An introduction in complex numbers

Introduction

We have studied before different sets of numbers , the set of natural numbers "N" , the set of integers "Z" , the set of rational numbers "Q" , and the set of real numbers "IR" and today we will study the complex numbers "C"

Task

- At the end of the lesson the student should be able to :-

 

  1. Arithmetic Operations on Complex Numbers
  2. Square Roots of Negative Numbers
  3. Complex Solutions of Quadratic Equations 



The student can use this link to show on the lesson:-

https://www.youtube.com/watch?v=5hoGKfzTuOI

- we will use this aids :

  1. Board 
  2. Student book
  3. Coloured marker
Process

DEFINITION OF COMPLEX NUMBERS

A complex number is an expression of the form :- a+bi

where a and b are real numbers and i-1. The real part of this complex number is a. and the imaginary part is b. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

                                                                                The standard form :- a+bi

a:- is a real part

b :- is the imaginary part

 

Arithmetic Operations on Complex Numbers

ADDING, SUBTRACTING, AND MULTIPLYING COMPLEX NUMBERS :-

 

Definition Description

Addition

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

   to add complex numbers, add the real parts and add the imaginary parts

Subtraction

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

   to subtract complex numbers, subtract the real parts and subtract the imaginary parts

Multiplication

(a+bi) . (c+di) = (ac-bd) + (ad + bc)i 

 

   multiply complex numbers like binomials, using i² = -1

ex :-  Express the following in the form :- a+bi

(a) :-  (3+5i) + (4-2i)

(b) :- (3+5i)(4-2i)

(c) :- i²³



DIVIDING COMPLEX NUMBERS

 - To simplify the quotient a+bi / c+di

multiply the numerator and the denominator  by the complex conjugate of the denominator.

a+bi / c+di = ((a+bi) / (c+di) ).( (c-di) /(c-di) ) = (ac+bd)+(bc-ad)i / c² + d²

ex :- Express the following in the form :- a+bi

(a) :- 3+5i / 1-2i

(b) :- 7+3i  / 4i

 

 

 

 

Evaluation

 

ex :-  Express the following in the form :- a+bi

(a) :-  (3+5i) + (4-2i)

(b) :- (3+5i)(4-2i)

(c) :- i²³

 

ex :- Express the following in the form :- a+bi

(a) :- 3+5i / 1-2i

(b) :- 7+3i  / 4i

Conclusion

we can solve complex numbers by using addition - subtraction - multiplication and dividing 

Teacher Page

adding page 15

subtraction page 16

multiplication and dividing in page 17