Rational and Irrational Numbers
Explained with examples and non examples
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
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)