Imaginary Number Calculator - Modulus, Argument, Conjugate

An imaginary number calculator turns z = a + bi into the four pieces of information you need: the magnitude, the phase angle, the complex conjugate, and the polar form.

• • Read the magnitude quickly: Confirm that |z| = sqrt(a^2 + b^2) for a specific (a, b) pair.
• • Get the argument in both units: See the phase angle in radians and degrees, with atan2 keeping the quadrant correct.
• • Pull the conjugate for division: Read a - bi in one row when you need to rationalize a denominator.
• • Write the polar form for trig work: Convert a + bi into r e^(i theta) for Euler's formula or phasor problems.

The term imaginary number usually means a real multiple of i, the square root of -1, but most readers reach for the wider form a + bi in practice. The calculator treats a = 0 (pure imaginary) and b = 0 (pure real) as normal inputs.

The four result rows update together as you type. The magnitude row tells you how big the number is, the phase rows tell you where it sits, the conjugate row is ready for rationalization, and the polar form row is ready for exponentiation.

When the next step is arithmetic between two complex numbers instead of properties of a single one, Complex Number Calculator handles the addition, subtraction, multiplication, and division of pairs in one panel.

The calculator reads a and b, squares each, sums the squares, takes the square root for the magnitude, and uses atan2(b, a) for the phase. The conjugate is a sign flip on b, and the polar form is a single string built from the magnitude and the degree phase.

• a: Real part of z, the number on the real axis. Can be positive, zero, or negative.
• b: Imaginary part (the coefficient of i). Sign decides which half-plane the point lives in.
• |z|: Magnitude of z, the distance from the origin to (a, b). Always non-negative.
• phi: Phase angle, measured from the positive real axis to (a, b). Returned in radians and degrees.

The magnitude formula is the same Pythagorean relation you would use on any (x, y) point in a coordinate plane, and the phase is the angle that line makes with the positive x-axis. atan2(b, a) keeps the sign information from a and b so the phase is correct in the second and third quadrants.

The conjugate row is a sign flip on b, and the polar form row packages the magnitude and the degree phase into a single readable string.

According to Britannica, an imaginary number is any real multiple of the imaginary unit i, and a complex number of the form a + bi has a - bi as its complex conjugate.

According to Wolfram MathWorld, the imaginary unit i is the square root of -1, and a complex number is any number of the form a + bi where a and b are real numbers.

When the phase angle has to be worked out by hand for a non-quadrant-1 input, Arctan Calculator covers the single-angle arctangent that the two-argument atan2 call in this page is built on.

Four ideas show up in every imaginary number problem and they line up with the four result rows. Knowing the definitions turns the result panel into a summary.

The polar form r e^(i phi) is the same number written with its magnitude r and phase phi instead of a and b. The two forms carry the same information.

Powers of i are the special case where a = 0 and b = 1, and the four-row cycle 1, i, -1, -i is the only sequence of powers of i that returns a real value. The imaginary number calculator covers the wider family of (a, b) pairs that the powers-of-i table does not.

When the input is given as a real multiple of i and has to be rewritten in standard a + bi form, A+Bi Form Calculator returns the real part, the imaginary part, and the magnitude in a single read.

Type the real part and the imaginary part on the left and read the four result rows on the right.

1. 1 Enter the real part a: Type the real part of z = a + bi in the first box. Use a negative sign if a is negative, and use 0 for a pure imaginary number such as i.
2. 2 Enter the imaginary part b: Type the imaginary part in the second box. Use 0 for a pure real, and use 1 or -1 for a clean i or -i.
3. 3 Read the magnitude and phase rows: Use the Magnitude |z| row for the size on the complex plane, and the Phase rows (radians or degrees) for the angle. atan2 keeps the answer in the right quadrant.
4. 4 Read the conjugate and polar form rows: Pull a - bi from the conjugate row to rationalize a denominator, and pull the polar form r e^(i phi) when a follow-up problem uses Euler's formula.

A student is asked for the magnitude and phase of z = -2 + 3i. They type -2 and 3, read Magnitude 3.6056, Phase 2.1588 rad / 123.69 deg, Conjugate -2 - 3i, and Polar form 3.6056 e^(i 123.69 deg).

When the polar form has to be checked against a coordinate pair, Cartesian to Polar Calculator converts a + bi into r e^(i theta) and back without re-doing the atan2 step.

The four result rows cover the four properties of an imaginary number that come up in homework, exams, and applied work.

• • All four properties in one panel: Magnitude, phase, conjugate, and polar form are computed together, so the (a, b) pair does not have to be re-typed into separate tools.
• • Correct quadrant every time: The phase uses atan2(b, a), so the angle is right in the second and third quadrants where a single-argument atan would silently return the wrong value.
• • Both angle units side by side: Phase in radians and phase in degrees are returned together, so the reader picks the unit that matches the problem.
• • Pure-imaginary and pure-real inputs handled: a = 0, b = 1 (the unit i) and a = 5, b = 0 (a real number) are normal inputs, and the four result rows return well-defined values for both.
• • Real-time updates on every keystroke: The result panel updates as you type, so stepping through a list of (a, b) pairs takes one keystroke per change.

Reading the four result rows together makes it easier to check answers. If a textbook gives a magnitude of 5 and a phase of 53.13 degrees for some (a, b) pair, the calculator returns the same two numbers in the same row order.

The polar form row is the most useful output for follow-up problems that use Euler's formula or phasor addition, because the r and theta are already in the form those formulas expect.

When the magnitude row has to be checked against the absolute value of a single real input, Absolute Value Calculator returns |x| in one step, which is the real-number counterpart of the |z| row.

A few choices about the input change what the four result rows mean. The same (a, b) pair can be informative in one context and misleading in another.

• • The calculator takes one (a, b) pair and returns the four properties of that single number. Arithmetic between two complex numbers is a different page.
• • The polar form row uses the degree value of the phase for readability. Readers who need the radian version of theta should pull the Phase (radians) row instead.

Even with the correct quadrant, the phase is not unique: adding or subtracting 360 degrees gives the same point. The calculator returns the principal value in (-180, 180] degrees, the textbook and engineering convention.

When the imaginary number represents a phasor, the magnitude row carries the length and the phase row carries the direction. The polar form row packages the two pieces the same way a phasor diagram would draw them.

According to Khan Academy, the magnitude of a complex number a + bi is the square root of a^2 + b^2, and the phase angle is the arctangent of b divided by a, with the correct quadrant chosen by the signs of a and b.

When the input is the pure imaginary unit i and the question is about i raised to an integer power, I Number Calculator covers the four-row cycle 1, i, -1, -i and the reduced exponent n mod 4 in one panel.