Complex Number Calculator: -3 / (2-i) ~ World News

z1 = + i z2 = + i. z1.|z-i|=|z+i| |z+i|=|z+1| 1<=|z-3|<=3 3+6i -5-1i 5+8i -5+3i 5+1i -1i 1+3^(1/2)i 3+3^(1/2)i 7+3^(1/2)i 3^(1/2)+. 3^(1/2)i 2^(1/2)+2^(1/2)i 5^(1/2)+5^(1/2)i |z|<=2 |z+2|>=4 |z+3i|<=2 |z+3-2i|<=4.Select rating 1 2 3 4 5.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.

Изображение комплексного числа на плоскости онлайн

Example 12 Find the conjugate of ((3 − 2i)(2 + 3i))/((1 + 2i)(2 − i) ) First we calculate ((3 − 2i)(2 + 3i))/((1 + 2i)(2 − i) ) then find its conjugate ((3 −2i)(2+3i))/((1+ 2i)(.In the process, distribute 1's numerator (3 -i2)(2 -i3) among the three major terms, and divide the resulting distribution by 1's denominator, again (3 -2i)(2 -i3). Obviously, (3 + 2i) is further on the real axis, and (2 + 3i) is further on the imaginary one.Решение #2. Вычислите: a) i( 1 + i); б) i(-3 + 2i); в) (4 - 3i)i; г) i(4 - 3i)i(4 + 3i).

Решение комплексных чисел | Онлайн калькулятор

Algebra. Simplify (3-2i)(3+2i).z - |z| = 2 + i.Найдем корень 12 + 16i подставив значение x и z в r1 and r2. r1 = x + zi = 4 + 2i r2 = -x — zi = -4 — 2i.

Rectangular shape:z = -1.2-0.6i(*3*)Angle notation (phasor): z = 1.3416408 ∠ -153°26'6″

(*3*)Polar shape:z = 1.3416408 × (cos (-153°26'6″) + i sin (-153°26'6″))

(*3*)Exponential form: z = 1.3416408 × ei (-0.8524164)

(*3*)Polar coordinates: r = |z| = 1.3416408 ... magnitude (modulus, absolute worth)θ = arg z = -2.677945 rad = -153.43495° = -153°26'6″ = -0.8524164π rad ... perspective (argument or phase)

(*3*)Cartesian coordinates: Cartesian form of imaginary number: z = -1.2-0.6iReal phase: x = Re z = -1.2Imaginary phase: y = Im z = -0.6

(*3*)

Calculation steps Complex number: 2-iDivide: -3 / the result of step No. 1 = -3 / (2-i) = -3/2-i = (-3)*(2+i)/(2-i)*(2+i) = i )/(2 * 2 + 2 * i + (-i) * 2 + (-i) * i )">-3 * 2 + (-3) * i /2 * 2 + 2 * i + (-i) * 2 + (-i) * i = i )/( 4+2i-2i-i2 )"> -6-3i / 4+2i-2i-i2 = i )/( 4+2i-2i+1 )"> -6-3i / 4+2i-2i+1 = i(-3))/( 4 + 1 +i(2 - 2))"> -6 +i(-3)/ 4 + 1 +i(2 - 2) = -6-3i/5 = -1.2-0.6i (*3*)To divide complicated numbers, you should multiply both (numerator and denominator) through the conjugate of the denominator. To in finding the conjugate of a fancy quantity, you change the sign in imaginary part. Distribute in both the numerator and denominator to take away the parenthesis and upload and simplify. Use rule .

3 × ei : 2.236068 × ei (-0.1475836) = (3 / 2.236068) × ei (-(-0.1475836)) = 1.3416408 × ei (-0.8524164) = -1.2-0.6i This calculator does elementary arithmetic on complicated numbers and evaluates expressions within the set of advanced numbers. As imaginary unit use i or j (in electric engineering), which satisfies fundamental equation i2 = −1 or j2 = −1. The calculator also converts a complex quantity into perspective notation (phasor notation), exponential, or polar coordinates (magnitude and perspective). Enter expression with complicated numbers like 5*(1+i)(-2-5i)^2(*3*)Complex numbers within the perspective notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is the attitude (phase) in degrees, for instance, 5L65 which is equal to 5*cis(65°). Example of multiplication of 2 imaginary numbers in the perspective/polar/phasor notation: 10L45 * 3L90.

