Imaginary Number Calculator

An imaginary number is a real number multiplied by the imaginary unit i, which is defined by the rule that i squared equals negative one. The numbers 3i, -2.5i, and i itself are all imaginary. They exist because no real number squared gives a negative result, so mathematicians added i to make equations such as x squared plus 9 equals zero solvable.

A complex number has a real part and an imaginary part, written a + bi, where a and b are real numbers. A purely imaginary number is the special case where the real part a is zero, such as 7i. Every real number is also complex, with an imaginary part of zero. This calculator accepts the full a + bi form, so it handles all three kinds.

For multiplication it expands the product with the distributive rule and replaces i squared with negative one, giving (ac - bd) + (ad + bc)i. For division it multiplies the top and bottom by the conjugate of the divisor, which turns the denominator into the real number c squared plus d squared. Each step appears in the step by step box so you can follow the algebra.

Because i to the fourth power equals 1. Starting from i, the powers run i, -1, -i, 1, and then the pattern begins again, since multiplying by 1 changes nothing. That means any whole number exponent only needs its remainder after division by 4. A remainder of 0 gives 1, 1 gives i, 2 gives -1, and 3 gives -i.

Divide the exponent by 4 and keep the remainder, treating negative exponents with the same rule after adding 4 until the remainder is positive. For example, i to the power of 47 has remainder 3, so it equals -i. The exponent minus 3 has remainder 1, so i to the power of -3 equals i. The Powers of i tab does this for any whole number you enter.

Pull out the factor of negative one as i, then simplify the remaining positive root. The square root of -72 becomes i times the square root of 72, and since 72 is 36 times 2, that simplifies to 6i times the square root of 2. The Square root tab shows this exact form for negative whole numbers and also gives the principal root of any complex number.

The modulus is the distance from zero to the point a + bi on the complex plane, found with the Pythagorean formula. The argument is the angle that point makes with the positive real axis, measured here in degrees. Together they give the polar form, written r with an angle, or r(cos θ + i sin θ). Polar form is handy for multiplying, dividing, and raising complex numbers to powers.

Results are rounded to six decimal places, and angles to two decimal places, so a value such as 0.1 + 0.7i appears cleanly. Calculations use standard double precision arithmetic, which is accurate for homework, engineering checks, and general study. Extremely large exponents or components can lose precision, so for formal proofs keep the exact algebra shown in the steps.