Complex Number to Polar Form Calculator (Modulus, Argument and Quadrant Steps with a Complex Plane Graph)

Choose the direction of the conversion and enter the numbers. For rectangular to polar, enter the real part a and the imaginary part b. For polar to rectangular, enter the absolute value r and the argument θ (and choose its unit).

Conversion

Real part a

Imaginary part b

Absolute value r

Argument θ

Unit of the argument

Decimals, negative numbers and fractions such as 3/4 can be used. A blank real or imaginary part is treated as 0.

Result and figure

Enter the numbers in the fields on the left and press "Calculate". The result and a graph on the complex plane will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Convert a complex number \(z = a + bi\) (rectangular form) on the spot to polar form \(r(\cos\theta + i\sin\theta)\), using the absolute value \(r\) and the argument \(\theta\). The absolute value is given as an exact value with the radical simplified, such as \(\sqrt{8} = 2\sqrt{2}\), and for special angles the argument is an exact fraction of π
The argument \(\theta\) is given mainly in the range \(0 \le \theta < 2\pi\), with the value in the range \(-\pi < \theta \le \pi\) (the principal argument) shown as well, so you can check your answer whichever convention your textbook uses
It also converts the other way (polar form to rectangular form). Enter the absolute value \(r\) and the argument \(\theta\) (in degrees, as a fraction of π, or as a decimal in radians), and \(a + bi\) is shown as an exact value such as \(1 + \sqrt{3}i\)
The steps show, one at a time, how the argument is found from \(\tan\theta = \dfrac{b}{a}\) and the quadrant the point is in (the quadrant adjustment)
You can also see the result on the complex plane, which shows at a glance where the absolute value \(r\) (the distance from the origin) and the argument \(\theta\) (the angle of rotation from the real axis) are

This page covers the polar form of complex numbers taught in Precalculus. The real and imaginary parts can be fractions or decimals (rational numbers); irrational numbers such as √3 cannot be entered directly. So the argument comes out as an exact fraction of π only when the point is on an axis or the angle is a multiple of 45° (for angles based on 30° and 60°, tan θ is an irrational number such as 1/√3, which cannot come from rational real and imaginary parts). In the other direction, from polar form to rectangular form, all special angles that are multiples of 30° or 45° are calculated exactly.

What is this calculation used for?

AC circuit calculations (electrical and electronic engineering)

The alternating current (AC) in a wall outlet has a voltage and current that swing like waves, carrying two pieces of information: the size of the wave and the shift in its timing (the phase). In electrical engineering, both are held in one complex number (absolute value = size of the wave, argument = phase), and the standard method is to multiply and divide in polar form to solve the circuit. Engineers call this phasor notation.
The design of the power grid from the power plant to your home, and of the circuit inside a phone charger, relies on these "absolute value and argument" calculations. It is the most common everyday use of complex numbers in engineering work, and it appears on engineering licensing exams such as the FE and PE exams.

Analyzing sound and radio waves (Fourier transform and signal processing)

When a computer handles sound or radio waves, it breaks them down with a calculation called the Fourier transform into "how much of each wave pitch is in it". The result is a list of complex numbers: the absolute value of each one is the strength of that wave (the amplitude), and the argument is its shift (the phase).
Many technologies that handle sound and radio waves, such as equalizers in music apps, noise canceling in earbuds and speech recognition, are built on this idea of reading a complex number by its absolute value and argument.

Rotating shapes (games and computer graphics)

The polar form rule "multiply the absolute values, add the arguments" is, geometrically, just "scaling and rotating". For example, multiplying by \(i\) (absolute value 1, argument 90°) rotates a point exactly 90° around the origin. In 2D games, the math for turning a character or a bullet can be written as complex multiplication.
Quaternions, used in 3D graphics and robot control, extend this idea of "rotation by complex numbers" to three dimensions.

The "distance and bearing" of radar and sonar (aviation and shipping)

Airport radar and ship sonar locate a target by a pair: "how far away it is" and "in which direction". This is the same idea as polar form (polar coordinates), which describes a point by \(r\) and \(\theta\). The same kind of conversion as on this page is done to turn it into the position shown on the screen (horizontal and vertical coordinates).
Whenever "distance and direction" is the natural way to describe a position, this conversion to and from rectangular coordinates shows up.

