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Addition of Complex Numbers

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Last updated date: 11th Aug 2024
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Introduction to Complex Number

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

1. Is the addition of complex numbers associative?

All complex numbers are Commutative and Associative under addition and multiplication, and Multiplication is distributive under addition.

2. For what complex numbers are used?

The complex number is used to easily find the square root of a negative number. Here we use the value of $i^2=-1$ to represent the negative sign of a number, which is helpful to find the square root easily. Here we have $\sqrt{ }-4=\sqrt4{i}^2 =\pm 2 i$. Further, to find the negative roots of the quadratic equation, we used complex numbers.

3. Why is it called a complex number?

Eventually, the modern terminology emerged in the 19th century: "complex numbers", meaning that they consist of two parts, real and imaginary