Arithmetic of Complex Numbers - Add, Subtract, Multiply

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:
bi + di becomes (b + d)i .

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

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

ADD: (6 + 4i) + (8 - 2i)
Express answer in a + bi form.

Adding parts: (6 + 4i) + ( 8 - 2i) = 6 + 4i + 8 - 2i = 6 + 8 + 4i - 2i = 14 + 2i

Addition Rule grouping: (6 + 4i) + ( 8 - 2i) = (6 + 8) + (4 - 2)i = 14 + 2i

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.
Just be careful of the signs when subtraction is involved.

ADD:
Express answer in a + bi form.

ex2n
Notice that the radicals were simplified before the addition began.

ADD:
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:
bi - di becomes (b - d)i .

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

Subtract Rule grouping: (10 + 3i) - (7 - 4i) = (10 - 7) + (3 - (-4))i = 3 + 7i

SUBTRACT:
Express answer in a + bi form.

ex4

SUBTRACT:
Express answer in a + bi form.

cpsub6

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.

Remember that
i² = -1

Distributive Multiplication

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 form will be 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.

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

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

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

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

For
calculator help with
complex
numbers.
click here.

For calculator help with
complex
numbers
click here.

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