Reciprocal (Inverse) of Complex Numbers
Calculation of \(\frac{1}{z}\) - the multiplicative inverse
Reciprocal Calculator
Reciprocal (Inverse)
The reciprocal \(\frac{1}{z}\) of a complex number is the multiplicative inverse: \(z \cdot \frac{1}{z} = 1\). It is calculated by multiplying with the conjugate number.
Reciprocal - Properties
Formula
Multiply by the conjugate number
Component form
Important Properties
- \(z \cdot \frac{1}{z} = 1\) (definition)
- \(\frac{1}{z} = z^{-1}\) (power notation)
- \(\frac{1}{\frac{1}{z}} = z\) (involution)
- \(\left|\frac{1}{z}\right| = \frac{1}{|z|}\) (magnitude)
Not defined for z = 0
The reciprocal is only defined for \(z \neq 0\). Division by zero is not possible!
With Polar Form
For \(z = re^{i\phi}\):
\[\frac{1}{z} = \frac{1}{r}e^{-i\phi}\]
Rule: invert magnitude, negate angle
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Formula for Calculating the Reciprocal
The reciprocal (inverse) of a complex number \(z = a + bi\) is calculated by multiplying with the conjugate number.
Derivation
Multiplying by the conjugate makes the denominator real
Final Formula
Divide real and imaginary parts by \(|z|^2 = a^2+b^2\)
Step-by-Step Example
Calculation: \(\frac{1}{3+5i}\)
Step 1: Multiply by conjugate
\(\frac{1}{3+5i} = \frac{1}{3+5i} \cdot \frac{3-5i}{3-5i}\)
Conjugate of \(3+5i\) is \(3-5i\)
Step 2: Numerator
Numerator: \(1 \cdot (3-5i) = 3-5i\)
Step 3: Denominator
\((3+5i)(3-5i) = 9 - 25i^2\)
\(= 9 - 25(-1) = 9 + 25 = 34\)
Step 4: Division
\(\frac{3-5i}{34} = \frac{3}{34} - \frac{5}{34}i\)
Step 5: Decimal value
\(\frac{3}{34} \approx 0.088\)
\(\frac{5}{34} \approx 0.147\)
\(\frac{1}{3+5i} \approx 0.088 - 0.147i\)
Verification
Check: \((3+5i)(0.088-0.147i)\)
\(= 0.264 - 0.441i + 0.440i - 0.735i^2\)
\(= 0.264 - 0.001i + 0.735\)
\(\approx 1.0\) ✓
Alternative calculation with formula
\[\frac{a}{a^2+b^2} = \frac{3}{3^2+5^2} = \frac{3}{34} \approx 0.088\]
\[-\frac{b}{a^2+b^2} = -\frac{5}{34} \approx -0.147\]
More Examples
Example 1: \(\frac{1}{i}\)
\(\frac{1}{i} = \frac{1}{i} \cdot \frac{-i}{-i}\)
\(= \frac{-i}{-i^2} = \frac{-i}{1}\)
\(= -i\)
Example 2: \(\frac{1}{1+i}\)
\(\frac{1}{1+i} = \frac{1-i}{(1+i)(1-i)}\)
\(= \frac{1-i}{1-i^2} = \frac{1-i}{2}\)
\(= 0.5 - 0.5i\)
Example 3: \(\frac{1}{2}\) (real)
\(\frac{1}{2+0i} = \frac{2}{2^2+0^2}\)
\(= \frac{2}{4}\)
\(= 0.5 + 0i\)
Example 4: With polar form
For \(z = 2e^{i\pi/3}\) (magnitude 2, angle 60°):
\(\frac{1}{z} = \frac{1}{2}e^{-i\pi/3}\)
Magnitude: 0.5, Angle: -60°
Example 5: \(\frac{1}{3-4i}\)
\(|3-4i|^2 = 9 + 16 = 25\)
\(\frac{1}{3-4i} = \frac{3+4i}{25}\)
\(= 0.12 + 0.16i\)
Reciprocal of Complex Numbers - Detailed Description
Multiplicative Inverse
The reciprocal \(\frac{1}{z}\) is the multiplicative inverse of the complex number \(z\).
\(z \cdot \frac{1}{z} = 1\) for all \(z \neq 0\)
Notation:
\(\frac{1}{z} = z^{-1}\) (power notation)
Calculation
The trick is to multiply by the conjugate number:
The denominator becomes real: \((a+bi)(a-bi) = a^2+b^2 = |z|^2\)
With Polar Form (easier!)
In polar form, the reciprocal is especially easy to calculate:
\[\frac{1}{z} = \frac{1}{r}e^{-i\phi}\] Rule:
• Magnitude: \(\frac{1}{r}\) (invert)
• Angle: \(-\phi\) (negate)
Practical Applications
• Division: \(\frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2}\)
• Electrical engineering: impedance calculations
• Control engineering: feedback systems
• Geometry: inversion at the unit circle
Important Properties
- \(\left|\frac{1}{z}\right| = \frac{1}{|z|}\) (magnitude inverts)
- \(\arg\left(\frac{1}{z}\right) = -\arg(z)\) (angle negates)
- \(\overline{\frac{1}{z}} = \frac{1}{\overline{z}}\) (conjugation commutes)
- \(\frac{1}{\frac{1}{z}} = z\) (involution)
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