Formulas and figures

Polar form (another way to write a complex number)

Figure

Standard notation (the usual math form)

\(z\) \(=\) \(r\) \(\left(\cos\theta + i\sin\theta\right)\)

In words (symbols replaced with words)

③ complex number \(z\) \(=\) ① absolute value \(r\) (distance from the origin) ② direction given by the argument \(\theta\)

The formula in words

① Take the absolute value \(r\) (the distance from the origin to the point)

② multiply it by the direction part \(\cos\theta + i\sin\theta\) for the argument \(\theta\)

③ and you get the complex number \(z\)

Quick example

Writing \(z = 1 + \sqrt{3}i\) in polar form (the absolute value is \(r = 2\) and the argument is \(\theta = \dfrac{\pi}{3}\), or 60°):

complex number \(1 + \sqrt{3}i\) \(=\) absolute value \(2\) \(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\)

\(r = \sqrt{1^{2} + \left(\sqrt{3}\right)^{2}} = \sqrt{4} = 2\)

\(1 + \sqrt{3}i = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)\)

Key idea

There are two ways to write the same complex number. Reading it as "the point \(a\) across and \(b\) up" gives rectangular form \(a + bi\). Reading it as "the point at distance \(r\) from the origin, in the direction at angle \(\theta\) from the real axis" gives polar form. It is like giving directions: rectangular form says "go 1 east and √3 north", while polar form says "turn 60° from east toward north and go 2". Addition and subtraction are easier in rectangular form, while multiplication, division and powers are easier in polar form (the formula for multiplying in polar form below shows why).

Absolute value \(r\) (distance from the origin)

Standard notation (the usual math form)

In words (symbols replaced with words)

\(r\) \(=\) \(\sqrt{a^{2} + b^{2}}\)

② absolute value \(r\) \(=\) ① square root of the sum of the squares of the real part \(a\) and the imaginary part \(b\)

The formula in words

① The square root of the real part \(a\) squared plus the imaginary part \(b\) squared, \(\sqrt{a^2 + b^2}\)

② is the absolute value \(r\)

Quick example

The absolute value of \(z = 3 + 4i\) is

absolute value \(r\) \(=\) \(\sqrt{3^2 + 4^2}\)

\(r = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\)

Key idea

This is just the Pythagorean theorem from Grade 8. On the complex plane, the point \(z = a + bi\) is \(a\) across and \(b\) up. Joining the origin, the point \((a,\ 0)\) and \(z\) makes a right triangle, and the distance from the origin to \(z\) (the hypotenuse) is \(\sqrt{a^2 + b^2}\). By convention, a square root in the answer is written in simplest radical form, with square factors taken out, as in \(\sqrt{8} = 2\sqrt{2}\) (this calculator shows it in that form too).

Argument \(\theta\) (found from tan θ and the quadrant)

Standard notation (the usual math form)

In words (symbols replaced with words)

\(\tan\theta\) \(=\) \(\dfrac{b}{a}\)

② tangent of the argument \(\theta\) \(=\) ① imaginary part \(b\) divided by real part \(a\)

The formula in words

① The imaginary part \(b\) divided by the real part \(a\)

② is the tangent of the argument, \(\tan\theta\) (this formula alone does not pin down a single \(\theta\), so the quadrant of the point \(z\) is used to choose one)

Quick example

The argument of \(z = -1 + i\) (the real part is negative and the imaginary part is positive, so the point is in Quadrant 2):

tangent \(\tan\theta\) \(=\) \(\dfrac{1}{-1} = -1\)

\(\tan\theta = \dfrac{1}{-1} = -1\)

\(\theta = \dfrac{3}{4}\pi \quad \left(a < 0,\ b > 0\right)\)

Key idea

In the range \(0 \le \theta < 2\pi\), there are two angles with \(\tan\theta = -1\): \(\dfrac{3}{4}\pi\) (135°) and \(\dfrac{7}{4}\pi\) (315°). Which one is right depends on the quadrant of the point \(z\). For \(z = -1 + i\), the real part is negative and the imaginary part is positive, which is Quadrant 2 (upper left), so we choose \(\theta = \dfrac{3}{4}\pi\). This is the quadrant adjustment. The \(\arctan\) (inverse tangent) key on a calculator or in a spreadsheet only returns angles between \(-\dfrac{\pi}{2}\) and \(\dfrac{\pi}{2}\), so for points in Quadrants 2 and 3 its answer is not the argument as it is. You need to adjust the acute angle you get from \(\arctan\), for example by adding \(\pi\), depending on the quadrant (this calculator shows the adjustment in the steps).

