complex numbers add and subtract complex numbers multiply and divide complex numbers
TRANSCRIPT
Complex Numbers
Add and Subtract complex numbers
Multiply and divide complex numbers
Lets solve x 2 = -1
To solve these problem, French mathematician René Descartes
proposed that a number i be defined such that i 2 = -1. iSo 1
Simplify
No Decimals
28
Simplify
No Decimals
7272
74128
ii
Simplify
No Decimals
3
3
32
32
yi
y
Simplify
No Decimals
yyi
yi
y
24
32
32
3
3
Powers of i values go a in a circle
Power of i are a cyclical function
1
1
1
4
3
2
i
ii
i
i
16128
15117
14106
1395
1;1
;
1;1
1,
iii
iiiii
iii
iiii
Powers of i values go a in a circle
Find the value the expression
1
1
1
4
3
2
i
ii
i
i
158
29438 1
i
iii
Powers of i values go a in a circle
Find the value the expression
1
1
1
4
3
2
i
ii
i
i
1
1
2394158
29438
iii
iii
Adding
Pure imaginary numbers are like terms.
iii 17512
Multiply Imaginary Numbers
Remember i2 equals -1
6
)1(6
6
23
2
i
ii
Multiply Imaginary Numbers
Pull the i out first
212
212
ii
Multiply Imaginary Numbers
Pull the i out first
62
62)1(
3222
212
212
2
i
ii
Solve
Remember when you take the square root of a number you must remember its positive and negative answer.
iy
y
y
y
y
2
4
4
205
0205
2
2
2
Complex number
Have two parts: Real and Imaginary
bia alRe aginaryIm
When adding complex numbers,
add the real numbers together and then the imaginary numbers.
ii 4253
When adding complex numbers,
add the real numbers together and then the imaginary numbers.
i
i
ii
5
4523
4253
When subtracting complex numbers,
add the real numbers together and then the imaginary numbers.
ii 7364
When subtracting complex numbers,
add the real numbers together and then the imaginary numbers.
ii
ii
7364
7364
When subtracting complex numbers,
add the real numbers together and then the imaginary numbers.
i
ii
ii
1
7364
7364
Multiplying Complex numbers
Its time to remember how to FOIL again.
2315420
534
iii
ii
Multiplying Complex numbers
Its time to remember how to FOIL again.
1315420
315420
534
2
ii
iii
ii
Multiplying Complex numbers
Its time to remember how to FOIL again.
i
i
ii
iii
ii
1123
31120
1315420
315420
534
2
Divide Complex numbers
Complex Conjugate work as with real numbers conjugates. Important must
break into fractions
13
1510
149
15110
49
1015
23
23
23
5
23
5
2
2
ii
i
ii
i
i
i
i
i
i
i13
15
13
10
Homework
Page 274
# 19 – 41 odd,
49 – 55 odd
Homework
Page 274
# 18 – 40 even,
48 – 54 even