Ordinary
numbers can be thought of as points on a line
or ruler, sometimes called the number line.
But there is no room for
this new imaginary number i. Gauss suggested making
a two dimensional map where i became a number
one step perpendicular to the number line.
Think of the number line
running east-west. Imagianry numbers are got to
by heading north-south.
Although imaginary numbers
had been around since Fermat's day, the picture
of these numbers as points on a map was a significant
breakthrough that was only made in the nineteenth
century. To represent numbers via pictures was
a relatively revolutionary proposal at the time.
People were suspicious of pictures. They had the
power to mislead and lacked the rigour of symbolic
manipulation. After all, the language of mathematics
had been introduced so as to tame the physical
world. And numbers above all were things one added
and multiplied, not drew pictures of. However,
the nineteenth century saw the walls between geometry,
arithmetic and algebra crumble culminating in
the work of Riemann.
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