Multiplying in polar form (multiply the absolute values, add the arguments)

Standard notation (the usual math form)

\(z_{1} z_{2}\) \(=\) \(r_{1} r_{2}\) \(\left\{\cos\left(\theta_{1} + \theta_{2}\right) + i\sin\left(\theta_{1} + \theta_{2}\right)\right\}\)

In words (symbols replaced with words)

③ product of the two complex numbers \(=\) ① product of the absolute values ② direction given by the sum of the arguments

The formula in words

① Take the product of the absolute values, \(r_1 r_2\)

② multiply it by the direction part for the sum of the arguments, \(\theta_1 + \theta_2\)

③ and you get the product of the two complex numbers, \(z_1 z_2\) (multiplying by \(z_2\) scales the length by \(r_2\) and rotates the direction by \(\theta_2\))

Quick example

The product of \(z_1 = 2\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)\) and \(z_2 = 3\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)\) is

product \(z_1 z_2\) \(=\) absolute values multiplied, \(2 \times 3\) \(\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}\)

\(z_1 z_2 = 2 \times 3\left\{\cos\left(\dfrac{\pi}{6} + \dfrac{\pi}{3}\right) + i\sin\left(\dfrac{\pi}{6} + \dfrac{\pi}{3}\right)\right\}\)

\(z_1 z_2 = 6\left(\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}\right) = 6i\)

Key idea

This is the biggest advantage of polar form. Multiplying in rectangular form needs the distributive property to expand four terms, but in polar form you just "multiply the absolute values and add the arguments". Geometrically, multiplying by \(z_2\) is "rotating by \(\theta_2\) around the origin and scaling by \(r_2\)". For example, multiplying by \(i\) (absolute value 1, argument 90°) rotates a point exactly 90° around the origin. Using this rule to multiply the same number \(n\) times gives De Moivre's theorem, \(\left(\cos\theta + i\sin\theta\right)^n = \cos n\theta + i\sin n\theta\), which leads on to \(n\)th powers and \(n\)th roots of complex numbers (Precalculus).

A complex number \(z = a + bi\) can also be written in polar form \(z = r(\cos\theta + i\sin\theta)\), using the absolute value \(r = \sqrt{a^2 + b^2}\) (the distance from the origin) and the argument \(\theta\) (the angle measured counterclockwise from the positive real axis). The argument is found from \(\tan\theta = \dfrac{b}{a}\) and the quadrant of the point. In polar form, multiplying complex numbers becomes a simple rule: multiply the absolute values and add the arguments.

Symbols and terms

Symbols

\(i\)	i	The imaginary unit, the number whose square is \(-1\) (\(i^2 = -1\)). The letter comes from "imaginary".
\(z\)	z	The letter usually used for a complex number. By custom, \(z\) and \(w\) are used for complex numbers, to tell them apart from real variables \(x,\ y\).
\(a,\ b\)	a, b	The real part (\(a\)) and the imaginary part (\(b\)) of \(z = a + bi\). Both are real numbers. On the complex plane, \(a\) is the position across (along the real axis) and \(b\) is the position up (along the imaginary axis).
\(r\)	r	The letter for the absolute value (the distance from the origin to the point \(z\)). It comes from "radius". In polar form it is a real number that is 0 or more.
\(\|z\|\)	absolute value of z	The symbol for the absolute value (modulus) of \(z\). It is written with two vertical bars, the same as for a real number, and \(\|a + bi\| = \sqrt{a^2 + b^2}\).
\(\theta\)	theta	The Greek letter used for the argument. In math it is the usual letter for an angle.
\(\arg z\)	arg z	The symbol for the argument of \(z\), from the first three letters of "argument". It is used as in \(\arg z = \theta\).
\(\cos\theta,\ \sin\theta\)	cosine theta, sine theta	Trigonometric functions. On the circle of radius 1 (the unit circle), the point in the direction of angle \(\theta\) has horizontal position \(\cos\theta\) and vertical position \(\sin\theta\). In polar form they are the part that gives the direction.
\(\tan\theta\)	tangent theta	A trigonometric function defined by \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\). On the complex plane, \(\tan\theta = \dfrac{b}{a}\) (rise over run, the slope), which is the key to finding the argument.
\(\arctan\)	arctangent	The inverse of tangent (also written \(\tan^{-1}\)). It returns "the angle \(\theta\) with \(\tan\theta = x\)". The angle it returns is always between \(-\dfrac{\pi}{2}\) and \(\dfrac{\pi}{2}\), so the quadrant adjustment is needed for arguments in Quadrants 2 and 3.
\(\pi\)	pi	The ratio of a circle's circumference to its diameter (about 3.14159). In radians, a half turn, \(180^{\circ}\), is exactly \(\pi\). The Greek letter \(\pi\) is said to come from the first letter of the Greek word for "perimeter".

