Find The Magnitude Of Complex Number Calculator

Calculate the Magnitude of a Complex Number (z = a + bi)

Enter the real and imaginary parts of your complex number to find its magnitude |z|.

The magnitude |z| of a complex number z = a + bi is calculated using the formula: |z| = √(a² + b²)

Visualization on the Complex Plane

Complex Number (z)	Real Part (a)	Imaginary Part (b)	Magnitude |z|
3 + 4i	3	4	5
5 – 12i	5	-12	13
-1 + i	-1	1	1.414
2	2	0	2
-3i	0	-3	3

Table of example complex numbers and their magnitudes.

What is the Magnitude of a Complex Number?

The magnitude of a complex number, also known as its modulus or absolute value, represents the distance of that number from the origin (0,0) in the complex plane (also called the Argand diagram). If you have a complex number z = a + bi, where ‘a’ is the real part and ‘b’ is the imaginary part, its magnitude, denoted as |z|, is a non-negative real number.

Geometrically, if you plot the complex number a + bi as the point (a, b) in the complex plane, the magnitude is the length of the line segment connecting the origin to this point. This is found using the Pythagorean theorem, as the real and imaginary parts form the two shorter sides of a right-angled triangle, and the magnitude is the hypotenuse.

Who Should Use This Calculator?

This magnitude of complex number calculator is useful for:

• Students studying algebra, pre-calculus, calculus, and complex analysis.
• Engineers (especially electrical engineers working with AC circuits and signal processing).
• Physicists dealing with wave functions and quantum mechanics.
• Mathematicians working with complex numbers.
• Anyone needing to find the “size” or length associated with a complex number.

Common Misconceptions

A common misconception is that the magnitude is simply the sum of the real and imaginary parts, or that it can be negative. The magnitude is always non-negative and is calculated using the square root of the sum of the squares of the real and imaginary parts, reflecting its nature as a distance.

Magnitude of a Complex Number Formula and Mathematical Explanation

For a complex number z = a + bi, where ‘a’ is the real part and ‘b’ is the imaginary part, the magnitude of the complex number |z| is given by the formula:

|z| = √(a² + b²)

This formula is derived directly from the Pythagorean theorem. In the complex plane:

1. The real part ‘a’ is the coordinate along the horizontal axis (real axis).
2. The imaginary part ‘b’ is the coordinate along the vertical axis (imaginary axis).
3. The complex number a + bi corresponds to the point (a, b).
4. The distance from the origin (0, 0) to the point (a, b) is the length of the hypotenuse of a right-angled triangle with sides ‘a’ and ‘b’.
5. By the Pythagorean theorem, (hypotenuse)² = a² + b², so hypotenuse = √(a² + b²), which is the magnitude |z|.

Variables Table

Variable	Meaning	Unit	Typical Range
z	The complex number	Dimensionless (or context-dependent, e.g., Volts, Amps)	Any complex number
a	Real part of z	Same as z	Any real number
b	Imaginary part of z	Same as z	Any real number
|z|	Magnitude of z	Same as z	Non-negative real numbers (≥ 0)

Practical Examples (Real-World Use Cases)

Example 1: Impedance in an AC Circuit

In electrical engineering, impedance (Z) in an AC circuit is often represented as a complex number: Z = R + jX, where R is resistance and X is reactance (j is used instead of i for the imaginary unit). If a circuit has a resistance R = 8 Ohms and an inductive reactance X = 6 Ohms, the impedance is Z = 8 + 6j Ohms.

To find the magnitude of the complex number representing impedance:

• Real part (a) = 8
• Imaginary part (b) = 6
• |Z| = √(8² + 6²) = √(64 + 36) = √100 = 10 Ohms

The magnitude |Z| = 10 Ohms represents the total opposition to current flow in the circuit.

Example 2: Displacement in 2D Space

While not strictly a complex number in all contexts, a 2D vector (3, -4) can be represented as 3 – 4i in the complex plane. The magnitude represents the length of the vector.

• Real part (a) = 3
• Imaginary part (b) = -4
• Magnitude = √(3² + (-4)²) = √(9 + 16) = √25 = 5 units

The length of the vector (or the magnitude of the complex number 3 – 4i) is 5 units.

