  The multiplicative inverse of an irrational number a is 1 / a since a · 1/a = 1, a ≠ 0. The set of irrational numbers is invertible with respect to multiplication. 
Group with respect to multiplication  Since 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. 
Cardinality  The cardinality of the irrational numbers is ℵ_{1} = 2^{ℵ0}. The set of irrational numbers is uncountable. 
Table 1: Properties 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:
Step  Equations/Diagrams  Discussion  

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  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 

Start with the Pythagorean theorem: A^{2} + B^{2} = C^{2}. 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 2b^{2} = c^{2}.  
4  2b^{2} = c^{2}  The definition of an even number is a number that has 2 as a factor. Since c^{2} = 2b^{2}, 2 is a factor of c^{2}. This means that c^{2} is an even number.  
5  c is even.  Since c^{2} has a factor of 2 and c^{2} = 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  4y^{2} = 2b^{2}  Since c^{2} = 2b^{2}, substitute 2y in for c. This gives the equation (2y)^{2} = 2b^{2}. Expand the exponent on the left side of the equation to get 4y^{2} = 2b^{2}.  
9  2y^{2} = b^{2}  Take the equation 4y^{2} = 2b^{2} and simplify. This gives the equation 2y^{2} = b^{2}.  
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. 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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