Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Imaginary Number Calculator.
Read the complete guideWhat Is an Imaginary Number?
An imaginary number has the form bi where b is a real number and i is the imaginary unit satisfying i² = −1. Combined with a real part, it forms a complex number a + bi. Imaginary numbers are essential in electrical engineering (where j is used instead of i), quantum mechanics, and signal processing.
Key Properties of i
The imaginary unit cycles through four values:
- i¹ = i
- i² = −1
- i³ = −i
- i⁴ = 1 (the cycle repeats)
Examples
Multiplying Purely Imaginary Numbers
What is (3i)(4i)?
(3i)(4i) = 12i² = 12(−1) = −12. The product of two purely imaginary numbers is a real number.
Key takeaway: Multiplying imaginary numbers follows the rule i² = −1, turning the product back into a real number.
Working with Imaginary Numbers
Key strategies for calculations involving i:
- Remember the cycle: i, −1, −i, 1 repeats every 4 powers
- Use polar form for powers and roots of imaginary numbers
- In engineering contexts, j is used instead of i to avoid confusion with current
Frequently Asked Questions about Imaginary Number Calculator
What is an imaginary number?
An imaginary number is a number of the form bi where b is real and i² = −1. For example, 3i and −5i are imaginary numbers.
Is i the same as √(−1)?
While often written that way informally, i is more precisely defined by the property i² = −1. The square root notation is ambiguous in the complex numbers.
What is i²?
i² = −1. This is the defining property of the imaginary unit.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Operation
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