CalcComplex Number Magnitude Calculator
Calculate the magnitude (absolute value) of a complex number.
What Is Complex Number Magnitude Calculator?
Complex Number Magnitude Calculator finds the absolute value (modulus) of a complex number — its distance from zero in the complex plane. Complex numbers combine real and imaginary parts, written as a + bi where i = √(-1). The magnitude tells you the "size" of the complex number, essential for electrical engineering, quantum mechanics, signal processing, and control systems. Whether analyzing AC circuits, studying wave functions, or designing filters, complex magnitude is fundamental.
Consider the complex number 3 + 4i. Its real part is 3, imaginary part is 4. On the complex plane, this plots as a point 3 units right and 4 units up from the origin. The magnitude is the straight-line distance from (0,0) to (3,4), calculated using the Pythagorean theorem: |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5. This famous 3-4-5 right triangle demonstrates how complex magnitude works geometrically.
Complex numbers aren't "imaginary" in the sense of being unreal — they're indispensable tools for modeling oscillations, rotations, and wave phenomena. The magnitude represents amplitude in signal processing, impedance in circuits, and probability amplitude in quantum mechanics. Understanding complex magnitude unlocks applications across science and engineering.
How Complex Number Magnitude Calculator Works: Formulas Explained
Magnitude formula: |a + bi| = √(a² + b²), where a is the real part and b is the imaginary part. This extends the Pythagorean theorem to the complex plane. Example: |5 + 12i| = √(5² + 12²) = √(25 + 144) = √169 = 13. Another: |-8 + 6i| = √((-8)² + 6²) = √(64 + 36) = √100 = 10. The magnitude is always a non-negative real number.
Polar form conversion: Any complex number can be written as r(cos θ + i sin θ) or re^(iθ), where r = |z| is the magnitude and θ = arg(z) is the angle (argument). For z = 3 + 4i: r = 5, θ = arctan(4/3) ≈ 53.13°. Polar form: 5(cos 53.13° + i sin 53.13°) or 5e^(i·53.13°). Polar form simplifies multiplication and division of complex numbers.
Magnitude of product: |z₁ × z₂| = |z₁| × |z₂|. The magnitude of a product equals the product of magnitudes. Example: z₁ = 3 + 4i (|z₁| = 5), z₂ = 1 + i (|z₂| = √2). Product: (3+4i)(1+i) = 3 + 3i + 4i + 4i² = 3 + 7i - 4 = -1 + 7i. Magnitude: |-1 + 7i| = √(1 + 49) = √50 = 5√2. Verify: |z₁| × |z₂| = 5 × √2 = 5√2 ✓.
Magnitude properties: Key properties simplify calculations: |z| ≥ 0 (always non-negative). |z| = 0 if and only if z = 0. |-z| = |z| (negation doesn't change magnitude). |conjugate(z)| = |z| (conjugate has same magnitude). |zⁿ| = |z|ⁿ (magnitude of power equals power of magnitude). These properties enable algebraic manipulation without converting to rectangular form.
Working through complete examples: Find |7 - 24i|. Real part a = 7, imaginary part b = -24. |7 - 24i| = √(7² + (-24)²) = √(49 + 576) = √625 = 25. Find |-5 - 12i|. a = -5, b = -12. |-5 - 12i| = √(25 + 144) = √169 = 13. Find |6i| (pure imaginary). a = 0, b = 6. |0 + 6i| = √(0 + 36) = 6. Find |10| (pure real). a = 10, b = 0. |10 + 0i| = √(100 + 0) = 10. Pure real and pure imaginary numbers follow the same formula.
Step-by-Step Guide to Finding Complex Number Magnitude
- Identify the real part (a). The real part is the term without i. In 8 + 3i, the real part is 8. In -5 - 7i, the real part is -5. In 6i, the real part is 0 (no real term written). Write down a explicitly, including the sign.
- Identify the imaginary part (b). The imaginary part is the coefficient of i (not including i itself). In 8 + 3i, b = 3. In -5 - 7i, b = -7 (the minus sign is part of b). In 6i, b = 6. In 10 (pure real), b = 0. Be careful: the imaginary part is the number multiplying i, not including i.
