Irrational Numbers


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It takes all kinds of numbers to make mathematics. One kind is the rather artistic group of numbers - they are called irrational. They are artistic because they appear in all kinds of ways in art, and some of these ways you can investigate in the worksheets that are attached to this page.

To start understanding how the irrational numbers differ from other numbers, we will first talk about rational numbers. A number is said to be rational if it can be written as a ratio of any two whole numbers (integers). So any number you can write as

is rational as long as both p and q are integers.

A rational number is any number that you can write as a fraction.

Irrational numbers have some common features - their decimals go on for ever, and there is no pattern in the decimal part of these numbers. So - although they all have infinitely many decimal places, there is no pattern in the way these appear. For example, 1/3 is a rational number although if you want to express it as a decimal you will get a reoccurring decimal. But that is beside the point: it doesn't matter if the decimal that you get from a fraction is reoccurring or not; as long as you can write that number as a fraction the number is not irrational.

So it follows that the numbers that cannot be written as fractions of two integers are irrational numbers. Some are more irrational than others, but that is another story...

Here are some examples of irrational numbers:  

, , , , , .

Click on some of the numbers above to see pages linke to them.

There is no use in trying to see whether a number is irrational by using a calculator. Your calculator will not tell you whether the number is a never-ending non-reoccurring decimal. Your calculator will simply give you the answer to the number of decimal places that it can cope with.  

Can you guess which of the mentioned irrational numbers is used in the construction of the pentagram?



Golden Rectangle can be constructed using the number

Which you can use to construct a logarithmic spiral

To learn how to do this, you can download Fibonacci worksheet no.3.

Or see what the Fibonacci worksheet no. 1 and no. 2 are all about.

This is the way to construct the - to learn about all the steps of this construction, go to Irrationals worksheet.


See also the page on famous numbers.


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