Equation Solver solves a system of equations with respect to a given set of variables. It supports polynomial equations as well as some equations with exponents, logarithms and trigonometric functions. Equation solver can find both numerical and parametric solutions of equations. The final result of solving the equation is simplified so it could be in a different form than what you expect. Both equations with complex solutions and complex equations are supported. Functions of complex variable are supported.
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This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As an imaginary unit, use i or j (in electrical engineering), which satisfies the basic equation i2 = −1 or j2 = −1. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2
Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the angle (phase) in
degrees, for example, 5L65 which is the same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 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 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 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
c+dia+bi=(c+di)(c−di)(a+bi)(c−di)=c2+d2ac+bd+i(bc−ad)=c2+d2ac+bd+c2+d2bc−adi
(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 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 go with 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
sqrtSquare Root of a value or expression.sinthe sine of a value or expression. Autodetect radians/degrees. costhe cosine of a value or expression. Autodetect radians/degrees. tantangent of a value or expression. Autodetect radians/degrees. expe (the Euler Constant) raised to the power of a value or expression powPower one complex number to another integer/real/complex number ln The natural logarithm of a value or expression logThe base-10 logarithm of a value or expression abs or |1+i|The absolute value of a value or expression phasePhase (angle) of a complex number cisis less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+iconjconjugate of complex number - example: conj(4i+5) = 5-4iComplex numbers in word problems:
- Complex
conjugate What is the conjugate of the expression 5√6 + 6√5 i? A.) -5√6 + 6√5 i B.) 5√6 - 6√5 i C.) -5√6 - 6√5 i D.) 6√5 - 5√6i
- Complex number coordinates Which coordinates show the location of -2+3i
- Evaluate 18 Evaluate the expression (-4-7i)-(-6-9i) and write the result in the form a+bi (Real + i* Imaginary).
- ReIm notation Let z = 6 + 5i and w = 3 - i. Compute the following and express your answer in a + bi form. w + 3z
- Is complex Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex?
- De Moivre's formula
There are two distinct complex numbers, such that z³ is equal to 1 and z is not equal to 1. Calculate the sum of these two numbers.
- Im>0? Is -10i a positive number?
- Complex plane mapping Show that the mapping w = z +c/z, where z = x+iy, w = u+iv and c is a real number, maps the circle |z| = 1 in the z-plane into an ellipse in the (u, v) plane.
- 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
- Reciprocal
Calculate the reciprocal of z=0.8-1.8i:
- Linear imaginary equation Given that 2(z+i)=i(z+i) "this is z star" Find the value of the complex number z.
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