# Adding and Subtracting Complex Numbers

## Step by step guide to add and subtract the Complex Numbers

• A complex number is expressed in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers, and $$i$$, which is called an imaginary number, is $$a$$ solution of the equation $$x^2=-1$$
• For adding complex numbers: $$(a+bi)+(c+di)=(a+c)+(b+d)i$$
• For subtracting complex numbers: $$(a+bi)-(c+di)=(a-c)+(b-d)i$$

### Example 1:

Solve: $$10+(-5-3i)-2$$

Solution:

Remove parentheses: $$10+(-5-3i)-2→10-5-3i-2$$
Combine like terms: $$10-5-3i-2=3-3i$$

### Example 2:

Solve: $$-3+(4i)+(9-2i)$$

Solution:

Remove parentheses: $$-3+(4i)+(9-2i)→-3+4i+9-2i$$
Group like terms: $$-3+4i+9-2i→6+2i$$

### Example 3:

Solve: $$(-8+2i)+(-8+6i)$$

Solution:

Remove parentheses: $$(-8+2i)+(-8+6i)=-8+2i-8+6i$$
Group like terms: $$-8+2i-8+6i→-8-8+2i+6i$$
Add similar terms: $$-8-8+2i+6i=-16+8i$$

### Example 4:

Solve: $$(6-3i)-(-8-7i)$$

Solution:

Remove parentheses by multiplying the negative sign to the second parenthesis: $$(6-3i)-(-8-7i)=6-3i+8+7i$$
Combine like terms: $$6-3i+8+7i=14+4i$$

## Exercises

### Simplify.

• $$\color{blue}{– 8 + (2i) + (– 8 + 6i)}$$
• $$\color{blue}{12 – (5i) + (4 – 14i)}$$
• $$\color{blue}{– 2 + (– 8 – 7i) – 9}$$
• $$\color{blue}{(– 18 – 3i) + (11 + 5i)}$$
• $$\color{blue}{(3 + 5i) – (8 + 3i)}$$
• $$\color{blue}{(8 – 3i) – (4 + i)}$$

• $$\color{blue}{– 16 + 8i}$$
• $$\color{blue}{16 – 19i}$$
• $$\color{blue}{–19 – 7i}$$
• $$\color{blue}{–7 + 2i}$$
• $$\color{blue}{-5 + 2i}$$
• $$\color{blue}{4 – 4i}$$