Adding, Subtracting, Multiplying
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Adding, Subtracting, Multiplying. Complex Numbers: z Î C. (4 + 3 i ) + (2 – 7 i ). Example: add. add the real part. 4 + 3 i + 2 – 7 i. add the imaginary part. 4 + 2 + 3 i – 7 i. – 4 i. = 6. Example: Subtract. (5 + 2 i ) – (7 – 3 i ). remove brackets. add the real part.

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Adding, Subtracting, Multiplying

Complex Numbers: z ÎC

(4 + 3i) + (2 – 7i)

Example: add

add the real part

4 + 3i + 2 – 7i

add the imaginary part.

4 + 2+ 3i – 7i

– 4i

= 6


Example: Subtract

(5 + 2i) – (7 – 3i)

remove brackets

add the real part

= 5 + 2i – 7 + 3i

add the imaginary part.

= 5 – 7+ 2i + 3i

= – 2

+ 5i


Multiplying Complex Numbers

Example: Multiply the following

(2 + 5i)(4 – 3i)

(use distributive property)

= 8 – 6i + 20i – 15i2

(i2 = –1)

= 8 + 14i – 15(–1)

= 8 + 14i + 15

= 23 + 14i


= –11 – 60i

(5 – 6i)2

Simplify the following:

(5 – 6i)(5 – 6i)

= 25 – 30i – 30i + 36i2

= 25 – 60i + 36i2

(i2 = –1)

= 25 – 60i + 36(–1)

= 25 – 60i – 36


Conjugates

5 – 2i

5 + 2i

3 + 4i

3 – 4i

–5 + 2i

–5 – 2i

a + bi

a – bi


Simplify the following (conjugates):

(5 – 6i)(5 + 6i)

= 25 + 30i – 30i – 36i2

= 25 – 36i2

(i2 = –1)

= 25 – 36(–1)

= 25 + 36

= 61


Multiplying conjugates:

Example:

(a – bi)(a + bi)

(2 – 7i)(2 + 7i)

= a2 + abi – abi – b2i2

a

b

(i2 = –1)

= a2 – b2i2

= 22 + 72

= 4 + 49

= a2 – b2(–1)

= 53

= a2 + b2

Product of conjugates is always real.


Squaring Complex Numbers:

Example:

(3 + 5i)2

(a + bi)2

= (a + bi)(a + bi)

a

b

= 32– 52 + 2(3)(5)i

= a2 + abi + abi + b2i2

= 9 – 25 + 30i

= a2 + b2i2 + 2abi

= – 16 + 30i

= a2 – b2 + 2abi

(3 – 5i)2

(a + bi)2 = a2 – b2 + 2abi

= – 16 – 30i

(a – bi)2 = a2 – b2 – 2abi


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