- Square both parts. Calculate a² and b². Example: z = -8 + 15i. a = -8, b = 15. a² = (-8)² = 64. b² = 15² = 225. Remember: squaring a negative gives a positive. (-8)² = 64, not -64. This is why magnitude is always non-negative.
- Add the squares. Compute a² + b². Continuing the example: 64 + 225 = 289. This sum represents the squared distance from the origin in the complex plane. By the Pythagorean theorem, this equals the square of the magnitude.
- Take the square root. |z| = √(a² + b²). Example: √289 = 17. The magnitude of -8 + 15i is 17. Use a calculator for non-perfect squares: |3 + 5i| = √(9 + 25) = √34 ≈ 5.831. Round to appropriate precision for your application.
- Verify reasonableness. The magnitude should be at least as large as the larger of |a| and |b|. For 3 + 4i, magnitude 5 is larger than both 3 and 4 ✓. For 10 + 0i, magnitude equals 10 ✓. If your magnitude is smaller than either |a| or |b|, recalculate — something went wrong.
Real-World Complex Number Magnitude Examples
Example 1: AC Circuit Impedance. An AC circuit has resistance R = 3 ohms and inductive reactance X_L = 4 ohms. Total impedance Z = 3 + 4i ohms (resistance is real, reactance is imaginary). Magnitude |Z| = √(3² + 4²) = 5 ohms. This is the effective resistance the circuit presents to AC current. With 120V AC applied, current I = V/|Z| = 120/5 = 24 amperes. The magnitude determines actual current flow, not the complex impedance itself.
Example 2: Signal Processing Amplitude. A signal is represented as the complex number 7 - 24i in the frequency domain. The magnitude |7 - 24i| = √(49 + 576) = √625 = 25 represents the signal's amplitude (strength). The phase angle θ = arctan(-24/7) ≈ -73.7° represents the phase shift. Engineers care about both: amplitude determines loudness/voltage, phase determines timing relationships.
Example 3: Quantum Mechanics Probability. A quantum state has probability amplitude ψ = 0.6 + 0.8i. The probability of measuring this state is |ψ|² = (0.6)² + (0.8)² = 0.36 + 0.64 = 1.0 = 100%. This is properly normalized. Another state φ = 0.3 + 0.4i has |φ|² = 0.09 + 0.16 = 0.25 = 25% probability. The magnitude squared gives measurable probability — complex numbers are essential to quantum theory.
Example 4: Control System Stability. A system's transfer function has a pole at s = -2 + 3i in the complex plane. The distance from the origin is |s| = √(4 + 9) = √13 ≈ 3.61. The real part (-2) indicates exponential decay; the imaginary part (3) indicates oscillation at 3 rad/s. The magnitude 3.61 relates to the system's natural frequency. Control engineers analyze pole magnitudes to predict system behavior.
Example 5: 2D Vector Magnitude. A force vector in 2D has components F_x = 12 N (horizontal) and F_y = -5 N (vertical). Represent as complex number 12 - 5i. Magnitude |12 - 5i| = √(144 + 25) = √169 = 13 N. This is the total force strength. Direction: θ = arctan(-5/12) ≈ -22.6° (22.6° below horizontal). Complex numbers provide elegant 2D vector calculations.
Common Mistakes with Complex Number Magnitude
Forgetting to square the imaginary coefficient, not i itself. Wrong: |3 + 4i| = √(3² + 4i²) = √(9 - 4) = √5. Correct: |3 + 4i| = √(3² + 4²) = √(9 + 16) = 5. The formula uses b², not (bi)². The i is not part of what gets squared — b is the coefficient. Think of it as the Pythagorean theorem: real and imaginary parts are perpendicular "legs" of a right triangle.
Dropping the negative sign when squaring. For z = -5 - 12i, both parts are negative. Correct: |z| = √((-5)² + (-12)²) = √(25 + 144) = 13. Error: |z| = √(5² + 12²) — this accidentally gives the right answer because squaring eliminates the sign anyway. But for the argument (angle), signs matter critically. Always track signs correctly throughout.
