Product Of Complex Numbers Calculator

Circuitous number calculator

This calculator does basic arithmetics on complex numbers and evaluates expressions in the set of complex numbers. Equally an imaginary unit, use i or j (in electrical engineering), which satisfies the bones equation i^two = −one or jⁱⁱ = −1. The figurer also converts a complex number into angle note (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers similar v*(1+i)(-2-5i)^two

Complex numbers in the bending notation or phasor (polar coordinates r, θ) may y'all write equally rLθ where r is magnitude/amplitude/radius, and θ is the angle (stage) in degrees, for example, 5L65 which is the aforementioned every bit v*cis(65°).
Instance of multiplication of two imaginary numbers in the angle/polar/phasor note: 10L45 * 3L90.

For use in education (for example, calculations of alternating currents at high school), you need a quick and precise complex number calculator.

Basic operations with complex numbers

We hope that working with the complex number is quite piece of cake because you lot tin can work with imaginary unit i every bit a variable. And use definition i^two = -1 to simplify circuitous expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

Very unproblematic, add up the existent parts (without i) and add up the imaginary parts (with i):
This is equal to utilise dominion: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = seven-ivi
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

Subtraction

Once again very simple, subtract the real parts and subtract the imaginary parts (with i):
This is equal to utilize rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+half dozeni
-1/2 - (half dozen-5i) = -half-dozen.5+vi
(10-5i) - (-5+5i) = fifteen-10i

Multiplication

To multiply two complex numbers, use distributive law, avoid binomials, and utilize iⁱⁱ = -1.
This is equal to utilize rule: (a+bi)(c+di) = (air-conditioning-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+i*5i+i*3+i*5i = iii+5i+3i-5 = -2+viiii
-i/2 * (vi-5i) = -iii+2.5i
(10-5i) * (-five+5i) = -25+75i

Sectionalisation

The division of ii complex numbers can be achieved by multiplying the numerator and denominator by the denominator's complex conjugate. This approach avoids imaginary unit i from the denominator. If the denominator is c+di, to make it without i (or make it existent), multiply with conjugate c-di:

(c+di)(c-di) = c²+d²

(10-5i) / (one+i) = ii.5-vii.5i
-3 / (2-i) = -i.2-0.half dozeni
6i / (4+3i) = 0.72+0.96i

Accented value or modulus

The accented value or modulus is the altitude of the prototype of a circuitous number from the origin in the plane. The figurer uses the Pythagorean theorem to find this distance. Very uncomplicated, see examples: |3+4i| = 5
|1-i| = 1.4142136
|6i| = 6
abs(two+5i) = v.3851648

Square root

Square root of complex number (a+bi) is z, if z^two = (a+bi). Hither ends simplicity. Because of the cardinal theorem of algebra, you lot volition ever have ii different foursquare roots for a given number. If you desire to discover out the possible values, the easiest fashion is to get with De Moivre's formula. Our calculator is on border because the square root is not a well-divers role on a circuitous number. Nosotros calculate all complex roots from whatsoever number - fifty-fifty in expressions:

sqrt(9i) = 2.1213203+2.1213203i
sqrt(ten-6i) = 3.2910412-0.9115656i
pw(-32,i/five)/v = -0.4
pow(1+2i,1/3)*sqrt(four) = two.439233+0.9434225i
prisoner of war(-5i,one/viii)*pow(8,1/3) = 2.3986959-0.4771303i

Square, power, circuitous exponentiation

Our calculator can power any complex number to an integer (positive, negative), real, or fifty-fifty complex number. In other words, we summate 'circuitous number to a complex power' or 'complex number raised to a power'...
Famous example:

$i^{i} = e^{- π / two}$

i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.1377428-2284.5557378i
(6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224i
i^i = 0.2078795764
pw(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: tangent of a value or expression. Autodetect radians/degrees.
exp: due east (the Euler Constant) raised to the power of a value or expression
pow: Ability ane complex number to another integer/real/complex number
ln: The natural logarithm of a value or expression
log: The base-x logarithm of a value or expression
abs or |ane+i|: The absolute value of a value or expression
stage: Stage (angle) of a complex number
cis: is less known note: cis(x) = cos(10)+ i sin(x); instance: cis (pi/ii) + three = iii+i
conj: cohabit of complex number - instance: conj(4i+5) = 5-4i

Complex numbers in word problems:

Circuitous number coordinates
Which coordinates bear witness the location of -2+3i
ReIm notation
Let z = vi + 5i and due west = 3 - i. Compute the following and express your answer in a + bi course. w + 3z
Evaluate 18
Evaluate the expression (-4-7i)-(-vi-9i) and write the result in the form a+bi (Existent + i* Imaginary).
Complex cohabit
What is the conjugate of the expression 5√vi + 6√five i? A.) -5√half dozen + half dozen√5 i B.) 5√6 - 6√v i C.) -5√6 - 6√5 i D.) 6√five - v√6i
Is complex
Are these numbers 2i, 4i, 2i + 1, 8i, 2i + three, 4 + 7i, 8i, 8i + iv, 5i, 6i, 3i circuitous?
De Moivre's formula
There are two singled-out complex numbers, such that z³ is equal to 1 and z is not equal to one. Calculate the sum of these 2 numbers.
Mappings of complex numbers
Observe the images of the following points under mappings: z=3-2j w=2zj+j-1
Subtracting complex in polar
Given w =√2(cosine (p/iv) + i sine (pi/iv) ) and z = ii (cosine (pi/2) + i sine (pi/2) ), what is w - z expressed in polar form?
Circuitous plane mapping
Show that the mapping w = z +c/z, where z = x+iy, westward = u+four and c is a existent number, maps the circle |z| = 1 in the z-plane into an ellipse in the (u, v) airplane.
Reciprocal
Summate the reciprocal of z=0.8-i.8i:
Im>0?
Is -10i a positive number?