Terms

complex number	A number of the form \(a + bi\), where \(a\) and \(b\) are real numbers. It is a number system that brings the real and imaginary numbers together, taught in Algebra 2.
imaginary unit	The number \(i\) whose square is \(-1\). No real number has a negative square, so it was introduced as a new number.
real part	The \(a\) in the complex number \(z = a + bi\). On the complex plane, it is the horizontal coordinate.
imaginary part	The \(b\) in the complex number \(z = a + bi\). It is the real number in front of \(i\), without the \(i\) itself (the imaginary part of \(1 + 2i\) is \(2\), not \(2i\)).
complex plane	The plane where \(a + bi\) is shown as the point \((a,\ b)\). The horizontal axis is the real axis and the vertical axis is the imaginary axis. A drawing of it is also called an Argand diagram. It is taught in Precalculus.
rectangular form	Writing a complex number as \(a + bi\) (the coordinates \(a\) across and \(b\) up). Also called standard form. Addition and subtraction are easiest in this form.
polar form	Writing a complex number with its distance \(r\) from the origin and its angle of rotation \(\theta\) from the real axis, as \(r(\cos\theta + i\sin\theta)\). Also called trigonometric form. Multiplication, division and powers are easiest in this form.
absolute value (modulus)	The distance from the origin to the point \(z\), calculated as \(\|z\| = \sqrt{a^2 + b^2}\). It extends the absolute value of a real number (its distance from 0 on the number line) to the plane.
argument	The angle of the direction from the origin to the point \(z\), measured counterclockwise from the positive real axis. The symbol is \(\arg z\). Adding a full turn (\(2\pi\)) gives the same direction, so by convention the answer is given in a set range such as \(0 \le \theta < 2\pi\) or \(-\pi < \theta \le \pi\) (the principal argument).
quadrant	One of the four regions the two axes divide the coordinate plane into. The upper right is Quadrant 1 (often written I), and counting counterclockwise from there come Quadrants 2, 3 and 4. Which quadrant a point is in is the key to pinning down its argument.
radian measure (radians)	Measuring an angle by the length of the arc it cuts on a circle of radius 1. \(180^{\circ} = \pi\) radians. From trigonometric functions in Precalculus on, radians are the standard unit rather than degrees.
special angles	\(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\) and related angles (these plus multiples of \(90^{\circ}\)). Their trig values can be written exactly, as \(\dfrac{1}{2}\), \(\dfrac{\sqrt{2}}{2}\) and \(\dfrac{\sqrt{3}}{2}\), and most textbook and test problems use these angles.
quadrant adjustment	Within one full turn there are two angles with \(\tan\theta = \dfrac{b}{a}\), so you choose the one that matches the quadrant of the point (for example, by adding \(\pi\) to the acute angle from \(\arctan\)).
De Moivre's theorem	The theorem \(\left(\cos\theta + i\sin\theta\right)^n = \cos n\theta + i\sin n\theta\). It is the result of repeating "multiplying adds the arguments" \(n\) times, and it is used to find \(n\)th powers and \(n\)th roots of complex numbers (Precalculus).

