Example Question - complex number division
Here are examples of questions we've helped users solve.
Solving Complex Number Division and Identifying Imaginary Part
Para resolver este problema necesitamos simplificar la expresión compleja y luego identificar la parte imaginaria del resultado. La expresión dada es: \[ z = \frac{3+i}{2-i} \] Para eliminar el número complejo del denominador, multiplicamos tanto el numerador como el denominador por el conjugado del denominador. El conjugado de un número complejo \( a + bi \) es \( a - bi \). Por lo tanto, el conjugado de \( 2 - i \) es \( 2 + i \). Multiplicamos tanto el numerador como el denominador por \( 2 + i \): \[ z = \frac{(3+i)(2+i)}{(2-i)(2+i)} \] Expandimos ambos el numerador y el denominador: \[ z = \frac{6 + 3i + 2i + i^2}{4 + 2i - 2i - i^2} \] Recordamos que \( i^2 = -1 \), entonces simplificamos: \[ z = \frac{6 + 5i - 1}{4 - (-1)} \] \[ z = \frac{5 + 5i}{5} \] Ahora, dividimos tanto la parte real como la parte imaginaria entre 5: \[ z = 1 + i \] La parte imaginaria de \( z \) es el coeficiente de \( i \), que en este caso es 1. Por lo tanto, la parte imaginaria del número complejo \( z \) es 1.
Complex Number Division and Its Parts
The problem asks for the real and imaginary parts of the complex number \(\frac{z_1}{z_2}\), where \(z_1 = x + 4j\) and \(z_2 = x + 2j\). To find \(\frac{z_1}{z_2}\), you can divide \(z_1\) by \(z_2\). Here's how you do it: \[ \frac{z_1}{z_2} = \frac{x + 4j}{x + 2j} \] To divide two complex numbers, you can multiply the numerator and the denominator by the conjugate of the denominator: \[ \frac{x + 4j}{x + 2j} \cdot \frac{x - 2j}{x - 2j} = \frac{(x + 4j)(x - 2j)}{x^2 - (2j)^2} \] Computing the products in the numerator and simplifying the denominator (using \(j^2 = -1\)): \[ \frac{x^2 - 2jx + 4xj - 8j^2}{x^2 - 4j^2} = \frac{x^2 + 2xj - 8(-1)}{x^2 - 4(-1)} \] Simplify further: \[ \frac{x^2 + 2xj + 8}{x^2 + 4} \] Now you have the complex number in fractional form. Let's separate it into real and imaginary parts: The real part is: \[ \frac{x^2 + 8}{x^2 + 4} \] The imaginary part is: \[ \frac{2xj}{x^2 + 4} \] Hence, the real part of \(\frac{z_1}{z_2}\) is \(\frac{x^2 + 8}{x^2 + 4}\) and the imaginary part is \(\frac{2x}{x^2 + 4}j\).
Solving Complex Number Division
Certainly! To find \( \frac{z_1}{z_2} \) given that \( z_1 = \cos \theta + j \sin \theta \) and \( z_2 = \cos \theta - j \sin \theta \), we can directly divide the complex numbers as follows: \[ \frac{z_1}{z_2} = \frac{\cos \theta + j \sin \theta}{\cos \theta - j \sin \theta} \] To divide these two complex numbers, we can multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary parts from the denominator. The conjugate of \( z_2 \) is \( \cos \theta + j \sin \theta \), which is actually the same as \( z_1 \). So, we multiply both top and bottom by this conjugate: \[ \frac{z_1}{z_2} = \frac{(\cos \theta + j \sin \theta)(\cos \theta + j \sin \theta)}{(\cos \theta - j \sin \theta)(\cos \theta + j \sin \theta)} \] Multiplying out the numerator, we get: Numerator: \( (\cos^2 \theta + 2j \cos \theta \sin \theta - \sin^2 \theta) \) (using \( j^2 = -1 \)) Multiplying out the denominator, we get: Denominator: \( (\cos^2 \theta - (\sin^2 \theta)) \) Simplify the denominator using \( \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \): Denominator: \( \cos(2\theta) \) However, the denominator simplifies even further because \( \cos^2 \theta - \sin^2 \theta = 1 \) (from the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \)). Therefore, the denominator actually is: Denominator: \( 1 \) Returning to the numerator, we use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) again: Numerator: \( 1 + 2j \cos \theta \sin \theta \) The numerator simplifies using the double angle identity for sine: \( 2\sin \theta \cos \theta = \sin(2\theta) \): Numerator: \( 1 + j \sin(2\theta) \) Putting it all together, we have: \[ \frac{z_1}{z_2} = 1 + j \sin(2\theta) \] This is the final simplified form of \( \frac{z_1}{z_2} \).
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com