Basic Operations with Complex Numbers

 We hope that work with the complex number is quite easy because you can work with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

Very simple, add up the real parts (without i) and add up the imaginary parts (with i):

This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i


(1+i) + (6-5i) = 7-4i

12 + 6-5i = 18-5i

(10-5i) + (-5+5i) = 5


Subtraction

Again very simple, subtract the real parts and subtract the imaginary parts (with i):

This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i


(1+i) - (3-5i) = -2+6i

-1/2 - (6-5i) = -6.5+5i

(10-5i) - (-5+5i) = 15-10i

Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and apply i2 = -1.

This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i


(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i

-1/2 * (6-5i) = -3+2.5i

(10-5i) * (-5+5i) = -25+75i

Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator. This avoid imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it real), just multiply with conjugate c-di:


(c+di)(c-di) = c2+d2



(10-5i) / (1+i) = 2.5-7.5i

-3 / (2-i) = -1.2-0.6i

6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

The absolute value or modulus is the distance of the image of a complex number from the origin in the plane. The calculator uses the Pythagorean theorem to find this distance. Very simple, see examples: |3+4i| = 5

|1-i| = 1.4142136

|6i| = 6

abs(2+5i) = 5.3851648

Square root

Square root of the complex number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula. Here our calculator is on edge, because square root is not a well defined function on complex numbers. We calculate all complex roots from any number - even in expressions:


sqrt(9i) = 2.1213203+2.1213203i

sqrt(10-6i) = 3.2910412-0.9115656i

pow(-32,1/5)/5 = -0.4

pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225i

pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i



Calculate Using Online Calculator: https://www.allmath.com/complex-number.php

Square, power, complex exponentiation

Our calculator can power any complex number to any integer (positive, negative), real, or even complex number. In other words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...


Famous example:

 

i^2 = -1

i^61 = i

(6-2i)^6 = -22528-59904i

(6-i)^4.5 = 2486.1377428-2284.5557378i

(6-5i)^(-3+32i) = 2929449.03994-9022199.58262i

i^i = 0.2078795764

pow(1+i,3) = -2+2i

Functions

sqrt

Square Root of a value or expression.

sin

the sine of a value or expression. Autodetect radians/degrees.

cos

the cosine of a value or expression. Autodetect radians/degrees.

tan/tg

tangent of a value or expression. Autodetect radians/degrees.

exp

e (the Euler Constant) raised to the power of a value or expression

pow

Power one complex number to another integer/real/complex number

ln

The natural logarithm of a value or expression

log

The base-10 logarithm of a value or expression

abs or |1+i|

The absolute value of a value or expression

phase

Phase (angle) of a complex number

cis

is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i

conj

conjugate of complex number - example: conj(4i+5) = 5-4i

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