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Square roots and radicals (Grades 8–9)	Knowing that \(\sqrt{2}\) is "the positive number whose square is 2" Being able to simplify a radical by taking out square factors, as in \(\sqrt{8} = 2\sqrt{2}\)
The Pythagorean theorem (Grade 8)	Knowing that the hypotenuse of a right triangle is \(\sqrt{a^2 + b^2}\) when the other two sides are \(a\) and \(b\) Being able to find the distance between two points on the coordinate plane with the Pythagorean theorem
Complex numbers (Algebra 2)	Knowing that the imaginary unit \(i\) is the number with \(i^2 = -1\) Being able to read the real part \(a\) and the imaginary part \(b\) from \(a + bi\)
Trigonometry and the unit circle (Geometry, Algebra 2)	Knowing that \(\sin\), \(\cos\) and \(\tan\) are the height, the horizontal position and the slope for a point on the unit circle Knowing the trig values of \(30^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\) (\(\dfrac{1}{2}\), \(\dfrac{\sqrt{2}}{2}\) and \(\dfrac{\sqrt{3}}{2}\)) Being able to find trig values of angles over \(90^{\circ}\) with the unit circle (example: \(\cos 135^{\circ} = -\dfrac{\sqrt{2}}{2}\))
Radian measure (Algebra 2, Precalculus)	Being able to convert between degrees and radians with \(180^{\circ} = \pi\) radians (example: \(60^{\circ} = \dfrac{\pi}{3}\))
The complex plane (Precalculus)	Being able to show a complex number \(a + bi\) as the point \((a,\ b)\) on the plane Knowing that the horizontal axis is called the real axis and the vertical axis the imaginary axis

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table for rectangular form to polar form

Real part a	3
Imaginary part b	4
Absolute value r	=SQRT(B1^2+B2^2)
Argument θ (radians, 0 ≤ θ < 2π)	=MOD(ATAN2(B1,B2),2*PI())
Argument θ (degrees)	=DEGREES(MOD(ATAN2(B1,B2),2*PI()))

Table for polar form to rectangular form

Absolute value r	2
Argument θ (degrees)	60
Real part a = r cos θ	=B1*COS(RADIANS(B2))
Imaginary part b = r sin θ	=B1*SIN(RADIANS(B2))

After pasting, the upper rows are your inputs and the lower rows are calculated automatically.
In the first table, ATAN2(x-coordinate, y-coordinate) returns the argument with the quadrant adjustment already done (greater than −π and up to π). Taking MOD with 2π turns it into an argument from 0 up to (but not including) 2π. For the example z = 3 + 4i, r = 5 and θ ≈ 0.9273 (about 53.13 degrees).
In the second table, RADIANS converts degrees to radians. For the example r = 2 and θ = 60 degrees, a = 1 and b ≈ 1.7320508 (= √3). Excel gives decimals, so if you want exact values such as √2 or fractions of π, use the calculator on this page.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table for rectangular form to polar form

Real part a	3
Imaginary part b	4
Absolute value r	=SQRT(B1^2+B2^2)
Argument θ (radians, 0 ≤ θ < 2π)	=MOD(ATAN2(B1,B2),2*PI())
Argument θ (degrees)	=DEGREES(MOD(ATAN2(B1,B2),2*PI()))

Table for polar form to rectangular form

Absolute value r	2
Argument θ (degrees)	60
Real part a = r cos θ	=B1*COS(RADIANS(B2))
Imaginary part b = r sin θ	=B1*SIN(RADIANS(B2))

The same formulas as in Excel work as is (ATAN2 also takes its inputs in the same order, "x-coordinate, y-coordinate"). Copy the whole table, paste it into cell A1, and replace the numbers with your own.

How to calculate it in Python

import math

# Rectangular form -> polar form
real_part = 3        # real part a
imaginary_part = 4   # imaginary part b
absolute_value = math.sqrt(real_part ** 2 + imaginary_part ** 2)
argument = math.atan2(imaginary_part, real_part)   # includes the quadrant adjustment (-π < θ ≤ π)
if argument < 0:
    argument += 2 * math.pi                        # move it into the range 0 ≤ θ < 2π
print(f"Absolute value r = {absolute_value}")
print(f"Argument θ = {argument} rad ({math.degrees(argument)} degrees)")

# Polar form -> rectangular form
r = 2
theta_degree = 60
theta = math.radians(theta_degree)
print(f"Real part a = {r * math.cos(theta)}")
print(f"Imaginary part b = {r * math.sin(theta)}")

math.atan2(y, x) returns the angle with the quadrant adjustment already done, so unlike arctan it gives the right direction even for points in Quadrants 2 and 3. The first half of this example converts z = 3 + 4i, and running it prints r = 5.0 and θ ≈ 0.9272952180016122 rad (about 53.13 degrees). The second half converts back from r = 2 and θ = 60°, giving a ≈ 1.0000000000000002 and b ≈ 1.7320508075688772 (= √3). The value of a is not exactly 1 because of rounding errors in decimals (floating-point numbers); for exact values you need to calculate with symbols, as the calculator on this page does. You can also get the absolute value and the angle (−π < θ ≤ π) at once with cmath.polar(complex(3, 4)) from the standard cmath module.

