The complex numbers To make many of the rules of mathematics apply universally we need to enlarge our number field. If we desire that every integer has.

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Presentation transcript:

The complex numbers To make many of the rules of mathematics apply universally we need to enlarge our number field. If we desire that every integer has an inverse element, we accept the existence of rational numbers. If we desire every polynomial equation to have root(s) equal in number to its highest variable power, we must extend the real number field R to a larger field C of complex numbers.

Where was i hiding? You may remember being told that you can't take the square root of a negative number. That's because you had no numbers that, when squared, were negative. Squaring a negative number always gives you a positive. So you couldn't very well square-root a negative and expect to come up with anything sensible.

Where was i hiding? Now, however, you can take the square root of a negative number, but it involves using a new number to do it. At one time, nobody believed that any "real world" use would be found for this new number, other than easing the computations involved in solving certain equations, so the new number was viewed as being a pretend number invented for convenience sake.

A complex number A complex number is an ordered pair of real numbers (a,b). We call a the real part and b the imaginary part of the complex number. We write that new number as a + bi. The '+' is used to indicate the sign of the imaginary number part. The real number part represented by a which can be either positive or negative. Examples : 2 - 4i i / 4 i These are examples of numbers that we say are strictly complex.

VENN DIAGRAM Representation Since all number belong to the Complex number field, C, all number can be classified as complex. The Real number field, R, and the imaginary numbers, i, are subsets of this field as illustrated below. Real Numbers a + 0i Imaginary Numbers 0 + bi Complex Numbers a + bi

Graphical representation of a complex number A complex number has a representation in a plane. Simply take the x-axis as the real numbers and an y-axis as the imaginary numbers. Thus, giving the complex number a + bi the representation as point P with coordinates (a,b).

Graphing a Complex Number Therefore, complex numbers can be represented by a two dimensional graph. Here we see the graph of the complex number 3 – 2i.