Numbers Are Really Geometric Transformations
I want you to forget everything you think you know about numbers and common operations like addition and multiplication. In this article we will start over by thinking of things a little differently but as you will soon see, by this way of thinking, the real and complex numbers will be as natural as the natural numbers are. This will show why complex numbers are not as imaginary as some people think they are. We will see why we need the complex numbers, how they arrive quite naturally, and why we, in some sense, don’t need any other numbers.
By symmetry, we mean some kind of operation that leaves an object unchanged. If you rotate a square by 90 degrees, it looks as if you did nothing. So a square has a rotational symmetry of 90 degrees. Likewise, a circle has infinitely many rotational symmetries. If you shift a line by some length in the direction of the line, it also looks exactly like we did nothing to it. This is called translational symmetry. You can also zoom in or out on any geometric figure in the plane and the figure will look the same. This is called dilation. The actions of rotating, translating, reflecting, and dilating shapes are the main geometric transformations of the plane.
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