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Topic: Re: Number line
Replies: 1   Last Post: Feb 19, 2009 5:16 PM

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James A. Landau

Posts: 17
Registered: 10/15/08
Re: Number line
Posted: Feb 19, 2009 3:36 PM
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> --- wrote:
> From: "Kutler, Samuel" <>
> Subject: RE: Sir Andrew Wiles
> Date: Mon, 16 Feb 2009 06:23:13 -0500
> When did the term real number line appear, and when did someone say that
> there is a one-to-one correspondence between the real numbers and the points on the real number line?

I have not been able to find a use of the exact English phrase "number line" before 1899, but the concept certainly existed before then. For example,


New York: MACMILLAN AND CO., 1893

Page 41
"In the course of this revision it became necessary to place the number-concepts of Algebra upon a basis independent of, but consistent with, Geometry. If a zero-point be selected on a straight line and also a fixed length, measured on this line, be chosen as the unit of length, any real number a can be represented by a point on this line at a distance from the zero-point equal to a units of length. Conversely, each point on the line is at a distance from the origin equal to a units of length, where a is a real number. This theorem may seem evident, but a little reflexion will show that it cannot be true unless the word number is so defined as to make the number-system continuous instead of discrete.

Definitions which will fulfil this requirement have been given by Weierstrass, G. Cantor, Heine, Dedekind, and others. Much of the theory of the higher Arithmetic is due to \Voierstrass and was communicated to the world in University lectures at Berlin."

(from Google Books,M1)

Let's go much further back. Classical Greek mathematicians invented the stereographic transformation:

Draw a line that is tangent to a circle at point O. Let point P be the other end of the diameter through O.
For any point X other than P on the circle, draw the line PX which intersects the tangent line at point Y. We have now created a one-to-one correspondence between the points X on the circle (other than P) and the points Y on the line.

(The stereographic transformation is still used today as one of the main methods for drawing a map of the earth on a flat sheet of paper.)

P is referred to as the "point at infinity".

Any point X (other than P) on the circle can be uniquely identified by the angle OPX. If angle OPX is measured from -pi to pi (or from -90 degrees to plus 90 degrees), then the ratio OY/OP is the tangent of angle OPX, or alternatively angle OPX is the arctangent of OY/OP. If we let OP = 1, then OY = tan OPX and OPX = arctan OY.

We have now associated every point Y on the tangent line with the number tan OPX = tan OPY, and conversely with every real number tan OPY we have associated a point. In other words, we have a one-to-one correspondence between the points of the tangent line and the real numbers, and we have done so using only two concepts, both known to the ancient Greeks: the stereographic transformation and the existence of the tangent ratio of opposite over adjacent.

If you want to make the above argument more rigorous, you need two more items, both known to Madhava of Sangamagramma in India circa 1400: that every real number can be represented as a Cauchy sequence (specifically as a decimal fraction,which is a Cauchy sequence), and the series formulae for tangent and arctangent.

James A. Landau
Northrop-Grumman Information Technology
8025 Black Horse Pike, Suite 300
West Atlantic City NJ 08232 USA

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