Complex Number Calculator (Add, Subtract, Multiply and Divide with Steps, Conjugate, Modulus and Complex Plane)

Enter the real and imaginary parts of the two complex numbers z₁ = a + bi and z₂ = c + di, and choose the operation (+ − × ÷). The expression below is linked to the input fields, so you can also edit the numbers directly in it.

Real part a of z₁

Imaginary part b of z₁

Operation

Real part c of z₂

Imaginary part d of z₂

Decimals, negative numbers and fractions such as 3/4 can be used. A blank real or imaginary part is treated as 0 (for example, if only the imaginary part of z₁ is blank, z₁ is the real number a).

Result and graph

Enter the real and imaginary parts in the fields on the left, choose an operation (+ − × ÷) 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

Add, subtract, multiply or divide two complex numbers \(z_1 = a+bi\) and \(z_2 = c+di\) on the spot. Just enter the real and imaginary parts and choose the operation
The answer is shown in \(a+bi\) form with fractions in lowest terms (exact values), such as \(\dfrac{11}{25} + \dfrac{2}{25}i\). For division, the steps show how multiplying by the complex conjugate of the denominator makes the denominator real. For multiplication, they show the expansion and how \(i^2\) is replaced with \(-1\)
As a bonus, the complex conjugate \(\overline{z}\) and the absolute value \(|z|\) of the result are also shown. The absolute value is given as an exact value with the radical simplified, such as \(|3+4i| = 5\) or \(|1+i| = \sqrt{2}\)
You can also see the result on the complex plane (Argand diagram), which shows at a glance where \(z_1\), \(z_2\) and the result lie
The real and imaginary parts can be decimals, negative numbers or fractions such as 3/4

This page covers the four basic operations on complex numbers (plus the complex conjugate and absolute value), as taught in Algebra 2. Polar form (writing a number with its modulus and argument) and De Moivre's theorem (Precalculus) are not covered here.

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, and different parts of a circuit shift the timing of the wave (the phase). In electrical engineering, the standard method is to combine this "size" and "shift" into one complex number (the impedance), so that a whole circuit can be solved with nothing more than complex arithmetic.
Complex calculations are part of every electrical engineering course and appear on engineering licensing exams such as the FE exam. Everywhere AC flows, from a phone charger to a power plant, the design relies on complex numbers.

Signal analysis in communications and audio (Fourier transform)

Waves such as sound and radio can be broken down with the Fourier transform, a calculation that uses complex numbers, into "how much of each pitch (frequency) is in it". Phone networks (Wi-Fi, 5G), music compression, noise reduction and speech recognition all run huge numbers of these complex calculations inside.
The strength of complex numbers is that one number carries both the size and the timing of a wave, so in signal processing complex numbers are used even more than real numbers. The Fourier transform itself is college-level, but what happens inside it is the complex multiplication and addition taught in Algebra 2.

Controlling airplanes and robots (checking stability)

When designing machines that must damp out shaking and stay steady, such as autopilots, drones and robots, engineers check stability by looking at where the solutions (complex numbers) of the system's equation lie on the complex plane. If the real parts of the solutions are negative, the shaking dies down; if any is positive, it grows.
In this field, the positions of the solutions on the complex plane directly decide whether a machine can fly safely. Control engineering, which covers this stability test, is studied in college.

Rotation in games and computer graphics

If you think of a point \((x,\ y)\) on the plane as the complex number \(x+yi\), multiplying by a complex number with absolute value 1 rotates the point around the origin. For example, multiplying by \(i\) rotates it by 90 degrees. This is exactly what 2D games need to turn characters or bullets.
Quaternions, used in 3D games and in phones to track their orientation, extend this idea of "rotation by complex numbers" to three dimensions.