(*3*)Why the following advanced numbers calculator when we have WolframAlpha? Because Wolfram tool is slow and a few options corresponding to step-by-step are charged premium carrier. For use in education (for example, calculations of alternating currents at high school), you wish to have a quick and actual complicated number calculator.

Basic operations with complex numbers

We hope that paintings with the advanced quantity is slightly easy because you can paintings with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the similar as operations with two-dimensional vectors. Addition Very easy, add up the true portions (with out i) and add up the imaginary portions (with i):This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i (*3*)(1+i) + (6-5i) = 7-4i12 + 6-5i = 18-5i(10-5i) + (-5+5i) = 5

Subtraction Again quite simple, subtract the actual portions and subtract the imaginary parts (with i):This is the same as use rule: (a+bi)+(c+di) = (a-c) + (b-d)i (*3*)(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 regulation, avoid binomials, and apply i2 = -1.This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i (*3*)(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 also be completed by way of multiplying the numerator and denominator by way of the denominator's advanced conjugate. This avoids imaginary unit i from the denominator. If the denominator is c+di, to make it with out i (or make it real), multiply with conjugate c-di:(*3*)(c+di)(c-di) = c2+d2

(*3*)(10-5i) / (1+i) = 2.5-7.5i-3 / (2-i) = -1.2-0.6i6i / (4+3i) = 0.72+0.96i

Absolute price or modulus The absolute worth or modulus is the gap of the image of a fancy quantity from the origin in the plane. The calculator uses the Pythagorean theorem to search out this distance. Very simple, see examples: |3+4i| = 5|1-i| = 1.4142136|6i| = 6abs(2+5i) = 5.3851648Square root Square root of complicated number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you'll at all times have two other sq. roots for a given quantity. If you wish to have to determine the imaginable values, one of the simplest ways is most definitely to go with De Moivre's formulation. Here our calculator is on edge, as a result of sq. root isn't a well defined serve as on complex number. We calculate all advanced roots from any number - even in expressions: (*3*)sqrt(9i) = 2.1213203+2.1213203isqrt(10-6i) = 3.2910412-0.9115656ipow(-32,1/5)/5 = -0.4pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225ipow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303i

Square, power, complicated exponentiation Our calculator can energy any complicated quantity to any integer (certain, adverse), actual, and even complicated number. In different phrases, we calculate 'complex number to a fancy power' or 'advanced number raised to an influence'... Famous instance:(*3*)ii=e−π/2ii=e−π/2

i^2 = -1i^61 = i(6-2i)^6 = -22528-59904i(6-i)^4.5 = 2486.1377428-2284.5557378i(6-5i)^(-3+32i) = 2929449.03994-9022199.58262ii^i = 0.2078795764pow(1+i,3) = -2+2iFunctions sqrtSquare Root of a value or expression. sinthe sine of a value or expression. Autodetect radians/degrees. costhe cosine of a price or expression. Autodetect radians/degrees. tantangent of a worth or expression. Autodetect radians/levels. expe (the Euler Constant) raised to the facility of a worth or expression powPower one advanced quantity to another integer/real/complicated number lnThe herbal 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 (perspective) of a posh number cisis less recognized notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i conjconjugate of complex quantity - example: conj(4i+5) = 5-4i Examples: • cube root: cuberoot(1-27i)• roots of Complex Numbers: pow(1+i,1/7)• phase, complex quantity attitude: phase(1+i)• cis form complicated 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 fabricated from 3-4i and its conjugate.: (3-4i)*conj(3-4i)• operations with advanced numbers: (3-i)^3

Complex numbers in phrase problems:

Reciprocal Calculate reciprocal of z=0.8-1.8i:Let z1=x1+y1i Let z1=x1+y1i and z2=x2+y2i Find: a = Im (z1z2) b = Re (z1/z2)(*3*)subsequent math problems »