Irrational Numbers

Just as rational numbers were made up by ratios, the irrational numbers cannot be expressed as a ratio. For example the square root of 2 does not equal one integer divided by another. You can get closer and closer with fractions, but you will never exactly equal √2.

This also means that the decimal expansion of an irrational number goes on forever.

Square roots, cube roots and other roots can all be irrational numbers. Transcendental numbers like pi and e are also irrational.

Raising an integer to a fraction will give an irrational number (unless it comes out exactly and gives an integer.) For example:

2^1/2 = √2 = 1.41421356...
5^3/2 = (√5)³ = 11.180339877...

Continued fractions

Any whole number or simple fraction can be written as a continued fraction. Continued fractions appear a little strange and look like this:

        1
a0 + ----------------------
            1
     a1 + ---------------
               1
          a2 + --------
               a3 + ...

To make it easier to write down, the above fraction is written as:

[a0 ; a1, a2, a3 ...]

For example, the continued fraction for 40/33 is:

        1
 1 + ----------------------
            1
      4 + ---------------
                1
          1 + ---------
                   1
              2 + ---
                   2

= [1; 4, 1, 2, 2]

You can check this on a calculator by starting at the bottom of the fraction and working upwards. First calculate 1/2, then add 2, then do 1/x , then add 1, then do 1/x again, then add 4, do 1/x again and finally add one to get 1.21212... = 40/33.

To work out the numbers that go in a continued fraction you go in the opposite direction. Start with 40/33. = 1.21212.

1.2121212

Subtract the whole number part, in this case 1

value remaining	continued fraction so far
0.2121212	1 + ...

Take 1/x

value remaining	continued fraction so far
4.7142857	1 + 1/ ...

Subtract the whole number part, in this case 4

value remaining	continued fraction so far
0.7142857	1 + 1/( 4 + ...)

Take 1/x

value remaining	continued fraction so far
1.4000000	1 + 1/( 4 + 1/ ...)

Subtract the whole number part, in this case 1

value remaining	continued fraction so far
0.4000000	1 + 1/( 4 + 1/ (1 +...))

Take 1/x

value remaining	continued fraction so far
2.5000000	1 + 1/( 4 + 1/ (1 + 1/ ...))

Subtract the whole number part, in this case 2

value remaining	continued fraction so far
0.5000000	1 + 1/( 4 + 1/ (1 + 1/ (2 + ...))

Take 1/x

value remaining	continued fraction so far
2.0000000	1 + 1/( 4 + 1/ (1 + 1/ (2 + 1/...))

We are left with a whole number so put this number in our continued fraction and we have reached the end.

40/33 = 1 + 1/( 4 + 1/ (1 + 1/ (2 + 1/2))

You can use the same method to work out continued fractions for square roots. Then the continued fraction goes on forever (unless the square root is a whole number eg √4 = 2):

           1
√2 = 1 + ---------------------
               1
         2 + --------------
                  1
             2 + --------
                  2 + ...

= [1; 2, 2, 2, 2, ...]

For square roots, the continued fraction soon forms a repeating pattern.

√2 = [1; 2, 2, ...]
√3 = [1; 1, 2, 1, 2, ...]
√5 = [2; 4, 4, ...]
√6 = [2; 2, 4, 2, 4, ...]
√7 = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]

For other numbers, such as π, the continued fraction does not form a pattern:

π = [3; 7, 15, 1, 292, 1, 1, 1, 2. ...]

By using the start of a continued fraction we can determine a fraction that is close to the irrational number. For the case of π:

[3;]	=	3/1	=	3	=	π - 0.14
[3; 7]	=	22/7	=	3.1428	=	π + 0.0012
[3; 7, 15]	=	333/106	=	3.141509	=	π - 0.000083
[3; 7, 15, 1]	=	355/113	=	3.14159292	=	π + 0.00000027
[3; 7, 15, 1, 292]	=	103993/33102	=	3.14159265301	=	π - 0.00000000058

The 292 in the continued fraction for π is quite large. This indicates that we will have to use much larger numbers in the faction to estimate π, before we find a better approximation than 355/113. So 355/113 will be quite a good approximation for π.