How to Use This Magnitude of Complex Number Calculator

1. Enter the Real Part (a): In the “Real Part (a)” field, type the real component of your complex number.
2. Enter the Imaginary Part (b): In the “Imaginary Part (b)” field, type the coefficient of ‘i’ (the imaginary component). Do not include ‘i’. For example, if the number is 5 – 2i, enter -2.
3. View the Results: The calculator will automatically update and display the magnitude |z|, along with intermediate steps (a², b², and a² + b²). The complex plane visualization will also update.
4. Reset: Click the “Reset” button to clear the inputs and set them to default values (3 and 4).
5. Copy Results: Click “Copy Results” to copy the magnitude and intermediate values to your clipboard.

The primary result is the magnitude of the complex number you entered. The visualization helps you see the number plotted on the complex plane and its distance from the origin.

Key Factors That Affect the Magnitude of a Complex Number

The magnitude of a complex number is directly affected by:

1. The Real Part (a): As the absolute value of ‘a’ increases, the magnitude |z| generally increases, as a² contributes to the sum under the square root.
2. The Imaginary Part (b): Similarly, as the absolute value of ‘b’ increases, the magnitude |z| generally increases due to b².
3. The Signs of a and b: The signs of ‘a’ and ‘b’ determine the quadrant in which the complex number lies, but they do not affect the magnitude directly because a² and b² are always non-negative. For example, 3+4i and 3-4i have the same magnitude.
4. Whether a or b is Zero: If b=0, the number is purely real (z=a), and |z|=|a|. If a=0, the number is purely imaginary (z=bi), and |z|=|b|.
5. Scaling: If you multiply a complex number by a real scalar k, its magnitude is scaled by |k|. |k(a+bi)| = |ka + kbi| = √((ka)² + (kb)²) = √(k²(a²+b²)) = |k|√(a²+b²) = |k||z|.
6. Operations with Other Complex Numbers: When multiplying two complex numbers, the magnitude of the product is the product of their magnitudes: |z1 * z2| = |z1| * |z2|. When dividing, |z1 / z2| = |z1| / |z2|.

Frequently Asked Questions (FAQ)

1. What is the magnitude of a complex number also called?

The magnitude of a complex number is also known as its modulus or absolute value.

2. Can the magnitude of a complex number be negative?

No, the magnitude represents a distance, so it is always non-negative (zero or positive).

3. What is the magnitude of z = 0 + 0i?

The magnitude of 0 + 0i is √(0² + 0²) = 0. It is the only complex number with a magnitude of zero.

4. How is the magnitude related to the complex conjugate?

If z = a + bi, its conjugate is z* = a – bi. The product z * z* = (a + bi)(a – bi) = a² – (bi)² = a² + b² = |z|². So, |z| = √(z * z*).

5. What does the magnitude represent in AC circuits?

In AC circuits, the magnitude of impedance (|Z|) represents the total opposition to current flow, and the magnitude of voltage (|V|) or current (|I|) represents their effective values (like RMS values for sinusoidal signals).

6. What if the imaginary part is zero?

If b=0, z = a + 0i = a (a real number). The magnitude |z| = √(a² + 0²) = √a² = |a| (the absolute value of ‘a’). Finding the magnitude of a complex number with b=0 gives the absolute value of the real number.

7. What if the real part is zero?

If a=0, z = 0 + bi = bi (a purely imaginary number). The magnitude |z| = √(0² + b²) = √b² = |b| (the absolute value of ‘b’).

8. Does 3 + 4i have the same magnitude as 4 + 3i?

Yes, |3 + 4i| = √(3² + 4²) = √25 = 5, and |4 + 3i| = √(4² + 3²) = √25 = 5. The order of a and b doesn’t change the sum a²+b².

Related Tools and Internal Resources

• Complex Number Addition Calculator: Add two complex numbers easily.
• Complex Number Multiplication Calculator: Multiply complex numbers and see the result.
• What is a Complex Number?: An introduction to complex numbers and their components.
• Phasors in AC Circuits: Learn how complex numbers represent AC quantities.
• Vector Magnitude Calculator: Calculate the magnitude of 2D or 3D vectors.
• Polar to Rectangular Calculator: Convert complex numbers between polar (magnitude and angle) and rectangular (a+bi) forms.