Addition & Subtraction of Complex Numbers(With Examples)

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A complex number is the combination of a real number and an imaginary number. It is of the form $a + ib$ and is usually represented by $z$. While adding two or more complex numbers, the real and imaginary parts of a complex number are added individually. Similarly, for the subtraction of complex numbers, the real and imaginary parts are subtracted separately.

Addition of Complex Numbers

While adding two or more complex numbers, you add real parts of complex numbers separately and similarly add imaginary parts of complex numbers separately. 

The formula used to add two complex numbers $z_{1} = a + ib$ and $z_{2} = c + id$ is 

$z_{1} + z_{2} = \left(a + ib \right) + \left(c + id \right) = \left(a + c \right) + \left(ib + id \right) = \left(a + c \right) + i \left(b + d \right)$. 

Hence we have $\left(a + ib \right) + \left(c + id \right) = \left(a + c \right) + i\left(b + d \right)$.

Steps for Adding Complex Numbers

Given below are the steps for adding complex numbers:

Step 1: Segregate the real and imaginary parts of the complex numbers

Step 2: Add the real parts of the complex numbers

Step 3: Add the imaginary parts of the complex numbers

Step 4: Write the final answer in $a + ib$ format

Examples

Let’s consider some examples to understand the process of the addition of complex numbers.

Ex 1: Add $2 + 3i$ and $5 + 4i$.

Here, $z_{1} = 2 + 3i$ and $z_{2} = 5 + 4i$

The real parts of the two complex numbers are $2$ and $5$ and the imaginary parts of the two complex numbers are $3i$ and $i$.

On adding real parts we get $2 + 5 = 7$ and on adding the imaginary parts we get $3i + 4i = 7i$.

Therefore, $\left(2 + 3i \right) + \left(5 + 4i \right) = 7 + 7i$.

Ex 2: Add $4 + 9i$ and $2 + 3i$.

The two complex numbers are $z_{1} = a + ib = 4 + 9i$ and $z_{2} = c + id = 2 + 3i$.

So, $a = 4$, $b = 9$, $c = 2$, and $d = 3$

According to the formula, $\left(a + ib \right) + \left(c + id \right) = \left(a + c \right) + i\left(b + d \right)$

Therefore, $\left(4 + 9i \right) + \left(2 + 3i \right) = \left(4 + 2 \right) + i \left(9 + 3 \right) = 6 + 12i$.

Ex 3: Add $-2 + 7i$ and $6 – 5i$.

$\left(-2 + 7i \right) + \left(6 – 5i \right) = \left(-2 + 6 \right) + \left(7 + \left(-5 \right) \right)i = 4 + 2i$.

Properties of Addition of Complex Numbers

Following are the properties of the addition of complex numbers:

• Closure Property: The sum of complex numbers is also a complex number. Hence, it holds the closure property.
• Commutative Property: The addition of complex numbers is commutative.
• Associative Property: The addition of complex numbers is associative.
• Additive Identity: $0$ is the additive identity of the complex numbers, i.e., for a complex number $z$, we have $z + 0 = 0 + z = z$.
• Additive Inverse: For a complex number $z$, the additive inverse in complex numbers is $-z$, i.e., $z + \left(-z \right) = 0$.

Subtraction of Complex Numbers

While subtracting two or more complex numbers, you subtract real parts of complex numbers separately and similarly subtract imaginary parts of complex numbers separately. 

The formula used to subtract two complex numbers $z_{1} = a + ib$ and $z_{2} = c + id$ is 

$z_{1} – z_{2} = \left(a + ib \right) – \left(c + id \right)$ = \left(a – c \right) + \left(ib – id \right) = \left(a – c \right) + i \left(b – d \right)$. 

Hence we have $\left(a + ib \right) – \left(c + id \right) = \left(a – c \right) + i\left(b – d \right)$.

Steps for Subtract Complex Numbers

Given below are the steps for subtracting complex numbers:

Step 1: Segregate the real and imaginary parts of the complex numbers

Step 2: Subtract the real parts of the complex numbers

Step 3: Subtract the imaginary parts of the complex numbers

Step 4: Write the final answer in $a + ib$ format

Examples

Let’s consider some examples to understand the process of the subtraction of complex numbers.

Ex 1: Subtract $3 + 2i$ from $7 + 11i$.

Here, $z_{1} = 3 + 2i$ and $z_{2} = 7 + 11i$

The real parts of the two complex numbers are $3$ and $7$ and the imaginary parts of the two complex numbers are $2i$ and $11i$.

On subtracting real parts we get $7 – 3 = 4$ and on adding the imaginary parts we get $11i – 2i = 9i$.

Therefore, $\left(7 + 11i \right) – \left(3 + 2i \right) = 4 + 9i$.

Ex 2: Subtract $3 + 5i$ from $9 + 14i$.

The two complex numbers are $z_{1} = a + ib = 3 + 5i$ and $z_{2} = c + id = 9 + 14i$.

So, $a = 3$, $b = 5$, $c = 9$, and $d = 14$

According to the formula, $\left(c + id \right) – \left(a + ib \right) = \left(c – a \right) + i\left(d – b \right)$

Therefore, $\left(9 + 14i \right) – \left(3 + 5i \right) = \left(9 – 3 \right) + i \left(14 – 5 \right) = 6 + 9i$.

Ex 3: Subtract $7 + 3i$ from $4 – 2i$.

$\left(4 – 2i \right) – \left(7 + 3i \right) = \left(4 – 7 \right) + \left(3 – \left(-2 \right) \right)i = 3 + 5i$.

Properties of Subtraction of Complex Numbers

Following are the properties of the subtraction of complex numbers:

• Closure Property: The difference between complex numbers is also a complex number. Hence, it holds the closure property.
• Commutative Property: The subtraction of complex numbers is not commutative.
• Associative Property: The subtraction of complex numbers is not associative.

Conclusion

The addition and subtraction of two complex numbers are very similar to the addition and subtraction of two binomial expressions, where the real parts and imaginary parts are added or subtracted separately.

Practice Problems

1. Add the following complex numbers
   • $z_{1} = 6 + 3i$ and $z_{2} = 9 + 2i$
   • $z_{1} = 2 + 5i$ and $z_{2} = 3 + 4i$
   • $z_{1} = -5 + 7i$ and $z_{2} = 7 – i$
   • $z_{1} = 12 – 8i$ and $z_{2} = -19 + 5i$
   • $z_{1} = -4 + 7i$ and $z_{2} = -10 – 8i$
2. Subtract the first complex number from the second complex number
   • $z_{1} = 1 + 2i$ and $z_{2} = 7 + 5i$
   • $z_{1} = 11 + 19i$ and $z_{2} = 8 + 3i$
   • $z_{1} = -4 + 5i$ and $z_{2} = 8 + 2i$
   • $z_{1} = 5 + 8i$ and $z_{2} = 3 – 3i$
   • $z_{1} = -2 – 6i$ and $z_{2} = 8 + 19i$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)