Addition and Subtraction of Complex Numbers
Learn how to add and subtract complex numbers with step-by-step examples
Introduction to Operations on Complex Numbers
As discussed in the introduction to complex numbers, the principle of permanence states that calculation rules valid for real numbers should continue to apply to complex numbers.
This principle allows us to treat complex numbers using familiar arithmetic operations:
- Addition of complex numbers
- Subtraction of complex numbers
- Multiplication of complex numbers
- Division of complex numbers
The calculation rules that hold for real numbers are extended to complex numbers without modification, ensuring mathematical consistency and intuitive results.
Addition of Complex Numbers
When adding complex numbers, we treat them just like polynomials with variable \(i\). We combine real parts separately and imaginary parts separately.
For complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\):
\(\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i\)
The key insight is to group real and imaginary parts together:
- Real parts: Add \(a + c\)
- Imaginary parts: Add \(b + d\)
Step-by-Step Addition Process
1Write the addition problem
Write out the two complex numbers being added2Group real and imaginary parts
Combine like terms: real with real, imaginary with imaginary3Add the grouped parts
Perform arithmetic on real parts and on imaginary parts separately4Write the result
Express the answer in the form \(a + bi\)Example 1: Simple Addition
Add \((3 + i) + (1 - 2i)\)
Step 1: Write the addition
Step 2: Group like terms
Step 3: Combine the parts
Step 4: Simplify
Result: The sum is \(\displaystyle 4 - i\)
Example 2: More Complex Addition
Add \((2 + 3i) + (5 - 4i)\)
Real parts: \(2 + 5 = 7\)
Imaginary parts: \(3 + (-4) = -1\)
Subtraction of Complex Numbers
Subtraction of complex numbers follows the same principle as addition, with the important difference that we must distribute the negative sign to both the real and imaginary parts of the second number.
For complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\):
\(\displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i\)
When subtracting, the negative sign applies to BOTH terms:
\(\displaystyle (a + bi) - (c + di) = a + bi - c - di\)
Then group: \(\displaystyle (a - c) + (b - d)i\)
Step-by-Step Subtraction Process
1Write the subtraction problem
Write out the complex numbers with subtraction operator2Distribute the negative sign
Change signs of all terms in the second complex number3Group like terms
Combine real with real, imaginary with imaginary4Simplify
Perform arithmetic and write the answer in standard formExample 1: Simple Subtraction
Subtract \((3 + i) - (1 - 2i)\)
Step 1: Write the subtraction
Step 2: Distribute the negative sign
Step 3: Group like terms
Step 4: Combine
Result: The difference is \(\displaystyle 2 + 3i\)
Example 2: Subtraction with Negative Numbers
Subtract \((4 - 2i) - (3 + 5i)\)
Distribute negative: \(\displaystyle 4 - 2i - 3 - 5i\)
Real parts: \(4 - 3 = 1\)
Imaginary parts: \(-2 - 5 = -7\)
Addition vs. Subtraction Summary
| Operation | Formula | Example | Result |
|---|---|---|---|
| Addition | \(\displaystyle (a+bi)+(c+di)\) | \(\displaystyle (3+2i)+(1+4i)\) | \(\displaystyle 4+6i\) |
| Subtraction | \(\displaystyle (a+bi)-(c+di)\) | \(\displaystyle (3+2i)-(1+4i)\) | \(\displaystyle 2-2i\) |
| Key Difference | In subtraction, distribute the minus sign to both parts of the second number | ||
Properties of Addition and Subtraction
Commutative Property
\(\displaystyle z_1 + z_2 = z_2 + z_1\)The order doesn't matter in addition
Associative Property
\(\displaystyle (z_1+z_2)+z_3 = z_1+(z_2+z_3)\)Grouping doesn't affect the sum
Identity Element
\(\displaystyle z + 0 = z\)Adding zero leaves unchanged
Inverse Element
\(\displaystyle z + (-z) = 0\)Every number has an additive inverse
Geometric Interpretation in the Complex Plane
Addition and subtraction of complex numbers have geometric interpretations using the complex plane.
Addition as Vector Addition
In the complex plane, complex numbers can be represented as vectors from the origin. Adding two complex numbers corresponds to vector addition (parallelogram method).
Vector Addition in Complex Plane:
The sum \(z_1 + z_2\) is found by placing the tail of vector \(z_2\) at the head of vector \(z_1\). The resultant vector points from the origin to the endpoint.
Subtraction as Vector Subtraction
Subtracting complex numbers corresponds to vector subtraction. The difference \(z_1 - z_2\) is the vector from the head of \(z_2\) to the head of \(z_1\).
The geometric interpretation helps visualize why addition and subtraction of complex numbers follow the same rules as vector operations in two dimensions.
Common Mistakes to Avoid
WRONG: \(\displaystyle (3+2i)+(1+4i) = 3+2i+1+4i = 4+6i\) (written without grouping) ✗
RIGHT: \(\displaystyle = (3+1)+(2+4)i = 4+6i\) ✓
WRONG: \(\displaystyle (5+3i)-(2+2i) = 3+i\) (forgot to distribute minus to imaginary part) ✗
RIGHT: \(\displaystyle = 5+3i-2-2i = (5-2)+(3-2)i = 3+i\) ✓
WRONG: \(\displaystyle (2+3i)+(4+5i) = 6+8i = 14\) (can't add real to imaginary) ✗
RIGHT: \(\displaystyle = (2+4)+(3+5)i = 6+8i\) ✓
Practice Examples
Try These Addition Problems
Examples for Practice
- \(\displaystyle (2+5i)+(3+2i) = ?\) → Answer: \(5+7i\)
- \(\displaystyle (7-3i)+(1+4i) = ?\) → Answer: \(8+i\)
- \(\displaystyle (-2+6i)+(4-2i) = ?\) → Answer: \(2+4i\)
- \(\displaystyle (1+i)+(1+i) = ?\) → Answer: \(2+2i\)
Try These Subtraction Problems
Examples for Practice
- \(\displaystyle (8+5i)-(2+3i) = ?\) → Answer: \(6+2i\)
- \(\displaystyle (4-i)-(1+2i) = ?\) → Answer: \(3-3i\)
- \(\displaystyle (3+4i)-(3+4i) = ?\) → Answer: \(0\)
- \(\displaystyle (6-2i)-(-1+3i) = ?\) → Answer: \(7-5i\)
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