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Irrational Number

An irrational number is real number that cannot be expressed as a ratio of two integers. When an irrational number is written with a decimal point, the numbers after the decimal point continue infinitely with no repeatable pattern.

The number "pi" or π (3.14159...) is a common example of an irrational number since it has an infinite number of digits after the decimal point. Many square roots are also irrational since they cannot be reduced to fractions. For example, the √2 is close to 1.414, but the exact value is indeterminate since the digits after the decimal point continue infinitely: 1.414213562373095... This value cannot be expressed as a fraction, so the square root of 2 is irrational.

As of 2018, π has been calculated to 22 trillion digits and no pattern has been found.

If a number can be expressed as a ratio of two integers, it is rational. Below are some examples of irrational and rational numbers.

  • 2 - rational
  • √2 - irrational
  • 3.14 - rational
  • π - irrational
  • √3 - irrational
  • √4 - rational
  • 7/8 - rational
  • 1.333 (repeating) - rational
  • 1.567 (repeating) - rational
  • 1.567183906 (not repeating) - irrational

NOTE: When irrational numbers are encountered by a computer program, they must be estimated.

Updated: June 5, 2018

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