Arithmetic with Complex Numbers (Add, subtract, multiply and divide complex numbers)

Arithmetic with complex numbers involves operations like addition, subtraction, multiplication, and division. Let's explore each of these operations in detail:

1. Addition of Complex Numbers:

When adding two complex numbers, you add their real parts and imaginary parts separately.

Formula:

If you have two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, their sum is:

$$z_1 + z_2 = (a + bi) + (c + di) = (a + c) + (b + d)i$$

Example:

Add $z_1 = 3 + 4i$ and $z_2 = 1 + 2i$:

$$z_1 + z_2 = (3 + 1) + (4 + 2)i = 4 + 6i$$

So, the sum is $4 + 6i$.

2. Subtraction of Complex Numbers:

To subtract two complex numbers, subtract the real parts and the imaginary parts separately.

Formula:

If $z_1 = a + bi$ and $z_2 = c + di$, their difference is:

$$z_1 - z_2 = (a + bi) - (c + di) = (a - c) + (b - d)i$$

Example:

Subtract $z_2 = 1 + 2i$ from $z_1 = 3 + 4i$:

$$z_1 - z_2 = (3 - 1) + (4 - 2)i = 2 + 2i$$

So, the difference is $2 + 2i$.

3. Multiplication of Complex Numbers:

Multiplying complex numbers requires the distributive property and remembering that $i^2 = -1$.

Formula:

If $z_1 = a + bi$ and $z_2 = c + di$, their product is:

$$z_1 \times z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2$$

Since $i^2 = -1$, this simplifies to:

$$z_1 \times z_2 = (ac - bd) + (ad + bc)i$$

Example:

Multiply $z_1 = 3 + 4i$ and $z_2 = 1 + 2i$:

$$z_1 \times z_2 = (3 + 4i)(1 + 2i)$$

Using distributive property:

$$= 3(1) + 3(2i) + 4i(1) + 4i(2i)$$ $$= 3 + 6i + 4i + 8i^2$$

Since $i^2 = -1$, this becomes:

$$= 3 + 6i + 4i - 8$$

Now, combine the real and imaginary parts:

$$= (3 - 8) + (6i + 4i) = -5 + 10i$$

So, the product is $-5 + 10i$.

4. Division of Complex Numbers:

To divide complex numbers, multiply both the numerator and the denominator by the complex conjugate of the denominator. This process eliminates the imaginary part in the denominator.

Formula:

To divide $z_1 = a + bi$ by $z_2 = c + di$, multiply both the numerator and the denominator by the complex conjugate of $z_2$, which is $c - di$:

$$\frac{z_1}{z_2} = \frac{a + bi}{c + di} \times \frac{c - di}{c - di}$$

This results in:

$$\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$

The denominator simplifies as:

$$(c + di)(c - di) = c^2 + d^2$$

And the numerator is expanded as:

$$(a + bi)(c - di) = ac - adi + bci - bdi^2$$

Since $i^2 = -1$, this simplifies to:

$$= (ac + bd) + (bc - ad)i$$

So, the division is:

$$\frac{z_1}{z_2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

Example:

Divide $z_1 = 3 + 4i$ by $z_2 = 1 + 2i$:

$$\frac{z_1}{z_2} = \frac{3 + 4i}{1 + 2i}$$

Multiply by the conjugate of $1 + 2i$, which is $1 - 2i$:

$$\frac{z_1}{z_2} = \frac{(3 + 4i)(1 - 2i)}{(1 + 2i)(1 - 2i)}$$

First, compute the denominator:

$$(1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - (-4) = 5$$

Next, expand the numerator:

$$(3 + 4i)(1 - 2i) = 3(1) - 3(2i) + 4i(1) - 4i(2i)$$ $$= 3 - 6i + 4i - 8i^2$$

Since $i^2 = -1$, this becomes:

$$= 3 - 6i + 4i + 8$$

Combine the real and imaginary parts:

$$= (3 + 8) + (-6i + 4i) = 11 - 2i$$

So, the division is:

$$\frac{z_1}{z_2} = \frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i$$

Thus, the result is:

$$\frac{11}{5} - \frac{2}{5}i$$

Summary of Operations:

• Addition: Add real parts and imaginary parts separately.
• Subtraction: Subtract real parts and imaginary parts separately.
• Multiplication: Use distributive property and simplify with $i^2 = -1$.
• Division: Multiply numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.