Complex Number Calculator

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What is Complex Number?

This calculator adds, subtracts, multiplies, and divides complex numbers written as a + bi. It also finds a principal square root of a single complex number. Each operation treats i as the imaginary unit, for which i² = −1.

Enter coefficients rather than an entire typed expression. For example, 3 − 2i uses 3 in the real field and −2 in the imaginary field. The calculator returns a real component and an imaginary component, rounded to four decimal places. This makes it useful for checking a hand calculation while retaining the structure of a complex answer.

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How to Use This Calculator

Choose the operator first. Addition, Subtraction, Multiplication, and Division use two numbers: a + bi and c + di. Enter a, b, c, and d in their matching positions, including negative signs. The fields labeled bi and di take numeric coefficients, so type −5 rather than −5i.

For Root, only the first number is used and the second pair of fields is hidden. This option means square root; it does not ask for an arbitrary root index. After calculating, read the operation label as well as the answer so that a result from a previous operation is not mistaken for the current one.

Division requires a nonzero second number. A denominator of 0 + 0i has no quotient; enter a different denominator before calculating. The current result area may not present that validation error clearly, so an incomplete or invalid display is not a numerical answer.

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Formula Used

Addition: (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i.

Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The minus sign in the real component comes from replacing i² with −1.

Division: (a + bi)/(c + di) = (ac + bd)/(c² + d²) + [(bc − ad)/(c² + d²)]i. This comes from multiplying by the denominator's conjugate.

For a square root, set R = √(a² + b²). The real component is √((R + a)/2), and the imaginary component has magnitude √((R − a)/2). Its sign follows b; for a negative real input with b = 0, this calculator uses the positive imaginary root.

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Variables Explained

Symbol or inputMeaning in this calculator
a and bReal and imaginary coefficients of the first number; b includes its sign.
c and dCoefficients of the second number, used for the four binary operations.
OperatorChooses the arithmetic rule. Root uses one number and means principal square root.
RModulus of the first number, used internally in the square-root formula.
c² + d²Real denominator in complex division. It is zero only when both c and d are zero.
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Worked Example

Take z = 3 + 2i and w = 1 − 5i. Enter a = 3, b = 2, c = 1, and d = −5. Addition gives 4 − 3i. Subtraction gives 2 + 7i, because subtracting −5i adds 5i.

Multiplication gives (3 × 1 − 2 × (−5)) + (3 × (−5) + 2 × 1)i = 13 − 13i. For division, the common denominator is 1² + (−5)² = 26. The real numerator is 3 − 10 = −7, and the imaginary numerator is 2 + 15 = 17. The displayed quotient is therefore approximately −0.2692 + 0.6538i.

For a separate Root example, use 3 + 4i. Its modulus is 5, so the real component is √4 = 2 and the imaginary component is √1 = 1. The displayed root is 2 + i. Squaring that expression gives 4 + 4i − 1 = 3 + 4i.

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Why Use This Calculator?

The answer is a number with two components, not two competing answers. A result of 13 − 13i has real part 13 and imaginary coefficient −13. If one component is zero, you can simplify the notation: 4 + 0i is 4, and 0 + 6i is 6i.

Use an inverse operation to check your work. Adding w back to z − w should recover z. Multiplying a quotient by its denominator should recover the numerator, apart from rounding. For square roots, square the displayed result and compare with the original input.

The square-root result is one selected root. Except at zero, the negative of that result is the other square root. A request to solve x² = z therefore usually needs both values, even though the Root box displays just the principal one.

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Common Mistakes

    Keep the order of subtraction and division: z − w differs from w − z, and z/w generally differs from w/z. Do not multiply corresponding coefficients alone; multiplication includes cross terms and the i² term.

    Four-decimal rounding can hide a very small component. Retain exact fractions in formal work when available, especially if a later calculation subtracts nearly equal quantities. The calculator handles numeric coefficients, not symbolic expressions such as √2 entered directly into a number field.

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Frequently Asked Questions

Can I calculate with ordinary real numbers here?

Yes. Enter zero for both imaginary coefficients. The arithmetic then reduces to the familiar real-number operation, although the display may retain a zero imaginary term.

Why does multiplying two imaginary terms change the real part?

Their product contains i², which equals −1. For example, (2i)(−5i) = −10i² = 10. That real contribution is essential in the worked multiplication example.

Does Root return both square roots?

No. It returns the principal selection. For 3 + 4i that is 2 + i; the other root is −2 − i. Both square to the same original number.

What happens when the divisor is purely imaginary?

Division is allowed provided its imaginary coefficient is nonzero. For example, 1/i = −i because multiplying −i by i gives 1. Only a divisor with both components zero is undefined.

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Sources & References

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