Addition of Complex Numbers
A complex number is of the form \(x+iy\) and is usually represented by \(z\).
Every complex number indicates a point in the XY-plane.
For example, the complex number \(x+iy\) represents the point \((x,y)\) in the XY-plane.
You can see this in the following illustration. Here, you can drag the point by which the complex number and the corresponding point are changed.
But, how to calculate complex numbers? Let's learn how to add complex numbers in this sectoin.
Lesson Plan
What Do You Mean by Addition of Complex Numbers?
Geometrically, the addition of two complex numbers is the addition of corresponding position vectors using the parallelogram law of addition of vectors.
Consider two complex numbers: \[\begin{array}{l}
z_{1}=a_{1}+i b_{1} \\[0.2cm]
z_{2}=a_{2}+i b_{2}
\end{array}\]
We already know that every complex number can be represented as a point on the coordinate plane (which is also called as complex plane in case of complex numbers).
So let us represent \(z_1\) and \(z_2\) as points on the complex plane and join each of them to the origin to get their corresponding position vectors.
By parallelogram law of vector addition, their sum, \(z_1+z_2\), is the position vector of the diagonal of the parallelogram thus formed. For this,
- First, draw the parallelogram with \(z_1\) and \(z_2\) as opposite vertices.
- Draw the diagonal vector whose endpoints are NOT \(z_1\) and \(z_2\).
- The resultant vector is the sum \(z_1+z_2\).
i.e., the sum is the tip of the diagonal that doesn't join \(z_1\) and \(z_2\).
Example
Let us add two complex numbers:
\[\begin{array}{l}
z_{1}=3+3i\\[0.2cm]
z_{2}=-3+i
\end{array}\]
We know that:
- \(z_1=3+3i\) corresponds to the point (3, 3) and
- \(z_2=-3+i\) corresponds to the point (-3, 1).
We just plot these on the complex plane and apply the parallelogram law of vector addition (by which, the tip of the diagonal represents the sum) to find their sum.
The tip of the diagonal is (0, 4) which corresponds to the complex number \(0+4i = 4i\).
Hence, \[z_1+z_2 = 4i\]
You can visualize the geometrical addition of complex numbers using the following illustration:
What Are the Steps to Add Complex Numbers?
We already learned how to add complex numbers geometrically.
But isn't that process difficult?
Here is the easy process to add complex numbers.
To add two complex numbers:
- Group the real numbers and imaginary numbers of the complex numbers.
- Simplify them.
Example
Let us add the same complex numbers in the previous example using these steps.
\[\begin{array}{l}
z_{1}=3+3i\\[0.2cm]
z_{2}=-3+i
\end{array}\]
Their sum is \[z_1+z_2 = (3+3i)+(-3+i)\]
|
Group the real and imaginary numbers |
\((3-3)+(3i+i)\) |
|---|---|
|
Simplify |
\(0+4i=4i\) |
Thus, the sum of the given two complex numbers is: \[z_1+z_2= 4i\]
The set of complex numbers is:
- Closed, as the sum of two complex numbers is also a complex number.
- Commutative, as the sum of two complex numbers, remains the same even when they are interchanged.
- Associative, as the sum of three complex numbers, remains the same even when the grouping is changed.
- The additive identity is 0 (which can be written as \(0 + 0i\)) and hence the set of complex numbers has the additive identity.
- The additive inverse of a complex number \(x+iy\) is \(-x-iy\) because \[(x+iy)+(-x-iy)=0\]
Solved Examples
| Example 1 |
Can we help Andrea add the following complex numbers geometrically?
\[z_{1}=-1+3 i \text { and } z_{2}=4-i\]
Solution
Consider the following figure:
From the figure,
| \(\therefore\) \(z_{1}+z_{2}=3+2 i\) |
Can you try verifying this algebraically?
| Example 2 |
Can we help James find the sum of the following complex numbers algebraically?
\[z_1=-2+\sqrt{-16} \text { and } z_2=3-\sqrt{-25}\]
Solution
We will find the sum of given two complex numbers by combining the real and imaginary parts.
