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. Show Show rules of syntax Equation Solver ExamplesPlease let us know if you have any suggestions on how to make Equation Solver better. Math tools for your website Choose language: Deutsch English Español Français Italiano Nederlands Polski Português Русский 中文 日本語 한국어 Number Empire - powerful math tools for everyone | Contact webmaster By using this website, you signify your acceptance of Terms and Conditions and Privacy Policy.© 2022 numberempire.com All rights reserved 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°). 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 numbersWe 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): (1+i) + (6-5i) = 7-4i Subtraction Again very simple, subtract the real parts and subtract the imaginary parts (with i): (1+i) - (3-5i) = -2+6i Multiplication To multiply two complex numbers, use distributive law, avoid binomials, and apply i2 = -1. (1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i DivisionThe 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 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 Square rootSquare 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 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'... ii=e−π/2 i^2 = -1 FunctionssqrtSquare 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:
more math problems » How do you know how many solutions an equation has?If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one solution. If solving an equation yields a statement that is always true, like 3 = 3, then the equation has infinitely many solutions.
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