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# Puzzles, Games, & Tricks by Jerome S. Meyer

Author:Jerome S. Meyer
Language: eng
Format: epub
ISBN: 9781510727816
Publisher: Skyhorse Publishing
Published: 2017-08-02T16:00:00+00:00

All even powers of i are either −1 or +1 and whenever you see i2 you can be sure that it is –1, while i4 is +1 and so on. It follows from this that 8i2 is −8 and 29i4 is + 29, and so on. These imaginary numbers, which appeared at first to exist only in the imagination and to have no real significance, actually transform themselves into real numbers when raised to even powers.

A most fascinating discovery was that every number has n nth roots: 7, for example, has 3 cube roots, 5 fifth roots, and so on; and so has 16, for any number at all has n nth. roots. Maybe this is not news to you, but ask anyone you know (provided he is not a mathematician) what the cube root of 8 is. He will say “2” just like that. But he is only one third correct. There are two other cube roots of 8 and, while we can’t conceive them since they are what we call complex numbers, they exist nevertheless. This is extremely interesting. Here are two inconceivable numbers that we can’t use at all, yet if either of them is cubed it will give the answer of 8. To prove this let us examine the three cube roots of 8. They are: and . Let us cube this and see if it actually gives us 8. Remember that i2 = −1 and 3i2 = −3.