How to Add Complex Numbers Using Formula and Solved Examples
Complex numbers are helpful in finding the square root of negative numbers. The concept of complex numbers was first referred to in the 1st century by a greek mathematician, Hero of Alexandria, when he tried to find the square root of a negative number. Complex numbers have applications in much scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. So in this article, we will learn what complex numbers are and how to add complex numbers.
What are Complex Numbers?
A complex number is the sum of a real number and an imaginary number. A complex number is of the form a + ib and is usually represented by z. Here both a and b are real numbers. The value 'a' is called the real part denoted by Re(z), and 'b' is called the imaginary part Im(z). Also, ib is called an imaginary number.
Some of the examples of complex numbers are $2+3 i,-2-5 i, \dfrac{1}{2}+i \dfrac{3}{2}$, etc.
How to Add Complex Numbers?
The addition of complex numbers is similar to the addition of natural numbers. In complex numbers, the real part is added to the real part, and the imaginary part is added to the imaginary part.
For two complex numbers of the form $z_1=a+i d$ and $z_2=c+i d$, the sum of complex numbers $z_1+z_2=(a+c)+i(b+d)$.
The complex numbers follow all the following properties of addition.
Closure Law: The sum of two complex numbers is also a complex number. For two complex numbers, $z_1$ and $z_2$, the sum of $z_1+z_2$ is also a complex number.
Commutative Law: For two complex numbers $z_1, z_2$ we have $z_1+z_2=z_2+z_1$
Associative Law: For the given three complex numbers $z_1, z_2, z_3$ we have $z_1+\left(z_2+z_3\right)=\left(z_1+z_2\right)+z_3$.
Additive Identity: For a complex number $z=a+i b$, there exists $0=0$ $+i 0$, such that $z+0=0+z=0$.
Additive Inverse of Complex numbers: For the complex number $z=a+i b$, there exists a complex number $-z=-a-i b$ such that $z+(-z)=(-z)+z=0$. Here $-z$ is the additive inverse.
Solved Examples in Addition Of Complex Numbers
Let us see some solved examples in addition to complex numbers.
Q 1. Find the addition of two complex numbers $(2+3 i)$ and $(-9-2 i)$.
Ans: $(2+3 \mathrm{i})+(-9-2 \mathrm{i})$
$=2+3 \mathrm{i}-9-2 \mathrm{i}$
$=2-9+3 \mathrm{i}-2 \mathrm{i}$
$=-7+\mathrm{i}$
Q 2. Evaluate: $(2 \sqrt{ } 3+5 i)+(\sqrt{ } 3-7 i)$
Ans: $2 \sqrt{ } 3+5 i+\sqrt{3}-7 i$
$= 2 \sqrt{3}+\sqrt{ } 3+5 i-7 i$
$=3 \sqrt{ } 3-2 i$
Q 3. Express the complex number $(1-i)+(-1+6 i)$ in the standard form a $+$ ib.
Ans: $(1-i)+(-1+6 i)$
$=1-i-1+6 i$
$=1-1-i+6 i$
$=0+5 \mathrm{i}$, which is the required form.
Practice Question
Based on the explanation given above in this article, try to solve the following questions.
Q 1. Solve (4i-7)+(1-i)
Ans: 3i -6.
Q 2. Solve it. (i-1) + (1+i)
Ans: 2i
Q 3. (1+i)+ (1-i) is?
Ans: 2
Q 4. Find z+z, if z = 6i
Ans: 12i.
Summary
The' Complex' theme does not mean it is 'complicated'.
It means the two types of numbers, real and imaginary, together form a complex number, just like building a complex game on mobile or running programs on a computer. Thus we made the addition operation on complex numbers simple for you in this article. By studying the solved examples and practising the Practice Set, students will master the concept of Addition on Complex Numbers.
FAQs on Addition of Complex Numbers with Concept and Method
1. What is the addition of complex numbers?
The addition of complex numbers means adding their real parts together and their imaginary parts together.
If z₁ = a + bi and z₂ = c + di, then:
z₁ + z₂ = (a + c) + (b + d)i
This means:
- Add the real parts: a + c
- Add the imaginary parts: b + d
- Write the result in the form a + bi
2. What is the formula for adding two complex numbers?
The formula for adding two complex numbers is (a + bi) + (c + di) = (a + c) + (b + d)i.
Where:
- a and c are real parts
- b and d are imaginary parts
- i is the imaginary unit with i² = -1
3. How do you add complex numbers step by step?
To add complex numbers, combine like terms (real with real, imaginary with imaginary).
Steps:
- Write both numbers in the form a + bi
- Add the real parts
- Add the imaginary parts
- Simplify and write the final answer
(3 + 4i) + (2 − 5i)
= (3 + 2) + (4 − 5)i
= 5 − i
4. Can you give an example of addition of complex numbers?
An example of complex number addition is (1 + 6i) + (−3 + 2i) = −2 + 8i.
Solution:
- Add real parts: 1 + (−3) = −2
- Add imaginary parts: 6i + 2i = 8i
5. Why do we add real and imaginary parts separately?
We add real and imaginary parts separately because they are unlike terms.
A complex number has two components:
- Real part (a)
- Imaginary part (bi)
6. What are the properties of addition of complex numbers?
Addition of complex numbers satisfies the commutative, associative, and identity properties.
Important properties:
- Commutative: z₁ + z₂ = z₂ + z₁
- Associative: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)
- Additive identity: z + 0 = z, where 0 = 0 + 0i
7. What is the additive identity in complex numbers?
The additive identity in complex numbers is 0 + 0i.
When any complex number z = a + bi is added to 0 + 0i, the result remains the same:
(a + bi) + (0 + 0i) = a + bi
This means 0 + 0i does not change the value of a complex number.
8. What is the geometric interpretation of addition of complex numbers?
Geometrically, addition of complex numbers corresponds to vector addition on the Argand plane.
In the complex plane:
- The real part is plotted on the x-axis
- The imaginary part is plotted on the y-axis
- Addition follows the head-to-tail vector rule
9. Can you add more than two complex numbers?
Yes, you can add any number of complex numbers by combining all real parts and all imaginary parts separately.
Example:
(1 + 2i) + (3 − i) + (−2 + 4i)
- Real parts: 1 + 3 − 2 = 2
- Imaginary parts: 2 − 1 + 4 = 5
10. What are common mistakes when adding complex numbers?
A common mistake in adding complex numbers is mixing real and imaginary terms incorrectly.
Common errors include:
- Adding a real part to an imaginary part
- Forgetting the sign of the imaginary term
- Not writing the answer in a + bi form
