Irrational Numbers

Rational and Irrational Numbers
You can write any rational number as a fraction of integers p/q. You cannot write an irrational number as an integer fraction.

In mathematics, an irrational number is a number that cannot be expressed as a fraction or ratio of two integers. For example, there is no fraction that is the same as √ 2. The decimal value of an irrational number neither regularly repeats nor ends. In contrast, a rational number can be expressed as a fraction of two integers, p/q. Together, the set of rational and irrational numbers form the real numbers. The set of irrational numbers is an uncountably infinite set.

Irrational Numbers Symbol

The most common symbol for an irrational number is the capital letter “P”. Meanwhile, “R” represents a real number and “Q” represents a rational number. Sometimes the set of irrational numbers is R-Q or R|Q.

Examples of Irrational Numbers

Irrational numbers can be positive or negative. There are many examples of irrational numbers:

  • √ 2 (Pythagoras’ constant)
  • Another way of expressing √ 2 is the hypotenuse of a triangle that has two sides with a length of 1 or the diagonal of a square with sides having a length of 1.
  • Many other square roots and cube roots, such as √3 and √99. Note that some square and cube roots are rational (√4 = 2, √9 = 3).
  • π or pi, or the of a circle’s circumference to its radius
  • π
  • -√ 2
  • 6 – 3π 
  • e or Euler’s number
  • φ or the Golden Ratio
  • eπ
  • πe (probably, not proven)
  • ii, where i is the imaginary number, √-1
  • The transcendental real numbers, where transcendental numbers are not algebraic (not the root of a non-zero polynomial). Examples are π and e. Note that all transcendental numbers are irrational, but not all irrational numbers are transcendental.

Examples of Numbers That Are Not Irrational

Numbers that are not irrational include the rational numbers and the imaginary numbers. You can write a rational number as a fraction. As a decimal, it either ends or else repeats. An imaginary number is a real number multiplied by i, where i is the square root of -1.

Examples of numbers that are not irrational include:

  • 1/3, 2/3, and their decimal equivalents, which repeat
  • i
  • 0
  • -42
  • 13.2
  • 7
  • 5/4
  • -12

Properties of Irrational Numbers

The irrational numbers are a subset of the real numbers, so they have all the properties of real numbers. They also have properties that distinguish them from rational numbers.

  • Adding a rational and irrational number gives an irrational number.
  • Adding or multiplying two irrational numbers may or may not give a rational number. For example, multiplying √ 2x√ 2= 2. But, π x π = π2, which is irrational.
  • The set of irrational numbers is not closed under multiplication. The set of real numbers is closed under multiplication, meaning multiplying any two rational numbers gives you another rational number.
  • Multiplying an irrational number by any nonzero rational number gives an irrational number.
  • Any two irrational numbers may or may not share a least common multiple.


  • Bridger, Mark (2007). Real Analysis: A Constructive Approach through Interval Arithmetic. John Wiley & Sons. ISBN 978-1-470-45144-8.
  • Choike, James R. (1980) “The Pentagram and the Discovery of an Irrational Number.” The Two-Year College Mathematics Journal. 11(5): 312-316. doi:10.1080/00494925.1980.11972468
  • Hardy, G. H. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN 0-19-853171-0.
  • McCabe, Robert L. (1976). “Theodorus’ Irrationality Proofs.” Mathematics Magazine. 49(4): 201-203.
  • Selin, Helaine; D’Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer. ISBN 1-4020-0260-2.