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THE IMAGINARY NUMBER
THE COMPLEX NUMBERS
Numbers are so familiar to us that it might seem unimaginable that there was a time when the very concept
didn’t exist. Indeed the invention of numbers is lost in antiquity. Historians of mathematics speculate that
the origin of numbers was probably connected with real problems of life at the time, like describing whether
there was one animal, or more than one animal as food source (or a threat). A certain level of abstraction was
required to use numbers. Three rabbits, three stars and three rocks only share the common property of threeness. Manipulation of number – with no connection to physical objects – was a great intellectual leap.
BEYOND THE COUNTING NUMBERS
Negative numbers arrived on the scene much later. Trading and commerce meant that profit and loss should
be accounted for properly. Negative numbers were used to represent an absence or a loss. Despite that negative numbers were not immediately accepted by mathematicians. Early practitioners of algebra would often
discard negative values when they appeared as solutions. After all it’s easy to picture three people in a room. Or
two. Or one. Or even none. But what does minus one person in a room look like? One of my students recently
suggested it would be like a ghost. There may be grounds for rejecting negative numbers as the solution to a
particular problem but in other situations their use may be perfectly acceptable.
Negative numbers eventually found their place in our number system because they can be solutions of equations – just as valid as their positive namesakes. Likewise the history of zero is just as fraught with controversy
and confusion. Zero initially served as a placeholder in the representation of number. For example, it is the
zeros which tell you about the size of the numbers 15 and 105 and 1005. But zero as a number in its own right
took a long time to gain acceptance. Just like negative values, the solutions to some equations can be zero.
ICY SCIENCE | WINTER 2013- 2014