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Rational and Irrational Numbers

Explained with examples and non examples

Rational Numbers

Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero. Many people are surprised to know that a repeating decimal is a rational number.
Examples of Rational Numbers
  • 5 . You can express 5 as which is the quotient of the integer 5 and 1.
  • Picture of Rational NumbersSee reason above
  • Picture of rational numberSee first reason.
  • is rational because you can simplify the square root to 3 which is the quotient of the integer 3 and 1
  • All repeating decimals are rational. It's a little bit tricker to show why so I will do that elsewhere.
  • .9 is a rational number becuase it canbe expressed as 9/10 (All terminating decimals are rational numbers)
  • .73 is a rational number becuase it canbe expressed as 73/100

Irrational Numbers

Cannot be expressed as the quotient of two integers (ie a fraction) such that the denominator is not zero
Examples of Irrational Numbers
  • Picture of irrational numberYou cannot simply this square root (like we did with ) so this number is irrational. ALL irreducible square roots are irrational
  • .If a fraction has a denominator of zero, it is irrational
  • is irrational for the same reasons as the prior example
  • . Pi is probably the most well known irrational number out there .

Proof that repeating decimals are rational numbers

1) let x = .1
2) 10x = 1.1
3) 10x -1x = 1.1 -.1 ( ie subtract top equation from bottom equation)
2) 9x = 1
3) x = 1/9 . Yes, the repeating decimal .1 is equivalent to the fraction 1/9

Other Bases

All of the examples that we looked at involve our base ten number system, but irrational and rational numbers exist in other bases. For instance, the number .1 is irrational in binary. There is no way to write .1 as the quotient of two integers in binary ( base 2).
In fact, you have probably experienced the effects of irrational numbers if you use calculators and computers much. If you have ever noticed that your calculator's or computer' s results are just slightly off, the cause could very well be the irrationality of numbers like .1.. If your calculator has to work with .1 it will need to round the irrational binary representation of .1 and therefore the rounded number will be slightly inaccurate...leading to these kinds of very small calculation inaccuracies. (By the way, irrational binary numbers are not the only causes of this 'rounding errors' in computer calculations)