Solving quadratic and cubic equations completely (math)

A quadratic equation has no real solution when its discriminant \(D = b^2 - 4ac\) is negative. But once the number system is extended to the complex numbers, even \(x^2 = -1\) has the solutions \(x = \pm i\), and every quadratic or cubic equation has solutions (the fundamental theorem of algebra).
With no more "no solution" exceptions, not only mathematics but also the theories of physics and engineering are built on complex numbers. Complex arithmetic in Algebra 2 is the entry point.

Formulas and figures

Adding and subtracting complex numbers

Figure

Standard notation (the usual math form)

\((a+bi) + (c+di)\) \(=\) \((a+c)\) \(+\) \((b+d)i\)

In words (symbols replaced with words)

③ sum of \(z_1\) and \(z_2\) \(=\) ① real parts added \(+\) ② imaginary parts added, times \(i\)

The formula in words

① The real parts and the \(i\) parts can be handled separately. Take the sum of the real parts, \(a+c\)

② add the sum of the imaginary parts, \(b+d\) with \(i\) attached

③ and you get the sum of \(z_1\) and \(z_2\)

Quick example

Working out \((2+3i) + (4-i)\) (the imaginary part of \(4-i\) is \(-1\)):

sum \((2+3i)+(4-i)\) \(=\) real parts \(2+4\) \(+\) imaginary parts \(3-1\), times \(i\)

\((2+3i) + (4-i) = (2+4) + (3-1)i = 6 + 2i\)

Key idea

Subtraction works exactly the same way: subtract the real parts and the imaginary parts separately, so \((a+bi) - (c+di) = (a-c) + (b-d)i\). The trick is to treat \(i\) like a variable such as \(x\) and combine like terms. When you add \(2x + 3\) and \(4x - 1\), you combine the \(x\) terms and the number terms separately. In the same way, combine the \(i\) terms (the imaginary part) and the number terms (the real part) separately. On the complex plane, addition is "joining two arrows tip to tail", and the four points \(0\), \(z_1\), \(z_2\) and \(z_1+z_2\) form a parallelogram (see the figure above).

Multiplying complex numbers and \(i^2 = -1\)

Standard notation (the usual math form)

\((a+bi)(c+di)\) \(=\) \((ac-bd)\) \(+\) \((ad+bc)i\)

In words (symbols replaced with words)

③ product of \(z_1\) and \(z_2\) \(=\) ① real part: \(ac\) minus \(bd\) \(+\) ② imaginary part: \(ad\) plus \(bc\), times \(i\)

The formula in words

① Expand the parentheses as you would with variables and replace \(i^2\) with \(-1\). You get the real part \(ac - bd\)

② and the imaginary part \(ad + bc\) with \(i\) attached.

③ Add the two, and you get the product of \(z_1\) and \(z_2\)

Quick example

Working out \((1+2i)(3+4i)\):

product \((1+2i)(3+4i)\) \(=\) real part \(1 \times 3 - 2 \times 4\) \(+\) imaginary part \(1 \times 4 + 2 \times 3\), times \(i\)

\((1+2i)(3+4i) = 3 + 4i + 6i + 8i^{2}\)

\(= 3 + 10i - 8 = -5 + 10i\)

Key idea

You expand exactly as with variables: \((a+b)(c+d) = ac + ad + bc + bd\) (the FOIL method). There is only one rule special to complex numbers: whenever \(i \times i = i^2\) appears, replace it with \(-1\). This is because \(i\) is defined as the number whose square is \(-1\) (the imaginary unit). Among the real numbers, every square is 0 or more, so \(x^2 = -1\) has no solution. By adding this \(i\) to the numbers, we get a larger number system, the complex numbers, in which every quadratic equation has a solution. There is no need to memorize the formula. Just "expand, then turn \(i^2\) into \(-1\)" every time, and you get the same result.

Dividing complex numbers (multiplying by the conjugate)

Standard notation (the usual math form)

\(\dfrac{a+bi}{c+di}\) \(=\) \(\dfrac{ac+bd}{c^{2}+d^{2}}\) \(+\) \(\dfrac{bc-ad}{c^{2}+d^{2}}\,i\)

In words (symbols replaced with words)

③ quotient of \(z_1\) divided by \(z_2\) \(=\) ① real part: \(ac+bd\) divided by \(c^2+d^2\) \(+\) ② imaginary part: \(bc-ad\) divided by \(c^2+d^2\), times \(i\)

The formula in words

① Multiply the numerator and the denominator by \(c-di\), the conjugate of the denominator. The denominator becomes the real number \(c^2+d^2\), and you get the real part \(\dfrac{ac+bd}{c^2+d^2}\)

② and the imaginary part \(\dfrac{bc-ad}{c^2+d^2}\) with \(i\) attached.

③ Add the two, and you get the quotient of \(z_1\) divided by \(z_2\)

Quick example

Working out \(\dfrac{1+2i}{3+4i}\) (multiply the numerator and the denominator by \(3-4i\), the conjugate of the denominator):

quotient \(\dfrac{1+2i}{3+4i}\) \(=\) real part \(\dfrac{11}{25}\) \(+\) imaginary part \(\dfrac{2}{25}\), times \(i\)

\(\dfrac{1+2i}{3+4i} = \dfrac{(1+2i)(3-4i)}{(3+4i)(3-4i)} = \dfrac{3 - 4i + 6i - 8i^{2}}{3^{2}+4^{2}}\)

\(= \dfrac{11 + 2i}{25} = \dfrac{11}{25} + \dfrac{2}{25}i\)

Key idea

Why multiply by the complex conjugate? Because \((c+di)(c-di) = c^2 - (di)^2 = c^2 + d^2\), so \(i\) drops out cleanly. This combines the difference of squares \((x+y)(x-y) = x^2 - y^2\) with \(i^2 = -1\). While \(i\) is still in the denominator, you cannot read off the \(a+bi\) form (the real part and the imaginary part), so the first job is to get \(i\) out of the denominator. It is the same idea as rationalizing the denominator in Algebra 1, where you multiply \(\dfrac{1}{\sqrt{2}}\) by \(\sqrt{2}\) on top and bottom to remove the square root from the denominator. You cannot divide by \(z_2 = 0\) (real part and imaginary part both 0). The rule "you cannot divide by 0" holds for complex numbers too.

Absolute value (modulus) of a complex number \(|z|\)

Figure

Standard notation (the usual math form)

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

In words (symbols replaced with words)

② absolute value of \(z\) \(=\) ① square root of (real part squared + imaginary part squared)

The formula in words

① For a complex number \(z = a+bi\), the square root of the real part squared plus the imaginary part squared, \(\sqrt{a^2+b^2}\)

② is the absolute value of \(z\)

Quick example

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

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

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

\(|1+i| = \sqrt{1^{2}+1^{2}} = \sqrt{2}\)

Key idea

The absolute value \(|z|\) is the distance from the origin to \(z\) on the complex plane (see the figure above). The point is \(a\) across and \(b\) up, so the distance is just the Pythagorean theorem. You can think of it as the absolute value of a real number (its distance from 0 on the number line) extended to the plane. With the complex conjugate \(\overline{z} = a-bi\), there is the relation \(z \times \overline{z} = a^2 + b^2 = |z|^2\). This is why, in division, multiplying by the conjugate turns the denominator into \(|z_2|^2\), a positive real number. Sometimes the square root disappears, as in \(\sqrt{25} = 5\), and sometimes it stays, as in \(\sqrt{2}\). This calculator also simplifies radicals, such as \(\sqrt{48} = 4\sqrt{3}\), and answers with exact values.

The basic rule for complex numbers is to treat \(i\) like a variable, and replace \(i^2\) with \(-1\) whenever it appears. Only division needs an extra step: multiply the numerator and the denominator by the complex conjugate of the denominator to make the denominator real, then split the result into its real part and imaginary part.

Symbols and terms

Symbols

\(i\)	i	The imaginary unit. It is defined as the number whose square is \(-1\) (\(i^2 = -1\)). The letter comes from "imaginary". (In electrical engineering, \(j\) is used instead, because \(i\) stands for current.)
\(z\)	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 \(x\) and \(y\).
\(z_1,\ z_2\)	z sub 1, z sub 2	The first and second complex numbers. The small number at the lower right (the subscript) is just a label for "which one". It is not the same as an exponent, which is written at the upper right.
\(a,\ b,\ c,\ d\)	a, b, c, d	On this page, \(a\) is the real part and \(b\) the imaginary part of \(z_1 = a+bi\), and \(c\) is the real part and \(d\) the imaginary part of \(z_2 = c+di\). By custom, letters near the start of the alphabet stand for fixed numbers.
\(\overline{z}\)	z bar	The complex conjugate of \(z\). For \(z = a+bi\), it is \(a-bi\), with only the sign of the imaginary part changed. The bar on top marks "the partner". Some books write it as \(z^*\).
\(\|z\|\)	absolute value of z	The absolute value (modulus) of \(z\). It is the distance from the origin to \(z\) on the complex plane, calculated as \(\|a+bi\| = \sqrt{a^2+b^2}\). It is written with two vertical bars, the same as the absolute value of a real number.

Terms

complex number	A number of the form \(a+bi\), where \(a\) and \(b\) are real numbers. It is a larger number system that includes both real and imaginary numbers, and within it every quadratic equation has a solution.
imaginary number	A complex number that is not real, that is, \(a+bi\) with a nonzero imaginary part \(b\). (Some books use "imaginary number" only for pure imaginary numbers such as \(2i\).) Despite the name, these numbers are very useful in real calculations in electricity and physics.
imaginary unit	The number \(i\). It is defined by \(i^2 = -1\), and every imaginary number is written using \(i\).
real part	The \(a\) in \(a+bi\), the "real number part" of a complex number. On the complex plane, it is the coordinate along the horizontal axis (the real axis).
imaginary part	The \(b\) in \(a+bi\) (without the \(i\)). On the complex plane, it is the coordinate along the vertical axis (the imaginary axis). For example, the imaginary part of \(3-2i\) is \(-2\).
pure imaginary number	A complex number whose real part is 0 and whose imaginary part is not 0, such as \(2i\) or \(-3i\). On the complex plane, it lies on the imaginary axis.
complex conjugate	For \(a+bi\), the number \(a-bi\). Often just called "the conjugate". Its key property is that the product \((a+bi)(a-bi) = a^2+b^2\) is a real number that is 0 or more, which is why it is used to make the denominator real in division.
absolute value (modulus)	The distance \(\sqrt{a^2+b^2}\) from the origin on the complex plane. It extends the absolute value of a real number (its distance from 0 on the number line) to the plane. Also called the modulus.
complex plane	A plane with the real part on the horizontal axis and the imaginary part on the vertical axis, where \(a+bi\) is shown as the point \((a,\ b)\). A drawing of it is also called an Argand diagram. It is covered in detail in Precalculus, but this page uses it as a tool to make complex numbers "visible".
real axis	The horizontal axis of the complex plane, where the real numbers lie. A real number is a complex number with imaginary part 0, so it sits on the real axis.
imaginary axis	The vertical axis of the complex plane, where the pure imaginary numbers (complex numbers with real part 0, such as \(2i\)) lie.
making the denominator real	For a fraction with a complex denominator, multiplying the numerator and the denominator by the conjugate of the denominator so that the denominator becomes real. It is the complex-number version of rationalizing the denominator (removing a square root from the denominator).
expanding	Removing the parentheses from an expression and writing it as a sum or difference of terms. As in \((a+b)(c+d) = ac+ad+bc+bd\), each term in one set of parentheses is multiplied by every term in the other.
like terms	Terms with the same variable part. When adding or subtracting complex numbers, the terms with \(i\) (the imaginary part) and the terms without \(i\) (the real part) are each combined as like terms.
discriminant	The expression \(D = b^2-4ac\), which tells what kind of solutions the quadratic equation \(ax^2+bx+c=0\) has. When \(D < 0\) there is no real solution, but there are complex solutions \(x = \dfrac{-b \pm \sqrt{-D}\,i}{2a}\). This is usually where students first meet complex numbers.

