Development of Numbers 4

< 20.4 The Number line | Topic Index | 20.6 Classification of Numbers 1 >

We saw that &ldquo;unexpected&rdquo; results of the 4 operations on numbers necessitated the invention of various types of numbers. This process culminated in the formation of the Set of Real Numbers. The Real Number System includes all numbers which could be plotted on the number line.

An operation other than the 4 operations helped invent another kind of number.

Imaginary Numbers

Mathematicians were challenged to interpret the operation when applied to Negative Numbers which occurred as solutions to certain algebraic equations. They realized that the solution would hinge on giving meaning to which for convenience they denoted as &ldquo;i''&rdquo;. ''This was possibly was a short form of Imaginary, the name they gave such numbers.

Keeping in mind consistency of mathematical logic, the formulated the following properties for representing and operating with &ldquo;i&rdquo;

Representation

&ldquo;i&rdquo; was represented in a direction vertical to the real number line. In the upward direction it would read &ldquo;i&rdquo;, &ldquo;2i&rdquo;, &ldquo;3i&rdquo; etc. In the downward direction it would read &ldquo;-i&rdquo;, &ldquo;-2i&rdquo;, &ldquo;-3i&rdquo; etc.

Operations

Multiplication with &ldquo;i&rdquo; was equivalent to a 90-degree rotation in the counter clockwise direction.

Imagine number +3 as a line on the number line joining 0 &amp; 3. This line can be thought of as pointing to the right.

Multiplying it by &ldquo;i&rdquo; is equivalent to rotating the line anticlockwise by 90 degrees. Then the line becomes a vertical line joining 0 &amp; 3i. The line now terminates at +3i. This can also be explained a +3 X i = +3i.

Multiplying it once again by i rotates the line by another 90 degrees in the anticlockwise direction. Now the line points to the left of the number line joining 0 with -3. This can be explained as +3i X i = -3, since by definition i X I = -1.

So this rotation model of multi0plication by i is consistent with mathematical logic.

Complex Numbers                            

These are combinations of imaginary numbers with real numbers, like +4 + 3i. They are again examples of a number represented with more than one number.

A number like +4 + 3i is represented on the vertical plane above the number line. It is a point on this plane which is +4 units to the right of the origin along the number line and +3i units in the vertical direction. It can be thought of as a line joining the origin 0, with the point (4,3) on a Cartesian plan whose x axis is the Number Line.

With the invention of Imaginary &amp; complex numbers, our number system is complete. It is also closed with respect to all arithmetic operations.

The complete number system can be graphically represented as shown in chapters 20,6 &amp; 20.7.

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