EXAMPLE 4 A Group of Numbers
Show that if we regard numbers as symmetries, with multiplication as the way of combining them, the positive real numbers have the four properties of symmetries noted above.
- Multiplying two positive real numbers yields another positive real number.
- The positive real number 1 is an identity element.
- Any positive real number has an inverse () in the collection.
- In multiplying several numbers, it doesn’t matter if we first multiply some adjacent pairs of numbers; that is, it doesn’t matter how we group or parenthesize the multiplication. For instance, is equal to and also to .
Hence, we say that the positive real numbers form a group under multiplication.