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.

Expanding expressions and like terms (Algebra 1, Grades 8–9)	Being able to expand a product of parentheses, as in \((a+b)(c+d) = ac+ad+bc+bd\) Being able to use the difference of squares \((x+y)(x-y) = x^{2}-y^{2}\) (this is exactly why multiplying by the complex conjugate gives a real number) Being able to combine like terms, as in \(2x + 3x = 5x\)
Square roots and radicals (Grades 8–9)	Knowing that a square root is the positive number whose square is the number inside, as in \(\sqrt{25} = 5\) Being able to simplify a radical by taking out square factors, as in \(\sqrt{48} = 4\sqrt{3}\)
Rationalizing the denominator (Algebra 1 and 2)	Being able to remove a radical from the denominator by multiplying the top and bottom by the same number, as in \(\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}\) (making the denominator real is the complex-number version of this idea)
The coordinate plane and the Pythagorean theorem (Grades 6–8)	Being able to plot the point \((a,\ b)\) on the coordinate plane (the complex plane is a coordinate plane with the real part across and the imaginary part up) Knowing from the Pythagorean theorem that the distance between the origin and \((a,\ b)\) is \(\sqrt{a^{2}+b^{2}}\) (this is what the absolute value of a complex number really is)
Quadratic equations and the discriminant (Algebra 1 and 2)	Being able to use the quadratic formula \(x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) Knowing that when the discriminant \(D = b^{2}-4ac\) is negative, there is no real solution (this is where imaginary numbers come in)
Working with fractions (Grades 5–7)	Being able to reduce fractions and find common denominators (results are shown as fractions in lowest terms, such as \(\dfrac{11}{25}\)) Being able to turn a fraction into a decimal (\(\dfrac{11}{25}\) into 0.44)

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 the four operations on complex numbers

Real part a of z₁	1
Imaginary part b of z₁	2
Real part c of z₂	3
Imaginary part d of z₂	4
Complex number z₁	=COMPLEX(B1,B2)
Complex number z₂	=COMPLEX(B3,B4)
Sum z₁ + z₂	=IMSUM(B5,B6)
Difference z₁ − z₂	=IMSUB(B5,B6)
Product z₁ × z₂	=IMPRODUCT(B5,B6)
Quotient z₁ ÷ z₂	=IMDIV(B5,B6)

Table for the complex conjugate and absolute value

Real part a	3
Imaginary part b	4
Complex number z	=COMPLEX(B1,B2)
Complex conjugate	=IMCONJUGATE(B3)
Absolute value \|z\|	=IMABS(B3)

Table to check division (multiplying by the conjugate)

Real part a of z₁	1
Imaginary part b of z₁	2
Real part c of z₂	3
Imaginary part d of z₂	4
Denominator c²+d²	=B3^2+B4^2
Real part of quotient (ac+bd)÷(c²+d²)	=(B1B3+B2B4)/B5
Imaginary part of quotient (bc−ad)÷(c²+d²)	=(B2B3-B1B4)/B5

Excel has a full set of functions for complex numbers. COMPLEX makes a complex number (a text string such as "1+2i") from the real and imaginary parts, and you pass it to IMSUM (sum), IMSUB (difference), IMPRODUCT (product) or IMDIV (quotient).
The first table uses z₁ = 1+2i and z₂ = 3+4i. The sum shows "4+6i", the difference "-2-2i", the product "-5+10i" and the quotient "0.44+0.08i" (these results are text, not numbers).
The second table uses z = 3+4i. The complex conjugate is "3-4i" and the absolute value is 5 (only IMABS returns an ordinary number).
The third table checks division without the complex-number functions, following the conjugate formula directly. The real part of the quotient is 0.44 (= 11/25) and the imaginary part is 0.08 (= 2/25), which match the IMDIV result. "^" is a power, "*" is multiplication and "/" is division.

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 the four operations on complex numbers

Real part a of z₁	1
Imaginary part b of z₁	2
Real part c of z₂	3
Imaginary part d of z₂	4
Complex number z₁	=COMPLEX(B1,B2)
Complex number z₂	=COMPLEX(B3,B4)
Sum z₁ + z₂	=IMSUM(B5,B6)
Difference z₁ − z₂	=IMSUB(B5,B6)
Product z₁ × z₂	=IMPRODUCT(B5,B6)
Quotient z₁ ÷ z₂	=IMDIV(B5,B6)

Table for the complex conjugate and absolute value

Real part a	3
Imaginary part b	4
Complex number z	=COMPLEX(B1,B2)
Complex conjugate	=IMCONJUGATE(B3)
Absolute value \|z\|	=IMABS(B3)

Table to check division (multiplying by the conjugate)

Real part a of z₁	1
Imaginary part b of z₁	2
Real part c of z₂	3
Imaginary part d of z₂	4
Denominator c²+d²	=B3^2+B4^2
Real part of quotient (ac+bd)÷(c²+d²)	=(B1B3+B2B4)/B5
Imaginary part of quotient (bc−ad)÷(c²+d²)	=(B2B3-B1B4)/B5

Google Sheets has the same complex-number functions as Excel (COMPLEX, IMSUM, IMSUB, IMPRODUCT, IMDIV, IMCONJUGATE and IMABS), so the same formulas work as is. Copy the whole table, paste it into cell A1, and replace the real and imaginary parts with your own numbers.

How to calculate it in Python

from fractions import Fraction
import math

# Real and imaginary parts of z1 = a + bi and z2 = c + di (a fraction such as 3/4 can be written Fraction(3, 4))
z1_real, z1_imag = Fraction(1), Fraction(2)
z2_real, z2_imag = Fraction(3), Fraction(4)

# Sum and difference: combine the real parts and the imaginary parts separately
add_real, add_imag = z1_real + z2_real, z1_imag + z2_imag
sub_real, sub_imag = z1_real - z2_real, z1_imag - z2_imag

# Product: the formula (ac-bd) + (ad+bc)i, from expanding and substituting i^2 = -1
mul_real = z1_real * z2_real - z1_imag * z2_imag
mul_imag = z1_real * z2_imag + z1_imag * z2_real

# Quotient: multiply by the conjugate. The denominator becomes the real number c^2 + d^2
denominator = z2_real ** 2 + z2_imag ** 2
div_real = (z1_real * z2_real + z1_imag * z2_imag) / denominator
div_imag = (z1_imag * z2_real - z1_real * z2_imag) / denominator

# Absolute value of the quotient: square root of (real part squared + imaginary part squared)
abs_squared = div_real ** 2 + div_imag ** 2
abs_value = math.sqrt(abs_squared)

# Format as a+bi (when the imaginary part is negative, show it as a subtraction, such as -2 - 2i)
def to_complex_str(real, imag):
    sign = "-" if imag < 0 else "+"
    return f"{real} {sign} {abs(imag)}i"

print(f"Sum: {to_complex_str(add_real, add_imag)}")
print(f"Difference: {to_complex_str(sub_real, sub_imag)}")
print(f"Product: {to_complex_str(mul_real, mul_imag)}")
print(f"Quotient: {to_complex_str(div_real, div_imag)}")
print(f"Absolute value of the quotient, squared (fraction): {abs_squared}")
print(f"Absolute value of the quotient (decimal): {abs_value}")

With the fractions module from the standard library, you can calculate with exact fractions and no decimal rounding errors. This example uses z₁ = 1+2i and z₂ = 3+4i. The quotient prints as "11/25 + 2/25i", and the square of its absolute value as "1/5" (so the absolute value is √5/5 ≈ 0.447). Change the real and imaginary parts and run it. If you do not need exact values, Python also has a built-in complex type, so you can calculate directly with j, as in (1+2j)/(3+4j).

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

Adding and subtracting complex numbers

(a+bi) + (c+di) = (a+c) + (b+d)i

(a+bi) + (c+di) = (a+c) + (b+d)i

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>)</mo>
    <mo>+</mo>
    <mo>(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi><mo>)</mo>
    <mo>=</mo>
    <mo>(</mo><mi>a</mi><mo>+</mo><mi>c</mi><mo>)</mo>
    <mo>+</mo>
    <mo>(</mo><mi>b</mi><mo>+</mo><mi>d</mi><mo>)</mo><mi>i</mi>
  </mrow>
</math>

(a+bi) + (c+di) = (a+c) + (b+d)i

ComplexExpand[(a + b I) + (c + d I)]

evalc((a + b*I) + (c + d*I));

z = (a + b*1i) + (c + d*1i);

(a+bi)+(c+di)=(a+c)+(b+d)i

Multiplying complex numbers and \(i^2 = -1\)

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>)</mo>
    <mo>(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi><mo>)</mo>
    <mo>=</mo>
    <mo>(</mo><mi>a</mi><mi>c</mi><mo>&#x2212;</mo><mi>b</mi><mi>d</mi><mo>)</mo>
    <mo>+</mo>
    <mo>(</mo><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>)</mo><mi>i</mi>
  </mrow>
</math>

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

ComplexExpand[(a + b I) (c + d I)]

evalc((a + b*I) * (c + d*I));

z = (a + b*1i) * (c + d*1i);

(a+bi)(c+di)=(ac-bd)+(ad+bc)i

Dividing complex numbers (multiplying by the conjugate)

(a+bi)/(c+di) = (ac+bd)/(c²+d²) + {(bc−ad)/(c²+d²)}i

\frac{a+bi}{c+di} = \frac{ac+bd}{c^{2}+d^{2}} + \frac{bc-ad}{c^{2}+d^{2}}\,i

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mfrac>
      <mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow>
      <mrow><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi></mrow>
    </mfrac>
    <mo>=</mo>
    <mfrac>
      <mrow><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow>
      <mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow>
    </mfrac>
    <mo>+</mo>
    <mfrac>
      <mrow><mi>b</mi><mi>c</mi><mo>&#x2212;</mo><mi>a</mi><mi>d</mi></mrow>
      <mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow>
    </mfrac>
    <mi>i</mi>
  </mrow>
</math>

(a+bi)/(c+di) = (ac+bd)/(c^2+d^2) + ((bc-ad)/(c^2+d^2))i

ComplexExpand[(a + b I)/(c + d I)]

evalc((a + b*I) / (c + d*I));

z = (a + b*1i) / (c + d*1i);

(a+bi)/(c+di)=(ac+bd)/(c^2+d^2)+((bc-ad)/(c^2+d^2))i

Absolute value (modulus) of a complex number \(|z|\)

|a+bi| = √(a²+b²)

\lvert a+bi \rvert = \sqrt{a^{2}+b^{2}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>|</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</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>

|a+bi| = sqrt(a^2+b^2)

ComplexExpand[Abs[a + b I]]

abs(a + b*I);

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

|a+bi| = √(a^2+b^2)

How to have ChatGPT do the calculation

You are a math calculation assistant for complex numbers. 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).

For the two complex numbers z1 = 1 + 2i and z2 = 3 + 4i, show each of the following:
1. The sum z1 + z2, the difference z1 − z2 and the product z1 × z2 (in a+bi form)
2. The quotient z1 ÷ z2 (in a+bi form with fractions in lowest terms, with the steps of multiplying by the conjugate of the denominator)
3. The complex conjugate and the absolute value of the quotient (the absolute value both as an exact value with the radical simplified and as a decimal)

In Python, use the fractions module from the standard library to calculate the real and imaginary parts exactly, 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