Imaginary Number Calculator

Simplify the square root of a negative number into imaginary (bi) form, combine two pure imaginary numbers, and evaluate powers of i — all built on i² = -1.

Quick Facts

Definition
i = √-1, so i² = -1
Every imaginary number is a real multiple of i, written bi.
Negative square roots
√(-n) = i√n for n > 0
Pull out the -1 as i, then simplify the remaining radical.
Product is real
(ai)(bi) = -ab
Multiplying (or dividing) two imaginary numbers cancels the i.
Powers of i cycle
i¹=i, i²=-1, i³=-i, i⁴=1
The pattern repeats every 4 exponents — use n mod 4.

Your Results

Calculated
√(radicand)
-
Simplified imaginary or real result
b₁i [op] b₂i
-
Result of the chosen operation
iⁿ
-
Power of i, from the 4-cycle
Note
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What the arithmetic result means

Ready

Enter a radicand, two imaginary coefficients, an operation, and an exponent, then press Calculate.

Formula and Method for Imaginary Numbers

The imaginary unit i is defined by i = √-1, which means i² = -1. A number of the form bi (where b is any real number) is a pure imaginary number, and this identity is the key that unlocks every negative square root, imaginary-number operation, and power of i below.

How the calculation works

This calculator handles three related tasks built on i² = -1. First, it simplifies √(radicand): for a negative radicand -n, it applies √(-n) = √(-1) × √n = i√n, then pulls the largest perfect-square factor out of the remaining radical (for example √-48 = √(16 × 3) × i = 4√3 i); for a non-negative radicand it just returns the ordinary real square root. Second, it combines two pure imaginary numbers, b₁i and b₂i, using the operation you choose: addition and subtraction keep the result imaginary — (b₁ ± b₂)i — while multiplication and division convert i² into -1, so (b₁i)(b₂i) = -b₁b₂ and (b₁i) ÷ (b₂i) = b₁/b₂ are both real numbers. Third, it evaluates iⁿ for an integer exponent by reducing n modulo 4, since powers of i repeat in the fixed cycle i, -1, -i, 1.

Common mistakes

  • Treating √-1 as undefined: in the real numbers it is undefined, but in the complex numbers it is defined as i — that is the entire point of the imaginary unit.
  • Forgetting i² = -1 when multiplying: (3i)(4i) is not 12i; the two i's multiply to i² = -1, giving -12, a real number.
  • Mis-simplifying radicals: √-48 is not 48i or 6.93i — factor out the largest perfect square first (16), giving 4√3 i.
  • Losing track of the power-of-i cycle: i⁵ is not a new value; 5 mod 4 = 1, so i⁵ = i¹ = i.

Real-world applications

Imaginary and complex numbers describe quantities that a single real number line cannot, such as phase and impedance in AC electrical circuits, rotations and oscillations in signal processing, and the roots of polynomials that never cross the real axis (like x² + 1 = 0). Engineers and physicists write these as pure imaginary or complex values precisely so the algebra of i² = -1 can track rotation and phase shift automatically.

Frequently Asked Questions

What is an imaginary number?
The imaginary unit i is defined as the square root of -1, so i² = -1. A pure imaginary number has the form bi, where b is a real number called the coefficient.
How do you simplify the square root of a negative number?
Factor out -1 and apply √(-n) = √(-1) × √n = i√n for n > 0. For example, √-16 = 4i, and √-48 simplifies to 4√3 i after pulling the largest perfect-square factor out of the radical.
What happens when you multiply or divide two imaginary numbers?
Because i² = -1, multiplying two pure imaginary numbers gives a real result: (ai)(bi) = ab·i² = -ab. Dividing them is also real: (ai) ÷ (bi) = a/b, provided b is not zero.
How do you evaluate powers of i?
Powers of i repeat in a cycle of 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then it repeats. To evaluate iⁿ for any integer n, find n mod 4 and read off the matching value in that cycle.

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