Irrational Numbers
An Irrational Number is a real number that cannot be written as a simple fraction:
1.5 is rational, but π is irrational
Irrational means not Rational (no ratio)
Let’s look at what makes a number rational or irrational …
Rational Numbers
A Rational Number can be written as a Ratio of two integers (i.e. a simple fraction).
Examples:
- 1.5 is rational, because it can be written as the ratio 3/2
- 7 is rational, because it can be written as the ratio 7/1
- 0.333… (3 repeating) is also rational, because it can be written as the ratio 1/3
But some numbers cannot be written as a ratio of two integers, they are called Irrational Numbers.
Example:
π (Pi) is a famous irrational number.
π = 3.1415926535897932384626433832795… (and more)
We cannot write down a simple fraction that equals Pi.
The popular approximation of 22/7 = 3.1428571428571… is close but not accurate.
Another clue is that the decimal goes on forever without repeating.
Cannot Be Written as a Fraction
It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy!
So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.
Example:
9.5 can be written as a simple fraction like this:
9.5 = 192
So it is a rational number (and so is not irrational)
Famous Irrational Numbers
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Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more …) |
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The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more …) |
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The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820… (and more …) |
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Many square roots, cube roots, etc are also irrational numbers. Examples:
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But √4 = 2 is rational, and √9 = 3 is rational, so not all roots are irrational.