ƒ MATH · EIGHT CALCULATION MODES

Complex Number Calculator

Calculate with complex numbers in a + bi form. Multiply, divide and find magnitude, argument or conjugate with the imaginary unit i. How to use it ↓

Calculate with complex numbers in a + bi form. Multiply, divide and find magnitude, argument or conjugate with the imaginary unit i.

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Enter an expression, then press Enter or =.

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About this result

Results round to 12 significant digits. Exact arithmetic fractions are available where supported.

Expression reference

sin(30) · asin(0.5)
ln(e) · log(8,2)
root(27,3) · 2^3
5! · nCr(5,2)
sinh(1) · conj(3+4i)
re(3+4i) · arg(3+4i)
1e-5 · 200+10%

Use exp(x) for eˣ and mod(a,b) for the remainder. Roots and logarithms may return complex values.

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How to use the complex number calculator

Enter a complex expression, using i for the imaginary unit and parentheses around each number.
Select DEG or RAD for the argument and any trigonometric functions.
Press Enter or = to calculate the rectangular result. Expand the details to see magnitude, argument and conjugate.
Use re(), im(), conj() or abs() in an expression to calculate a particular component.

Formula and method

(a + bi)(c + di) = (ac − bd) + (ad + bc)i; i² = −1

Rectangular form and multiplication

A complex number has a real part a and an imaginary part b in a + bi. Expand a product as ordinary algebra, then replace i² with −1. For division, multiplying numerator and denominator by the denominator’s conjugate makes the denominator real.

Magnitude, argument and conjugate

The magnitude of a + bi is √(a² + b²). Its argument is atan2(b, a), which places the angle in the correct quadrant. The conjugate is a − bi. In the result details, the argument follows the displayed DEG/RAD setting; the zero complex number has no defined direction.

ONE TOOL, DIFFERENT TASKS

Choose the mode for your calculation

Scientific Calculator

Powers, roots, logarithms, trigonometry, factorials, combinations and permutations. Decimal and exact arithmetic fraction displays, Ans and memory.

Complex Number Calculator

Calculate with i. Read the rectangular result, magnitude, argument and conjugate. Angle units apply to the argument and trigonometric functions.

Matrix Calculator

Add, subtract and multiply matrices up to 4 × 4. Find a determinant, inverse or transpose. Matrix B is needed only for binary operations.

Numerical Calculus Calculator

Approximate a definite integral or a derivative at a real point. This is numerical calculus; it does not produce symbolic antiderivatives.

Equation Solver

Solve a quadratic, including complex roots, or a system of two or three simultaneous linear equations. Degenerate quadratic inputs are handled as linear equations.

Statistics Calculator

Find count, sum, mean, median, range endpoints, variance and standard deviation for up to 1,000 values. Choose sample or population statistics.

Base-N Calculator

Convert signed integers of up to 128 bits between binary, octal, decimal and hexadecimal. Bitwise operations use unsigned 32-bit operands.

Function Table Calculator

Evaluate a real-valued function over a range of x values. Choose start, end and step; each table supports up to 201 rows.

TRY IT YOURSELF

Complex Number Calculator worked examples

Multiply two complex numbers

(3 + 4i) × (2 − i)

10 + 5i

The four products give 6−3i+8i−4i². Since i²=−1, the real part is 10 and the imaginary part is 5i.

Find a complex magnitude

abs(3 + 4i)

The magnitude is √(3²+4²)=√25=5. It is a real, nonnegative number.

Calculate a negative square root

sqrt(−4)

The principal square root has square −4 because (2i)²=4i²=−4.

READ THE RESULT WITH CONTEXT

Accuracy, notation and limits

Complex arithmetic uses numerical values, rounded to the selected significant digits. The exact Fraction control applies to supported real rational arithmetic, not symbolic complex fractions or surds.

Division by zero is rejected. Roots and logarithms can produce complex values, and principal-value results are not an enumeration of every possible root.

Read our calculation methodology ↗

COMMON QUESTIONS

Complex Number Calculator FAQs

How do I enter the imaginary unit?

Type i, such as 3+4i or (3+4i)*(2-i). The Imaginary unit key is also available under More functions.

How do I calculate a conjugate or magnitude?

Enter conj(3+4i) for 3−4i, or abs(3+4i) for 5. Complex mode also provides these details underneath the main result.

Can the calculator use polar form?

The main result uses rectangular form. For a polar input, enter r*(cos(theta)+i*sin(theta)), using the correct DEG/RAD setting. Expand the details to read magnitude and argument.

Calculation library: math.js. This page is an independent CalcPebble tool.