How to Subtract Complex Numbers Using Formula and Solved Examples
In everyday life, addition and subtraction are the most used mathematical operations. We use it on a daily basis. If we go to the supermarket for shopping, we need an addition or subtraction of money to make the payment. We use addition or subtraction when we have some chocolates, and someone gives us more chocolates or if someone takes some chocolates from us.
Confused about complex numbers? Don't worry; all your doubts will be cleared after reading this article. Because in this article today, we are going to learn about what complex numbers are, what their properties are when it comes to subtraction, and what the steps are to solve complex numbers. So now, let's learn about the subtraction of complex numbers.
What are Complex Numbers?
Complex numbers are also numbers, but they are different from normal numbers in many ways. They are made with the help of two numbers that are combined.
And in the new combined number, the first number is the real number, and another one is the imaginary number. Like all complex numbers can be expressed in the form a + bi, here, a is the real number, and bi is the imaginary number.
Remember that Complex Numbers are denoted by the letter 'C'.
Complex numbers also use all mathematical operations: addition, subtraction, multiplication, and division. But today, we will learn about the subtraction of complex numbers, which is as easy as solving a normal number.
And the Subtraction of complex numbers requires a formula.
Do you know the formula for the subtraction of Complex Numbers?
So the formula is (a+ib) - (c + id) = (a-c) +i(b-d).
Complex Number
Properties of Subtracting Complex Numbers
Closure property - In the closure property of Subtracting Complex Numbers, the difference between complex numbers is also a complex number.
Commutative property- Subtraction of complex numbers is not commutative.
Associative property- Subtraction of complex numbers is not associative.
Remember that this general form of a complex number is different from the polar form. Similarly, the addition and subtraction of complex numbers in the polar form are different.
Steps to Subtract Complex Numbers
Subtraction of two complex numbers is easy if it's done in the right and systematic way. So let's see the steps of subtraction of complex numbers.
$\mathrm{z}_1=\mathrm{a}+\mathrm{ib}, \mathrm{z}_2=\mathrm{c}+\mathrm{id}$
Subtraction
$\mathrm{z}_1-\mathrm{z}_2+(a-c)-(c+i d)$
$\mathrm{z}_1-\mathrm{z}_2=(a-c)+(b-d) i$
where $a, b, c, d$ are real number and $i$ is imaginary number
Step 1: Disperse the negative
Step 2: Now combine the real and imaginary complex numbers in a single group.
Step 3: Now, you need to combine and simplify similar terms.
Step 4: The answer will be there.
Subtraction of two complex numbers example
Subtraction of Complex Numbers Examples
For the subtraction of two complex numbers, we need to Subtract the real number from the real and the imaginary numbers from the imaginary. Let's learn it through examples:
Q 1. Subtract $(4-3 i)$ from $(7+5 i)$
Ans. $=7+5 i-(4-3 i)$
$=7+5 i-4+3 i$
$=3+8 i$
Q 2. Subtract ( 8 - 15i ) from (15 - 34i )
Ans. $=15-34 i-(8-15 i)$
$=15-34 i-8+15 i$
$=7-19 i$
Q 3. Subtract $(2+2 i)$ from $(3+5 i)$
Ans. $=3+5 i-(2+2 i)$
$=1+3 i$
Points to remember for the Subtraction of Complex Numbers
Subtracting a complex number is like subtracting two binomials; we just need to combine the like terms.
The subtraction of two complex numbers does not hold in the commutative law.
All real numbers are complex numbers, but all complex numbers need not be real numbers.
Practice Questions
We know that maths is not learned just with reading. To understand it, we need to solve the question. So that's why we have provided a practice sheet for you. With the help of this practice sheet, you can better understand the subtraction of two complex numbers. Below are the addition and subtraction of complex numbers worksheets for your practice. Use the subtraction of complex numbers formula mentioned in this article to solve these problems.
Q1. $(16+5 i)+(8-3 i)$
Ans. $24+2 i$
Q2. $(5+7 i)-2 i$
Ans. $5+5 i$
Q3. $(4+3 i)-(4-3 i)$
Ans. 6i
Summary
In this article, we learned about the subtraction of complex numbers. With the help of this article, we learned what complex number is, how to subtract complex numbers, and what the properties of Subtracting Complex Numbers are.
We learn the steps we need to follow while subtracting a complex number. We also checked an example of subtraction of complex numbers, and at the end, there is a complex number worksheet with the help of which we can easily clear our concept of subtraction of complex numbers.
FAQs on Subtraction of Complex Numbers Explained Clearly
1. What is subtraction of complex numbers?
The subtraction of complex numbers means subtracting the real parts and imaginary parts separately. If z₁ = a + bi and z₂ = c + di, then z₁ − z₂ = (a − c) + (b − d)i.
- Subtract the real parts: a − c
- Subtract the imaginary parts: b − d
- Write the result in standard form a + bi
2. What is the formula for subtracting complex numbers?
The formula for subtracting complex numbers is (a + bi) − (c + di) = (a − c) + (b − d)i. Here:
- a and c are real parts
- b and d are imaginary coefficients
- i is the imaginary unit where i² = −1
3. How do you subtract complex numbers step by step?
To subtract complex numbers, subtract corresponding real and imaginary parts separately. For example, subtract (5 + 3i) − (2 + i):
- Step 1: Subtract real parts → 5 − 2 = 3
- Step 2: Subtract imaginary parts → 3 − 1 = 2
- Step 3: Write the answer → 3 + 2i
4. Can you give an example of subtraction of complex numbers?
Yes, an example of subtracting complex numbers is (7 − 4i) − (3 + 2i) = 4 − 6i. Calculation:
- Real parts: 7 − 3 = 4
- Imaginary parts: −4 − 2 = −6
- Final answer: 4 − 6i
5. What happens when you subtract a complex number from a real number?
When subtracting a complex number from a real number, treat the real number as having zero imaginary part. For example, 6 − (2 + 3i):
- Write 6 as 6 + 0i
- Subtract real parts: 6 − 2 = 4
- Subtract imaginary parts: 0 − 3 = −3
- Result: 4 − 3i
6. Is subtraction of complex numbers commutative?
No, subtraction of complex numbers is not commutative, meaning z₁ − z₂ ≠ z₂ − z₁ in general. For example:
- (4 + 2i) − (1 + i) = 3 + i
- (1 + i) − (4 + 2i) = −3 − i
7. What is the geometric interpretation of subtracting complex numbers?
Geometrically, subtracting complex numbers represents finding the vector difference on the complex plane. If z₁ and z₂ are points, then z₁ − z₂ gives the vector from z₂ to z₁.
- Plot both numbers on the Argand plane
- Subtract coordinates: (a − c, b − d)
- The result is another point in the plane
8. How do you subtract complex numbers in polar form?
To subtract complex numbers in polar form, first convert them to rectangular form a + bi, then subtract normally. Since subtraction is easier in rectangular form:
- Convert r(cosθ + i sinθ) to a + bi
- Apply (a + bi) − (c + di)
- Simplify the result
9. What are common mistakes when subtracting complex numbers?
Common mistakes in subtraction of complex numbers include sign errors and incorrect distribution of the negative sign. Watch out for:
- Forgetting to change signs inside parentheses
- Mixing real and imaginary parts
- Not writing the answer in standard form a + bi
10. What is the difference between addition and subtraction of complex numbers?
The difference between addition and subtraction of complex numbers lies in how the parts are combined: addition uses plus signs, while subtraction uses minus signs. Formulas:
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
- Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
