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Arithmetic of Complex Numbers
(Add, Subtract, Multiply)
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Add and Subtract Complex Numbers



When performing the arithmetic operations of adding or subtracting on complex numbers, remember to combine "similar" terms. Also check to see if the answer must be expressed in simplest a+ bi form.

Addition Rule:    (a + bi) + (c + di) = (a + c) + (b + d)i
Add the "real" portions, and add the "imaginary" portions of the complex numbers.
Notice the distributive property at work when adding the imaginary portions.

Additive Identity:    (a + bi) + (0 + 0i) = a + bi

Additive Inverse:    (a + bi) + (-a - bi) = (0 +0i)

ex1    ADD: (6 + 4i) + (8 - 2i)

(6 + 4i) + ( 8 - 2i) = (6 + 8) + (4 - 2)i = 14 + 2i

ex2   ADD: cpex2 Express answer in a + bi form.

cpesans2a


ex3   ADD: cpex3Express answer in a + bi form.

ASM3N

 

Subtraction Rule:    (a + bi) - (c + di) = (a - c) + (b - d)i
Subtract the "real" portions, and subtract the "imaginary" portions of the complex numbers.
Notice the distributive property at work when subtracting the imaginary portions.

ex1    SUBTRACT: (10 + 3i) - (7 - 4i)
Express answer in a + bi form.

(10 + 3i) - (7 - 4i) = (10 - 7) + (3 - (-4))i = 3 + 7i

ex2   SUBTRACT: cpsub2
Express answer in a + bi form.

cpsub3


ex3   SUBTRACT: cpsub4
Express answer in a + bi form.

cpsub6

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Multiply Complex Numbers

 


Multiplying two complex numbers is accomplished in a manner similar to multiplying two binomials.
The distributive multiplication process (sometimes referred to as FOIL) is used.

distributive pic

Remember that
i 2 = -1
Distributive Multiplication
distmultmath
Be sure to replace i2 with (-1).

Multiplication Rule:   The product of two complex numbers is a complex number.
complexcomplexrule
The final result is expressed in a + bi form and is a complex number.

Multiplicative Identity:    (a + bi) • (1 + 0i) = a + bi

Mutiplicative Inverse:    complexmi
The number (0 + 0i) has no multiplicative inverse.


The conjugate of a complex number a + bi is the complex number a - bi.
For example, the conjugate of 3 + 7i is 3 - 7i.

(Notice that only the sign of the bi term is changed.)

If a complex number is multiplied by its conjugate, the result will be a positive real number
(which, of course, is still a complex number where the b in a + bi is 0).

The product of a complex number and its conjugate is a real number,
and is always positive.

conjugatmult

This answer is a real number (no i's).
In addition, since both values are squared, the answer is positive.


ex1    Compute: (2 + 3i)•(1 + 5i)

(2 + 3i)• (1 + 5i) = 2(1 + 5i) + 3i(1 + 5i) = 2 + 10i + 3i + 15i2
= 2 + 13i + 15(-1) =
-13 + 13i

ex2   Compute: (2 + i)2

(2 + i)• (2 + i) = 2(2 + i) + i(2 + i) = 4 + 2i + 2i + i2
= 4 + 4i + (-1) =
3 + 4i


ex3   Compute: (3 - 2i)•(1 - 4i)

(3 - 2i)• (1 - 4i) = 3(1 - 4i) + (-2i)(1 - 4i) = 3 - 12i - 2i + 8i2
= 3 - 14i + 8(-1) =
-5 - 14i


ex4   Compute: (3 +4i)•(3 - 4i) (conjugates!)

(3 + 4i)• (3 - 4i) = 3(3 - 4i) + 4i(3 - 4i) = 9 - 12i + 12i - 16i2
= 9 - 16(-1) =
25 (a real number)

 

ti84
How to use your
TI-83+/84+
calculator with
complex numbers.
Click here.

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Topical Outline | Algebra 2 Outline | MathBitsNotebook.com | MathBits' Teacher Resources
Terms of Use
   Contact Person: Donna Roberts