COMPLEX NUMBERS

DEFINITION A complex number is made up of both real and imaginary components. It can be represented by an expression of the form (a+bi), where a and b are real numbers and i is imaginary. When defining i we say that i = . Then we can think of i2 as -1. In general, if c is any positive number, we would write:

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1

If we have a complex number z, where z=a+bithen a would be the real component (denoted: Re z) and b would represent the imaginary component of z (denoted Im z). Thus the real component of z=4+3i is 4 and the imaginary component would be 3. From this, it is obvious that two complex numbers (a+bi) and (c+di) are equal if and only if a=c and b=d, that is, the real and imaginary components are equal.

The complex number (a+bi) can also be represented by the ordered pair (a,b) and plotted on a special plane called the complex plane or the Argand Plane. On the ArgandPlane the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. This is shown in the figure on the next slide:

PROPERTIES OF THE COMPLEX NUMBERS

The SUM AND DIFFERENCE of complex numbers is defined by adding or subtracting their real components ie:

(a + bi)+(c + di) = (a + c) + (b + d)i(a - bi) - (c - di) = (a - c) - (b - d)i

Eg. (3 + i) + (1-7i) = (3 + 1 ) + (1 – 7)i = (4 – 6i)

The communitive and distributive properties hold for the PRODUCT of complex numbersie: (a + bi)x(c + di) = a(c + di) + bci + bdi2

= ac + adi + bci + bdi2

We know: i2 = -1Therefore giving us(a + bi)(c + di) = (ac - bd) + (ad + bc)i

CONJUGATESThe geometric interpretation of a complex conjugate is the reflection along the real axis. ExampleIf z = a+bi is a complex number, the conjugateof z would be………?

When DIVIDING two complex numbers you are basically rationalizing the denominator of a rational expression. If we have a complex number defined as z =a+bi then the conjugate would be See the following example:Example:

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