Groups (11)

And finally ...

Multiplying all whole numbers and fractions together will give us a group provided that we exclude that tricky number zero, which turns every other number it multiplies into zero as well, a sort of mathematical kiss of death.

There are many other kinds of infinite group. For example, repeating patterns and their shifts give infinite groups.
Here's an example of this. (you have to imagine that the pattern extends infinitely in both directions.)

Shifting the unit of repeat to the right or to the left works in exactly the same way as adding whole numbers.

Looking forward

The huge importance of groups in scientific applications comes from being able to apply their shared mathematical properties to the many different kinds of physical situation which share the same mathematical structure.
Groups have been found which are at the cutting edge of theories which describe the behaviour of the physical world, so this is exciting both scientifically and mathematically.

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