But before that Let us recall the value of \(i\) (iota) to be \( \sqrt{-1}\). Thus,
\[ \begin{align} \sqrt{-16} &= \sqrt{-1} \cdot \sqrt{16}= i(4)= 4i\\[0.2cm] \sqrt{-25} &= \sqrt{-1} \cdot \sqrt{25}= i(5)= 5i \end{align}\]
Hence,
\[ \begin{align} &z_1+z_2\\[0.2cm] &=(-2+\sqrt{-16})+(3-\sqrt{-25})\\[0.2cm] &= -2+ 4i + 3-5i \\[0.2cm] &=(-2+3)+(4i-5i)\\[0.2cm] &=1-i \end{align}\]
| \(\therefore z_1+z_2 = 1-i\) |
- The addition of complex numbers is just like adding two binomials. i.e., we just need to combine the like terms.
- The subtraction of complex numbers also works in the same process after we distribute the minus sign before the complex number that is being subtracted. For example,
\( \begin{align}&(3+2i)-(1+i)\\[0.2cm]& = 3+2i-1-i\\[0.2cm]& = (3-1)+(2i-i)\\[0.2cm]& = 2+i \end{align}\) - The multiplication of two complex numbers is NOT just multiplying the real and imaginary numbers. We have to multiply the complex numbers considering them as binomials. For example:
\( \begin{align} &(3+2i)(1+i) \\[0.2cm]&= 3+3i+2i+2i^2\\[0.2cm] &= 3+5i-2 \\[0.2cm] &=1+5i \end{align} \)
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
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FAQs on Addition of Complex Numbers
1. Is every complex number a real number?
No, every complex number is NOT a real number.
For example, \(4+ 3i\) is a complex number but NOT a real number.
2. Where are complex numbers used?
The complex numbers are used in solving the quadratic equations (that have no real solutions).
Also, they are used in advanced calculus.
3. How are complex numbers written?
The complex numbers are written in the form \(x+iy\) and they correspond to the points on the coordinate plane (or complex plane)
i.e., \(x+iy\) corresponds to \((x, y)\) in the complex plane.
4. What is the sum of complex numbers?
The sum of two complex numbers is a complex number whose real and imaginary parts are obtained by adding the corresponding parts of the given two complex numbers.
i.e., \[\begin{align}&(a_1+ib_1)+(a_2+ib_2)\\[0.2cm]& = (a_1+a_2) + i (b_1+b_2)\end{align}\]
5. Are complex numbers commutative?
Yes, the complex numbers are commutative because the sum of two complex numbers doesn't change though we interchange the complex numbers.
This is linked with the fact that the set of real numbers is commutative (as both real and imaginary parts of a complex number are real numbers).
6. How do you add and multiply complex numbers?
We add complex numbers just by grouping their real and imaginary parts. For example:
\[ \begin{align} &(3+2i)+(1+i) \\[0.2cm]&= (3+1)+(2i+i)\\[0.2cm] &= 4+3i \end{align}\]
We multiply complex numbers by considering them as binomials.
\[ \begin{align} &(3+2i)(1+i)\\[0.2cm] &= 3+3i+2i+2i^2\\[0.2cm] &= 3+5i-2 \\[0.2cm] &=1+5i \end{align} \]
7. Can the sum of two complex numbers be a real number?
Yes, the sum of two complex numbers can be a real number.
For example: \( (1+2i)+(3-2i) = 4\)
8. What properties apply to addition with complex numbers?
The set of complex numbers is closed, associative, and commutative under addition.
Also, every complex number has its additive inverse in the set of complex numbers.
The additive identity, 0 is also present in the set of complex numbers.
Distributive property can also be used for complex numbers.
9. Are complex numbers closed under addition?
Yes, because the sum of two complex numbers is a complex number.
10. What is the product of two complex numbers?
We multiply complex numbers by considering them as binomials.
\[ \begin{align} &(3+i)(1+2i)\\[0.2cm] &= 3+6i+i+2i^2\\[0.2cm] &= 3+7i-2 \\[0.2cm] &=1+7i \end{align} \]