Imaginary Number Calculator

What Is an Imaginary Number?

An imaginary number is a real number multiplied by the imaginary unit $i$, where $i$ is defined by the property:

$$i^2 = -1 \quad \text{or} \quad i = \sqrt{-1}$$

While no real number squared yields a negative result, imaginary numbers enable mathematicians, physicists, and electrical engineers to solve quadratic equations with negative discriminants, model alternating current (AC) circuits, analyze quantum wave mechanics, and process digital signals.

The Cyclic Powers of $i$

Because $i^2 = -1$, raising $i$ to successive integer powers produces a repeating 4-element cycle:

• $i^0 = 1$
• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = (-1) \cdot i = -i$
• $i^4 = (i^2)^2 = (-1)^2 = 1$
• $i^5 = i^4 \cdot i = 1 \cdot i = i$

How to Simplify Any Power $i^n$

To evaluate $i^n$ for any integer exponent $n$:

1. Divide the exponent $n$ by $4$ to find the integer quotient $q$ and remainder $r \in \{0, 1, 2, 3\}$.
2. Express $i^n$ as $(i^4)^q \cdot i^r = (1)^q \cdot i^r = i^r$.
3. Evaluate $i^r$ using the 4-step reference: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$.

For example, to evaluate $i^{27}$: since $27 = 4 \times 6 + 3$, the remainder is $3$, so $i^{27} = i^3 = -i$.

Simplifying Square Roots of Negative Numbers

For any positive real number $a > 0$, the square root of its negative is simplified by factoring out $\sqrt{-1} = i$:

$$\sqrt{-a} = \sqrt{a \cdot (-1)} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a} \, i$$

If $a$ is a perfect square (such as $49$), $\sqrt{-49} = 7i$. If $a$ is not a perfect square (such as $72 = 36 \times 2$), $\sqrt{-72} = \sqrt{36 \times 2} \, i = 6\sqrt{2} \, i \approx 8.4853i$.

Arithmetic with Pure Imaginary Numbers

To work with complete complex numbers combining real and imaginary parts, use our a+bi Form Calculator, Complex Root Calculator, and Conjugate Calculator.