Title: An introduction to complex numbers

Are the real numbers not sufficient?

If we desire that every integer has an inverse element, we have to invent rational numbers and many things become much simpler.

If we desire every polynomial equation to have a root, we have to extend the real number field R to a larger field C of 'complex numbers'.

Notation

A complex number is written as z = a + ib where a is the real part of z and b is the imaginary part of z.  Every complex number a + ib has a complex conjugate given by a - ib.

Addition and Subtraction

To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.

(a + bi) + (c + id) = (a + c) + (b + d)i.

(a + bi) - (c + id) = (a - c) + (b - d)i.

Example 1:

(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.

(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.

Note

These operations are the same as combining similar terms in expressions that have a variable. For example, if we were to simplify the expression (3 - 5x) + (6 + 7x) by combining similar terms, then the constants 3 and 6 would be combined, and the terms -5x and 7x would be combined to yield 9 + 2x.

Product of complex numbers

Complex multiplication is a more difficult operation to understand from either an algebraic or a geometric point of view. Let's do it algebraically first, and let's take specific complex numbers to multiply, say 3+2i and 1+4i. Each has two terms, so when we multiply them, we'll get four terms:

(3+2i)(1+4i) = 3 + 12i + 2i + 8i²

Now the 12i + 2i simplifies to 14i, of course. What about the 8i2? Remember we introduced i as an abbreviation for √–1, the square root of –1. In other words, i is something whose square is –1. Thus, 8i2 equals –8. Therefore, the product (3+2i)(1+4i) equals –5+14i.

If you generalize this example, you'll get the general rule for multiplication

(x+yi)(u+vi) = (xu–yv)+(xv+yu)i

Example 2:

(2 + 3i)(1 + 2i)=(-4 + 7i)

Later on we shall give a geometric interpretation of the multiplication of complex numbers. The importance of that strange product is connected with

Conjugate complex numbers

We define the conjugate of a + bi as

 Example 3:

Modulus of a complex number

We define modulus or absolute value of a + bi as

We write this modulus of a + bi as |a + bi|.

Example 4:

|3 + 4i| =

Division if complex numbers

Here is the complete division problem, with the result written in standard form.

Fundamental Theorem of Algebra

Every nth - order polynomial possess exactly n complex roots.

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