In mathematics, a complex variable opens a doorway

In mathematics, a complex variable opens a doorway to a fascinating world where numbers are more than just points on a plane. A complex variable, often denoted by \(z,\), combines a real part and an imaginary part, written as \(z=x+iy.\) Here, \(x\) is the real part, and \(y\) is the imaginary part, with \(i\) representing the square root of \(-1\). This combination creates a vibrant plane of possibilities, enriching our understanding and solving problems in many fields, from physics to engineering. Complex variables reveal the beauty and depth of mathematics in a truly unique way.

1. Complex Numbers10.7
1.Write imaginary numbers using i.
2.Perform arithmetic operations with complex numbers.
3.Raise ito powers.

3. Imaginary unit:
Imaginary number: A number that can be expressed
in the form bi, where bis 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 42i 1
1
2


i
i

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

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

7. Add Complex Numbers  ii 5435  1
1
2


i
i ii 5435 
Add the real parts –add the imaginary partsi89

8. Subtract 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 Numbersii74 1
1
2


i
i 2
28i 128 28

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

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

14. Multiply 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 Numbersi7
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 Numbersi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 form37
530i


19. Slide 10-19Copyright © 2011 Pearson Education, Inc.
Write in standard form.
a)
b)
c)
d) 4
23
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
23
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: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:

23. 432
iiii  Simplify: