Multiplication of Complex Numbers
Online calculator for multiplying complex numbers with step-by-step explanation
Multiplication Calculator
Multiplication of Complex Numbers
The multiplication of complex numbers is done by expanding the parentheses and using \(i^2 = -1\).
Multiplication - Properties
FOIL Method
First, Outer, Inner, Last
Systematic expansion of the parentheses
Formula
Important: \(i^2 = -1\)
When expanding you will encounter \(i^2\), which should be replaced by \(-1\).
Algebraic Rules
- Commutative: \(z_1 \cdot z_2 = z_2 \cdot z_1\)
- Associative: \((z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)\)
- Distributive: \(z_1(z_2 + z_3) = z_1z_2 + z_1z_3\)
- Magnitude: \(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)
In Polar Form
Easier: multiply magnitudes, add angles
\(r_1e^{i\phi_1} \cdot r_2e^{i\phi_2} = r_1r_2 \cdot e^{i(\phi_1+\phi_2)}\)
Formulas for Multiplication of Complex Numbers
The multiplication of two complex numbers is done by expanding according to the principle of permanence - the algebraic rules for real numbers remain valid.
Expansion
Multiply each term of the first parenthesis with each of the second
Simplification
Using \(i^2 = -1\) and separating real and imaginary parts
Step-by-Step Example
Calculation: \((3+i) \cdot (1-2i)\)
Step 1: Expand
\((3+i)(1-2i)\)
\(= 3 \cdot 1 + 3 \cdot (-2i) + i \cdot 1 + i \cdot (-2i)\)
\(= 3 - 6i + i - 2i^2\)
Step 2: Replace \(i^2\)
Since \(i^2 = -1\):
\(3 - 6i + i - 2i^2\)
\(= 3 - 6i + i - 2(-1)\)
Step 3: Simplify
\(3 - 6i + i + 2\)
Real part: \(3 + 2 = 5\)
Imaginary part: \(-6i + i = -5i\)
Step 4: Result
\((3+i)(1-2i) = 5 - 5i\)
Verification with formula
Real part: \(ac - bd\)
\(= 3 \cdot 1 - 1 \cdot (-2) = 3 + 2 = 5\) ✓
Imaginary part: \(ad + bc\)
\(= 3 \cdot (-2) + 1 \cdot 1 = -6 + 1 = -5\) ✓
FOIL Method in Detail
What is FOIL?
FOIL is a mnemonic for expanding two binomials:
First: multiply first terms
Outer: multiply outer terms
Inner: multiply inner terms
Last: multiply last terms
Example: (3+i)(1-2i)
F: \(3 \cdot 1 = 3\)
O: \(3 \cdot (-2i) = -6i\)
I: \(i \cdot 1 = i\)
L: \(i \cdot (-2i) = -2i^2 = 2\)
Sum: \(3 - 6i + i + 2 = 5 - 5i\)
Visualization
(3 + i) · (1 - 2i)
↓ ↓ ↓ ↓
F O I L
F: 3·1 = 3
O: 3·(-2i) = -6i
I: i·1 = i
L: i·(-2i) = -2i² = 2
Alternative: Distributive Law
\((a+bi)(c+di)\)
\(= a(c+di) + bi(c+di)\)
\(= ac + adi + bci + bdi^2\)
\(= (ac-bd) + (ad+bc)i\)
Multiplication of Complex Numbers - Detailed Description
Principle of Permanence
According to the principle of permanence the arithmetic rules for real numbers should also apply to complex numbers. Therefore we multiply complex numbers like binomials.
1. Expand the parentheses (as with real numbers)
2. Replace \(i^2\) by \(-1\)
3. Combine real and imaginary parts
4. Write the result in standard form \(a + bi\)
Important: \(i^2 = -1\)
The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This is the key to multiplying complex numbers.
Common mistake!
❌ Wrong: \(i^2 = 1\) or forgetting \(i^2\)
✅ Right: Use \(i^2 = -1\) consistently
Practical Applications
Multiplication of complex numbers is used in many technical fields:
• Electrical engineering: impedance calculations
• Signal processing: filter design
• Quantum mechanics: wave functions
• Control engineering: transfer functions
Polar Form (easier!)
In polar form multiplication is much simpler:
Rule: multiply magnitudes, add angles
Special cases
- Multiply by i: \(z \cdot i = (a+bi)i = -b+ai\) (90° rotation)
- Square: \(z^2 = (a+bi)^2 = a^2-b^2+2abi\)
- With conjugate: \(z \cdot \overline{z} = |z|^2\) (real!)
- Multiply by 1: \(z \cdot 1 = z\) (identity element)
|
|
More complex functions
Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •