# Irrational Numbers

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.

### References

• 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.