A presentation on complex numbers of mathematics

A brief introduction to complex numbers

1. Complex Numbers
10.710.7
1.Write imaginary numbers using i.
2.Perform arithmetic operations with complex numbers.
3.Raise i to powers.

3. Imaginary unit:

Imaginary number: A number that can be expressed
in the form bi, where b is a real number and i is the
imaginary unit.
i3 i9
i
5
2
1i
1
2
i

4. 16
21
32
1 16   4i  4i
1 21   21i
1 32   16 2i  4 2i
1
1
2


i
i

5. Complex number: A number that can be expressed in
the form a + bi, where a and b are real numbers and i
is the imaginary unit.
i34 i57 i
5
4
3
2

Examples:
real
im
a
g
in
a
ry

6. Complex Numbers: a + bComplex Numbers: a + bii
b = 0: Real numbers
a = 0: Imaginary numbers
real imaginary

7. Add Complex NumbersAdd Complex Numbers
  ii 5435 
1
1
2


i
i
ii 5435 
Add the real parts – add the imaginary parts
i89

8. Subtract Complex NumbersSubtract Complex Numbers
  ii 2138 
1
1
2


i
i
ii 2138 
i9

9. Slide 10- 9Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i
2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i

10. Slide 10- 10Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i
2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i

11. Multiply Complex NumbersMultiply Complex Numbers
ii74
1
1
2


i
i
2
28i
128
28

12. Multiply Complex NumbersMultiply Complex Numbers
 ii857
1
1
2


i
i
2
5635 ii
15635 i
5635i
i3556
standard a + bi form

13. Multiply Complex NumbersMultiply Complex Numbers
 ii425
1
1
2


i
i
2
28520 iii 
2
2320 ii
12320 i
2320 i
i322

14. Multiply Complex NumbersMultiply Complex Numbers
 
2
35i
1
1
2


i
i
2
9151525 iii 
193025 i
93025 i
i3016
Rewrite & Foil
  ii 3535 

15. Slide 10- 15Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i

16. Slide 10- 16Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i

17. Divide Complex NumbersDivide Complex Numbers
i7
6
1
1
2


i
i
i
i
i

7
6
2
7
6
i
i

17
6


i
7
6


i
7
6i

7
6i


18. Binomial denominator conjugate
Divide Complex NumbersDivide Complex Numbers
i6
5
1
1
2


i
i


i
i
i 


 6
6
6
5
2
36
530
i
i



136
530



i
i
37
5
37
30
standard a + bi form
37
530i


19. Slide 10- 19Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i


5 14
13 13
i

5 14
13 13
i

11 14
13 13
i

11 14
13 13
i


20. Slide 10- 20Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d)
4
2 3
i
i


5 14
13 13
i

5 14
13 13
i

11 14
13 13
i

11 14
13 13
i


21. Powers of i:Powers of i:
1
1
2


i
i
i
111
224
iii
iiiii  1
23
1
2
i



111
1
111
1
448
347
246
45




iii
iiiii
iii
iiiii

22. 
41
i ii
10
4
ii1

15
i 
3
3
4
ii ii1
i
111
224
iii
iiiii  1
23
1
2
i
Powers of i:Powers of i:

23. 432
iiii 
Simplify: