MTH632 Handouts Lectures 1-25

-1/2 Sqrt.2 -2 -1 0 1 2 Numbers are useful: 6x + 1 = 13 Do we need more numbers? X2 + 2x + 2 = 0 , x2 + 1 = 0

We need one more number : i = √−𝟏 Complex Numbers = a + ib : a, b ∈𝑅 “ Cardino was the first to introduce complex numbers into algebra, but had misgivings about it” “To divide 10 into two parts, the product of which is 40.

5 + √−5 , 5 - √−5 Rafael Bombelli authored I’Algebra(1572 and 1579) a set of three books. “At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found the proof”.

L.Euler (1707-1783) introduced the notation i = √−𝟏

Topic:002

What are Complex Numbers?

A complex number z can be defined as an ordered pair z = (x,y) where x and y are real numbers. Example: z1 = (2,1) , z2 = (1,2) z1≠ z2 Complex Numbers can interpreted as points in the Complex plane. imaginary axis

(4,3)

-5 -4 -3 -2 -1 (0,0) -1 real axis -2 -3 -4 -5

• When real numbers x are displayed as points (x,0) on the real axis the set of complex numbers includes the real numbers as a subset.
• Complex numbers of the form (0,y) correspond to points on the y-axis and are pure imaginary numbers when y≠0. The y-axis is then referred to as the imaginary axis.

For Complex Numbers z = (x,y), the real numbers x and y are known as the real and imaginary parts of z respectively; and we write: x = Re z , y = Im z

Equality of Complex Numbers:

Two complex Numbers z1 and z2 are equal whenever they have the same real parts and the same imaginary parts. Thus the statement z1 = z2 means that z1 and z2 correspond to the same point in the complex, or z plane.

Topic:003 · Computations in Complex Numbers · Addition of two Complex Numbers:

For:

The addition is defined as: