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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)
Express answer in a + bi form.

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

Or by rule grouping: (6 + 4i) + ( 8 - 2i) = (6 + 8) + (4 - 2)i = 14 + 2i

 

ex2    ADD: 3 + (-2 - 4i) + (5 + i) + (0 - 2i)
Express answer in a + bi form.

3 + (-2 - 4i) + (5 + i) + (0 - 2i) = 3 - 2 - 4i + 5 + i - 2i = 6 - 5i
It is not necessary to always show the "grouping" of terms unless you are asked to do so.

ex3   ADD: cpex2
Express answer in a + bi form.

ex2n


ex4   ADD: cpex3
Express answer in a + bi form.

ex3n

 

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

Or by rule grouping: (10 + 3i) - (7 - 4i) = (10 - 7) + (3 - (-4))i = 3 + 7i

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

ex4


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: (a + bi) • (c + di) = (ac - bd) + (ad + bc)i
This rule shows that the product of two complex numbers is a complex number.

When multiplying two complex numbers, it will be sufficient to simply multiply as you would two binomials. Notice how the simple binomial multiplying will yield this multiplication rule.  
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)
Express answer in a + bi form.

(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
Express answer in a + bi form.

(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)
Express answer in a + bi form.

(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!)
Express answer in a + bi form.

(3 + 4i) • (3 - 4i) = 3(3 - 4i) + 4i(3 - 4i) = 9 - 12i + 12i - 16i2
= 9 - 16(-1) = 25 (a real number)
If written in "a + bi" form, the answer is 25 + 0i

 

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

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