How to Add and Subtract Complex Numbers (a+bi Format)

Last Updated: August 31, 2021

If you ever encounter in your degrees of high school math problems dealing with another system of math involving solving for the square root of a negative number, do not fear. A special value - $i$ - can help solve this problem. However, if you have a problem where you have to solve an addition or subtraction problem involving these problems, this article will give you help to find the right answer.

Steps

1

Have an understanding of the system of imaginary numbers. If you were trying to calculate the value of the square root of -1, you'd get an error on any calculator. However, when you see that imaginary numbers lie within a set of numbers, not within the rational numbers, an early Italian mathematician - Rafaello Bombelli, came up with a system of numbers to format numbers that couldn't be solved in the real/rational system. His value $i={\sqrt {-1}}$ . ^{[1]
X
Research source
<b>Course III: Integrated Mathematics.</b> Third Edition, ISBN: 9781567655216}
2
Write numbers under numbers. Complex numbers take the form $a+bi$ as a complex number takes the form of a real number and an imaginary unit/imaginary number - always with a plus sign in between. When both sets of complex numbers take this format, it'll be easy to put numbers under numbers.
- Remember that if there's no "a" value, additive identities of a complex number yield $0+0i$ . Additive identities over complex numbers contain a more valid $0+0i$ value, but it's more tricky to learn unless you need to use it for some reason.
Advertisement
3
Look for some differences in some expressions.
- Look for common imaginary values in square roots. Common square roots matched up with "i" may be carried along the trail straight down with your complex number with $i$ to the end. (In essence, the ${\sqrt {-25}}=5i$ since -25 can be factored into square roots of 25 and -1 - and therefore split off into it's component parts as $5$ and $i$ .
- Look for commonality if the values of the negative square roots aren't the same. Either simplify or expand to get their values to equal the same. Uncommon common imaginary numbers can be carried over into the final answer and can be written.
4
Look for a quick solution.
- When adding , you'll notice that $(a+bi)+(c+di)=(a+c)+(b+d)i$ .
- When subtracting, you'll notice that $(a+bi)-(c+di)=(a-c)+(b-d)i$ .
Advertisement

Method 1

Method 1 of 2:

Adding the Expressions

1
Examine your problem. Learn what needs to be solved and write it down.
- Add: $3+7i$ and $2+8i$ would then produce showing:
  ${\begin{aligned}3+7i&\\2+8i\end{aligned}}$
2
Add the "a" values together and place this portion of the answer below a line-separator separating your problem from your calculated answer. Adding is easier written top to bottom, but you may also need to determine what sign is needed, as you bring things down to create your sum.
- Using the example above, solve the $a+c$ problem. In this example since $3+2=5$ , write the real number portion as only (at first) $5$ .
3
Figure out what sign needs to go in the position between a and b. Solutions in math with complex numbers must always be written with a plus sign in between, but in case the b and d values sum to a negative number, remember that the addition of a negative number is equivalent to a subtraction sign.
- Therefore, write your sign between them as a positive sign $+$ .
4
Add the "b" values and place this value underneath the value for the line separator for b's. Bring down any additional portions of a part of the square root.
- For the example above, add your b value to your d value. Hence, $7+8$ which all you'll need to write down is $15$ .
5

Carry down the "imaginary number constant with the script-like tail" - $i$ into your answer. It must be written alongside your summed b values in all answers where the variables are the same.
6
Recognize that complex numbers in a+bi form - when added together - will still give numbers in a+bi form. If you get anything different, you might have done something wrong, or it's just that the answer is too simplified or not simplified enough.
- With no need to simplify and factor out any extra factors in these problems, don't try to show items factored out.
- For the example above, the answer would then lead to $5+15i$ .
Advertisement

Method 2

Method 2 of 2:

Subtracting the Expressions

1
Examine your problem. Learn what needs to be solved and write it down.
- Subtract $2-13i$ from $7-5i$ .
2
Figure out what your values for complex numbers represent and write them down. Write numbers under numbers. If any parts of the solution require a subtraction sign, change the expression to be the addition of a negative value instead.
- For this problem, you'd see the problem written as:
Advertisement

${\begin{aligned}7-5i&\\2-13i\end{aligned}}$

1
- Remember, the from value is called the minuend, while the "to" value is called the subtrahend. ^{[2]
  X
  Research source}
2
Subtract your a and c values from the left-hand column - the column of real numbers, as well as the b and d values - in two separate calculations. If the operators between the a and b values are through subtraction, change subtraction to the addition of a negative and add the values together.
- With the example above, your $a-c=7-2=5$ and when the $-5$ gets changed to a $+-5i$ and $-13i$ changes to $+-13i$ , you'll subtract the two $-5i-(-13i)=8i$ (remember subtraction of a bigger negative from a smaller negative will turn that portion of the value positive and when you add -5 and 13, you'll get 8.
3

Show the answer. Since the b value is positive, you can use an addition sign. If the b value ever became negative, you'd use a subtraction sign, since the addition of a negative is the same as a subtraction expression. When you perform the subtraction, you'll get an answer of $5+8i$

Advertisement