How to write it in LaTeX and other math languages (copy and paste)

Polar form (another way to write a complex number)

z = r(cosθ + i sinθ)

z = r(\cos\theta + i\sin\theta)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>z</mi>
    <mo>=</mo>
    <mi>r</mi>
    <mo>(</mo>
    <mi>cos</mi><mi>&#x3B8;</mi>
    <mo>+</mo>
    <mi>i</mi><mi>sin</mi><mi>&#x3B8;</mi>
    <mo>)</mo>
  </mrow>
</math>

z = r(cos theta + i sin theta)

z == r (Cos[theta] + I Sin[theta])

z := r*(cos(theta) + I*sin(theta));

z = r*(cos(theta) + 1i*sin(theta));

z = r(cos θ + i sin θ)

Absolute value \(r\) (distance from the origin)

r = |z| = √(a² + b²)

r = |z| = \sqrt{a^{2} + b^{2}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>r</mi>
    <mo>=</mo>
    <mo>|</mo><mi>z</mi><mo>|</mo>
    <mo>=</mo>
    <msqrt>
      <mrow>
        <msup><mi>a</mi><mn>2</mn></msup>
        <mo>+</mo>
        <msup><mi>b</mi><mn>2</mn></msup>
      </mrow>
    </msqrt>
  </mrow>
</math>

r = |z| = sqrt(a^2 + b^2)

Abs[a + b I]

r := abs(a + b*I);

r = abs(a + b*1i);

r = √(a^2 + b^2)

Argument \(\theta\) (found from tan θ and the quadrant)

tanθ = b/a

\tan\theta = \dfrac{b}{a}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>tan</mi><mi>&#x3B8;</mi>
    <mo>=</mo>
    <mfrac><mi>b</mi><mi>a</mi></mfrac>
  </mrow>
</math>

tan theta = b/a

theta = Arg[a + b I]

theta := argument(a + b*I);

theta = angle(a + b*1i);

tan θ = b/a

Multiplying in polar form (multiply the absolute values, add the arguments)

z₁z₂ = r₁r₂{cos(θ₁+θ₂) + i sin(θ₁+θ₂)}

z_{1}z_{2} = r_{1}r_{2}\{\cos(\theta_{1}+\theta_{2}) + i\sin(\theta_{1}+\theta_{2})\}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>z</mi><mn>1</mn></msub>
    <msub><mi>z</mi><mn>2</mn></msub>
    <mo>=</mo>
    <msub><mi>r</mi><mn>1</mn></msub>
    <msub><mi>r</mi><mn>2</mn></msub>
    <mo>{</mo>
    <mi>cos</mi>
    <mo>(</mo>
    <msub><mi>&#x3B8;</mi><mn>1</mn></msub>
    <mo>+</mo>
    <msub><mi>&#x3B8;</mi><mn>2</mn></msub>
    <mo>)</mo>
    <mo>+</mo>
    <mi>i</mi>
    <mi>sin</mi>
    <mo>(</mo>
    <msub><mi>&#x3B8;</mi><mn>1</mn></msub>
    <mo>+</mo>
    <msub><mi>&#x3B8;</mi><mn>2</mn></msub>
    <mo>)</mo>
    <mo>}</mo>
  </mrow>
</math>

z_1 z_2 = r_1 r_2 (cos(theta_1 + theta_2) + i sin(theta_1 + theta_2))

r1 r2 (Cos[theta1 + theta2] + I Sin[theta1 + theta2])

z1z2 := r1*r2*(cos(theta1 + theta2) + I*sin(theta1 + theta2));

z12 = r1*r2*(cos(t1 + t2) + 1i*sin(t1 + t2));

z_1 z_2 = r_1 r_2 (cos(θ_1+θ_2) + i sin(θ_1+θ_2))

How to have ChatGPT do the calculation

You are a math calculation assistant for the complex plane. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

1. Convert the complex number z = 3 + 4i to polar form r(cos θ + i sin θ). Show the absolute value r and the argument θ (in the range 0 ≤ θ < 2π, in both radians and degrees).
2. Convert the complex number with absolute value r = 2 and argument θ = 60° to rectangular form a + bi. Show a and b as decimals, and also as exact values with square roots if possible.

In Python, use math.atan2 and math.sqrt, and show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula