Complex number calculator - calculation: 2i (2024)

Rectangular form (standard form):
z = 2i

Angle notation (phasor):
z = 2 ∠ 90°

Polar form:
z = 2 × (cos 90° + i sin 90°)

Exponential form:
z = 2 × e^{i 1.5707963} = 2 × e^{i π/2}

Polar coordinates:
r = |z| = 2 ... magnitude (modulus, absolute value)
θ = arg z = 1.5707963 rad = 90° = 0.5π = π/2 rad ... angle (argument or phase)

Cartesian coordinates:
Cartesian form of imaginary number: z = 2i
Real part: x = Re z = 0
Imaginary part: y = Im z = 2

Basic operations with complex numbers

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

Addition

It is 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 it is 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 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 real), 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

The square root of a 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 to use De Moivre's formula.Our calculator is on edge because the square root is not a well-defined function on a complex number. 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

Square, power, complex exponentiation

Our calculator can power any complex number to an 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:

ii=e−π/2

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

Examples:

• cube root: cuberoot(1 - 27i)
• roots of Complex Numbers: pow(1 + i,1/7)
• phase, complex number angle: phase(1 + i)
• cis form complex numbers: 5 * cis(45°)
• The polar form of complex numbers: 10L60
• complex conjugate calculator: conj(4 + 5i)
• equation with complex numbers: (z + i/2 )/(1 - i) = 4z + 5i
• system of equations with imaginary numbers: x - y = 4 + 6i; 3ix + 7y=x + iy
• De Moivre's theorem - equation: z ^ 4=1
• multiplication of three complex numbers: (1 + 3i)(3 + 4i)(−5 + 3i)
• Find the product of 3-4i and its conjugate.: (3 - 4i) * conj(3 - 4i)
• operations with complex numbers: (3 - i) ^ 3

Complex numbers in word problems:

Mistake in expression While attempting to multiply the expression (2 - 5i)(5 + 2i), a student made a mistake. (2 - 5i)(5 + 2i) = 10 + 4i - 25i - 10i2 = 10 + 4(-1) - 25(-1) - 10(1) = 10 - 4 + 25 - 10 = 21 Complete the explanation and correct the error. Hint: The student incorre
Quadratic 21643 Solve the quadratic equation: 2y²-8y + 12 = 0
Equation: 3726 Determine the real root of the equation: x^-3: x^-8 = 32
Subtracting complex in polar Given w =√2(cosine (pi/4) + i sine (pi/4) ) and z = 2 (cosine (pi/2) + i sine (pi/2) ). What is w - z expressed in polar form?

Complex expr with fractions Find 1½ of 16 ÷2⅓+(2¼ of ⅑). Use the correct order of operations. The order is PEMDAS: Parentheses, Exponents, Multiplication, Division (from left to right), addition, and Subtraction (from left to right).
Stadium A domed stadium is shaped like a spherical segment with a base radius of 150 m. The dome must contain a volume of 3500000 m³. Determine the dome's height at its center to the nearest tenth of a meter.
Conjugate equation Find the numbers of a and b; if (a - bi) (3 + 5i) is the Conjugate of (-6 - 24i)