How to Add Complex Numbers: Rules, Methods & Solved Questions
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 Explained with Examples
1. What is the fundamental rule for adding two complex numbers?
The fundamental rule for adding complex numbers is to combine the like parts separately. This means you add the real parts of the two complex numbers together and add the imaginary parts together. The result is a new complex number composed of the sum of the real parts and the sum of the imaginary parts.
2. What is the general formula for the addition of complex numbers?
If you have two complex numbers, z1 = (a + ib) and z2 = (c + id), their sum is found using the following formula: z1 + z2 = (a + c) + i(b + d). Here, 'a' and 'c' are the real parts, and 'ib' and 'id' are the imaginary parts.
3. Can you explain the addition of two complex numbers with a clear example?
Certainly. Let's add the complex numbers (4 + 3i) and (2 + 5i).
- Step 1: Identify and add the real parts: 4 + 2 = 6.
- Step 2: Identify and add the imaginary parts: 3i + 5i = (3 + 5)i = 8i.
- Step 3: Combine the results to form the new complex number.
4. What are the key algebraic properties of complex number addition as per the CBSE syllabus?
The addition of complex numbers follows several important algebraic properties that are fundamental in the NCERT curriculum. These include:
- The Closure Law: The sum of any two complex numbers is always another complex number.
- The Commutative Law: The order of addition does not matter. For any two complex numbers z1 and z2, z1 + z2 = z2 + z1.
- The Associative Law: When adding three or more complex numbers, the grouping does not affect the result. (z1 + z2) + z3 = z1 + (z2 + z3).
- Existence of Additive Identity: The complex number 0 (which is 0 + 0i) is the additive identity, as adding it to any complex number z results in z.
5. What is the concept of an additive inverse in complex numbers, and how does it relate to addition?
The additive inverse of a complex number z = a + ib is the number which, when added to z, results in the additive identity (0 + 0i). The inverse is denoted as -z and is equal to -a - ib. For example, the additive inverse of (3 - 7i) is (-3 + 7i). This concept is crucial because it defines the operation of subtraction, where subtracting a complex number is equivalent to adding its additive inverse.
6. How can the addition of complex numbers be represented geometrically on an Argand plane?
Geometrically, a complex number can be visualised as a vector from the origin to a point on the Argand plane. When you add two complex numbers, say z1 and z2, their sum z1 + z2 can be found using the parallelogram law of vector addition. If you draw vectors for z1 and z2 from the origin, the diagonal of the parallelogram formed by these two vectors (originating from the origin) represents the vector for their sum, z1 + z2.
7. How does the process of adding complex numbers differ from multiplying them?
The processes are fundamentally different. Addition is a straightforward combination of like terms: real parts are added to real parts, and imaginary parts are added to imaginary parts. Multiplication, on the other hand, involves the distributive property (similar to the FOIL method for binomials). When multiplying (a + ib) by (c + id), you must multiply each term in the first complex number by each term in the second and crucially, remember that i² = -1, which changes the structure of the final result.
