Adding and Subtracting Complex Numbers

What are Complex Numbers?

Adding and subtracting complex numbers are mathematical operations done on complex numbers. A complex number has the form a + ib and is typically denoted by the symbol z. Both a and b are genuine numbers in this case. The value ‘a is the real part, denoted by Re(z), and ‘b’ is the imaginary part, denoted by Im (z). ib is also known as an imaginary number.

Equality of Complex Numbers

Assume that z1 and z2 are the two complex numbers.

Here z1 = a1+i b1 and z2 = a2+ib2

If both the complex numbers z1 and z2 are equal (i.e) z1 = z2, then we can say that the real part of the first complex number is equal to the real part of the second complex number, whereas the imaginary part of the first complex number is equal to the imaginary part of the second complex number. Re means Real and Im means Imaginary 

(i.e) Re(z1) = Re(z2) and Im(z1) = Im(Z2)

Thus, the equality of complex numbers states that, 

if a1+ib1 = a2+ib2, then a1 = a2 and b1 = b2.

Operations on Complex Numbers

The basic algebraic operations on complex numbers discussed here are:

• Addition of Two Complex Numbers
• Subtraction(Difference) of Two Complex Numbers
• Multiplication of Two Complex Numbers
• Division of Two Complex Numbers.

Adding and Subtracting Complex Numbers

Adding and subtracting are mathematical operations that can be done on complex numbers. Before we get into the details of adding and taking away from complex numbers, let’s go over what complex numbers are. A complex number is made up of two real numbers and one made-up number. It looks like a+ib and is usually written as z. When adding complex numbers, the real and imagined parts are added together separately. In the same way, when we subtract complex numbers, we take away the real and imaginary parts of the numbers separately.

In this article, we will look at the idea of adding and taking away from complex numbers, as well as the rules and steps for doing so. We will also learn how to add and take away from complex numbers written in polar form.

What is Adding and Subtracting Complex Numbers?

The addition and subtraction of complex numbers are fundamental operations used with complex numbers. Similar to adding or subtracting polynomials, like terms are grouped together. Similarly, when adding and subtracting complex numbers, we mix the real and imaginary components before applying the operation. Consider the formula for the addition and subtraction of complex numbers. z1 = a + ib, whereas z2 = c + id, where a, b, c, and d are real values.

Adding Complex Numbers

While performing the operation of addition of complex numbers, we combine the real parts and imaginary parts of the complex numbers and add them. The formula for adding complex numbers is given by,

z1 + z2 = a + ib + c + id

= (a + c) + (ib + id)

= (a + c) + i(b + d)

Hence we have (a + ib) + (c + id) = (a + c) + i(b + d)

Subtracting Complex Numbers

For subtracting complex numbers, we divide the real and imaginary components and subtract the real and imaginary components of one complex number from the real and imaginary components of the other complex number, respectively. The formula for subtracting complex numbers is:

z1 – z2 = (a + ib) – (c + id)

= a + ib – c – id

= (a – c) + (ib – id)

= (a – c) + i(b – d)

Hence we have (a + ib) – (c + id) = (a – c) + i(b – d)

Steps and Rules for Adding and Subtracting Complex Numbers

Now we know the addition and subtraction formulas for complex numbers. Next, we shall analyse the procedure for the same in detail. The steps for adding and subtracting complex numbers are as follows:

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

Step 2: Add (subtract) the real parts of the complex numbers.

Step 3: Add (subtract) the imaginary parts of the complex numbers.

Step 4: Give the final answer in a + ib format.

Properties of Adding and Subtracting Complex Numbers

Listed below are the addition and subtraction properties of complex numbers:

Closure Property: The sum and difference of complex numbers is also a complex number. Hence, it holds the closure property.

Commutative Property: The addition of complex numbers is commutative but the subtraction of complex numbers is not commutative.

Associative Property: Adding complex numbers is associative but the subtraction of complex numbers is not 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 + (-z) = 0

Important Notes on Adding and Subtracting Complex Numbers

• Complex number addition and subtraction is identical to the addition and subtraction of two binomials. Thus, we only need to mix similar phrases.

• All real numbers are complex, but not all complex numbers are always real.

• Subtracting complex numbers violates the law of commutativity.

• For addition and subtraction of complex numbers in polar form, the complex numbers must first be converted to rectangle form. The final answer is then converted into polar form.

Examples of Adding and Subtracting Complex Numbers

Example 1: Add the two complex numbers z = 3 – 6i and w = -5 + 4i.

Solution: For adding complex numbers z and w, we will use the formula (a + ib) + (c + id) = (a + c) + i(b + d). Here a = 3, b = -6, c = -5, d = 4

z + w = (3 – 6i) + (-5 + 4i)

= (3 – 5) + i (-6 + 4)

= -2 + i(-2)

Answer: (3 – 4i) + (-5 + 7i) = -2 – 2i

Example 2: Subtract the complex numbers -12 + 6i and 7 + 5i.

Solution: For subtracting complex numbers, we will use the formula (a + ib) – (c + id) = (a – c) + i(b – d). Here a = -12, b = 6, c = 7, d = 5

(-12 + 6i) – (7 + 5i) = (-12 – 7) + i(6 – 5)

= -19 + i

Answer: (-12 + 6i) – (7 + 5i) = -19 + i

FAQs on Adding and Subtracting Complex Numbers