 ## How to Add and Subtract Complex Numbers

### Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers are two of the most basic things you can do with them. Like when we add or subtract a polynomial, we put like terms together. In the same way, when we add or subtract complex numbers, we put together the real and imaginary parts of the numbers and then use the operation. Let's look at the equation for adding and subtracting complex numbers.

Consider $$z_1 \ = \ a \ + \ ib$$ and $$z_2 \ = \ c \ + \ id$$ as complex numbers, where $$a, \ b, \ c,$$ and $$d$$ are real numbers.

Addition of complex numbers can be done by,

$$z_1 \ + \ z_2 \ = \ a \ + \ ib \ + \ c \ + \ id \ = \ (a \ + \ c) \ + \ (ib \ + \ id) \ = \ (a \ + \ c) \ + \ i(b \ + \ d)$$

### Subtraction of Complex Numbers

To subtract two complex numbers, we look at their real and imaginary parts separately and then take the real and imaginary parts of one complex number and subtract them from the real and imaginary parts of the other complex number. Complex numbers can be subtracted using the formula,

$$z_1 \ - \ z_2 \ = \ (a \ + \ ib) \ - \ (c \ + \ id) \ = \ a \ + \ ib \ - \ c \ – \ id \ = \ (a \ - \ c) \ + \ (ib \ - \ id) \ =$$ $$(a \ - \ c) \ + \ i(b \ - \ d)$$

### Properties of Addition and Subtraction of Complex Numbers

Here is a list of the properties of adding and subtracting complex numbers:

• The Closure Property says that the sum and difference of two complex numbers is also a complex number.
• Commutative Property: Complex numbers are commutative when added, but they are not commutative when subtracted.
• Associative Property: It is associative to add complex numbers, but it is not associative to subtract complex numbers.
• Additive Identity: $$0$$ is the additive identity of complex numbers, which means that $$z \ + \ 0 \ = \ 0 \ + \ z \ = \ z$$ for any complex number $$z$$.
• Additive Inverse: The additive inverse of a complex number, $$z$$, is $$-z$$. This means that $$z \ + \ (-z) \ = \ 0$$.

### Example1:

Find the answer to $$(3 \ - \ i) \ + \ (-8 \ + \ 5i)$$

Solution:

Get rid of the parentheses: $$3 \ - \ i \ - \ 8 \ + \ 5i$$

Combine like terms: $$3 \ - \ 8 \ - \ i \ + \ 5i \ = \ -5 \ + \ i(-1 \ + \ 5) \ = \ -5 \ + \ 4i$$

### Example2:

Find the answer to $$(9 \ + \ 4i) \ - \ (6 \ - \ 8i)$$

Solution:

Multiply the negative sign by the second parentheses to get rid of the parentheses: $$9 \ + \ 4i \ - \ 6 \ + \ 8i$$

Combine like terms: $$9 \ - \ 6 \ + \ 4i \ + \ 8i \ = \ 3 \ + \ i(4 \ + \ 8) \ = \ 3 \ + \ 12i$$

### Exercises for Adding and Subtracting Complex Numbers

1) Find the answer: $$(1 \ + \ 2i) \ + \ 5i$$

2) Find the answer: $$(3 \ - \ i) \ + \ 7i$$

3) Find the answer: $$(-2 \ + \ 4i) \ + \ (11 \ + \ 6i)$$

4) Find the answer: $$(7 \ - \ 6i) \ + \ (5 \ + \ 7i)$$

5) Find the answer: $$(-3 \ - \ 9i) \ + \ (4 \ - \ 12i)$$

6) Find the answer: $$(5 \ + \ 2i) \ - \ (7 \ - \ 4i)$$

7) Find the answer: $$(8 \ - \ 9i) \ - \ (3 \ + \ 11i)$$

8) Find the answer: $$(-12 \ - \ 17i) \ - \ (-3 \ + \ 5i)$$

9) Find the answer: $$(-6 \ + \ 18i) \ - \ (13 \ + \ 7i)$$

10)  Find the answer: $$(-3 \ - \ 14i) \ - \ (21 \ - \ 4i)$$

1) Find the answer: $$(1 \ + \ 2i) \ + \ 5i$$

$$\color{red}{1 \ + \ (2i \ + \ 5i) \ = \ 1 \ + \ 7i}$$

2) Find the answer: $$(3 \ - \ i) \ + \ 7i$$

$$\color{red}{3 \ + \ (-i \ + \ 7i) \ = \ 3 \ + \ 6i}$$

3) Find the answer: $$(-2 \ + \ 4i) \ + \ (11 \ + \ 6i)$$

$$\color{red}{(-2 \ + \ 11) \ + \ (4i \ + \ 6i) \ = \ 9 \ + \ 10i}$$

4) Find the answer: $$(7 \ - \ 6i) \ + \ (5 \ + \ 7i)$$

$$\color{red}{(7 \ + \ 5) \ + \ (-6i \ + \ 7i) \ = \ 12 \ + \ i}$$

5) Find the answer: $$(-3 \ - \ 9i) \ + \ (4 \ - \ 12i)$$

$$\color{red}{(-3 \ + \ 4) \ + \ (-9i \ - \ 12i) \ = \ 1 \ - \ 21i}$$

6) Find the answer: $$(5 \ + \ 2i) \ - \ (7 \ - \ 4i)$$

$$\color{red}{(5 \ - \ 7) \ + \ (2i \ - \ (-4i)) \ = \ -2 \ + \ 6i}$$

7) Find the answer: $$(8 \ - \ 9i) \ - \ (3 \ + \ 11i)$$

$$\color{red}{(8 \ - \ 3) \ + \ (-9i \ - \ 11i) \ = \ 5 \ + \ 20i}$$

8) Find the answer: $$(-12 \ - \ 17i) \ - \ (-3 \ + \ 5i)$$

$$\color{red}{(-12 \ + \ (-3)) \ + \ (-17i \ - \ 5i) \ = \ -15 \ - \ 22i}$$

9) Find the answer: $$(-6 \ + \ 18i) \ - \ (13 \ + \ 7i)$$

$$\color{red}{(-6 \ - \ 13) \ + \ (18i \ - \ 7i) \ = \ -19 \ + \ 11i}$$

10) Find the answer: $$(-3 \ - \ 14i) \ - \ (21 \ - \ 4i)$$

$$\color{red}{(-3 \ - \ 21) \ + \ (-14i \ - \ (-4i)) \ = \ -24 \ - \ 10i}$$

### Related Topics

How to Multiply and Divide Complex Numbers
How to Graph Complex Numbers
How to Solve Rationalizing Imaginary Denominators

## Adding and Subtracting Complex Numbers Practice Quiz

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