Confusing magnitude with the complex number itself. Magnitude is a real number (distance), not a complex number. |3 + 4i| = 5, not 5 + 0i or just "5i". The magnitude operation |z| outputs a non-negative real scalar. Don't write |3 + 4i| = 5i or |3 + 4i| = 3 + 4i — both are wrong. Magnitude strips away direction, leaving only size.
Misidentifying the imaginary part. In z = 5 + 3i, the imaginary part is 3, not 3i. In z = 7 - 2i, the imaginary part is -2, not -2i or 2. The imaginary part b is the real coefficient of i. Formula: |a + bi| = √(a² + b²), where b is a real number. Writing b = 3i and computing √(a² + (3i)²) = √(a² - 9) is incorrect.
Pro Tips for Complex Number Magnitude
Use conjugate multiplication for division. To find |z₁/z₂|, use |z₁/z₂| = |z₁|/|z₂| instead of dividing complex numbers directly. Example: z₁ = 8 + 6i (|z₁| = 10), z₂ = 3 - 4i (|z₂| = 5). |z₁/z₂| = 10/5 = 2. Much simpler than computing (8+6i)/(3-4i) = (8+6i)(3+4i)/(9+16) = (24+32i+18i-24)/25 = 50i/25 = 2i, then |2i| = 2.
Recognize Pythagorean triples for quick calculation. Common Pythagorean triples give integer magnitudes: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. Complex numbers 3+4i, 5+12i, 8+15i have magnitudes 5, 13, 17 respectively. Memorizing these triples speeds up manual calculations and helps verify calculator results.
Convert to polar form for powers and roots. For zⁿ, use polar form: if z = re^(iθ), then zⁿ = rⁿe^(inθ). Magnitude |zⁿ| = rⁿ = |z|ⁿ. Example: (3+4i)³ has magnitude 5³ = 125. Computing (3+4i)³ directly: (3+4i)² = 9+24i-16 = -7+24i. Then (-7+24i)(3+4i) = -21-28i+72i-96 = -117+44i. Magnitude: √(117²+44²) = √(13689+1936) = √15625 = 125 ✓.
Apply the triangle inequality. For any complex numbers: |z₁ + z₂| ≤ |z₁| + |z₂|. The magnitude of a sum is at most the sum of magnitudes. Example: z₁ = 3+4i (|z₁|=5), z₂ = 5-12i (|z₂|=13). z₁+z₂ = 8-8i, |z₁+z₂| = √128 ≈ 11.31. Sum of magnitudes: 5+13 = 18. Indeed, 11.31 ≤ 18. Equality holds only when z₁ and z₂ point in the same direction.
Frequently Asked Questions
For pure real numbers (b=0): |a + 0i| = √(a² + 0) = |a|, the ordinary absolute value. For pure imaginary numbers (a=0): |0 + bi| = √(0 + b²) = |b|. Examples: |7| = 7, |-5| = 5, |6i| = 6, |-9i| = 9. Complex magnitude generalizes the real absolute value to two dimensions.
No. Magnitude is a distance, and distance is always non-negative. |z| ≥ 0 for all complex numbers z. The only complex number with magnitude 0 is z = 0 (both real and imaginary parts are zero). If your calculation produces a negative magnitude, you've made an error — likely forgot to take the square root or mishandled signs.
In polar form z = r(cos θ + i sin θ) or z = re^(iθ), the magnitude is simply r — it's built into the representation. For z = 5e^(i·π/3), the magnitude is 5. For z = 2.5(cos 60° + i sin 60°), the magnitude is 2.5. Polar form explicitly separates magnitude (r) from direction (θ), making magnitude immediately visible.
On the complex plane (real axis horizontal, imaginary axis vertical), a complex number a + bi plots as the point (a, b). The magnitude |a + bi| = √(a² + b²) is the Euclidean distance from the origin (0,0) to the point (a, b). All complex numbers with the same magnitude lie on a circle centered at the origin — this circle is called a "level curve" of constant magnitude.
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