Example Question - real and imaginary parts
Here are examples of questions we've helped users solve.
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 a Complex Equation with Imaginary Numbers
The image displays a mathematical problem: \[ \text{Find the value of a and b if } bi^{2} + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}. \] To solve this, we need to realize that \(i^2 = -1\) and group real and imaginary parts separately. \[ bi^{2} + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}, \] \[ b(-1) + \frac{4}{3}i - \sqrt{3} = a + bi\sqrt{3}. \] Now, group real parts together and imaginary parts together: Real parts: \(-b - \sqrt{3} = a,\) Imaginary parts: \(\frac{4}{3}i = bi\sqrt{3}.\) Now let's isolate \(b\) from the imaginary part: \[ b = \frac{\frac{4}{3}}{\sqrt{3}} = \frac{4}{3\sqrt{3}} = \frac{4}{3}\cdot\frac{\sqrt{3}}{3} = \frac{4\sqrt{3}}{9}.\] Then substitute \(b\) into the real part to find \(a\): \[ -\left(\frac{4\sqrt{3}}{9}\right) - \sqrt{3} = a,\] \[ a = -\left(\frac{4\sqrt{3}}{9} + \frac{9\sqrt{3}}{9}\right), \] \[ a = -\left(\frac{4\sqrt{3} + 9\sqrt{3}}{9}\right), \] \[ a = -\left(\frac{13\sqrt{3}}{9}\right). \] Thus, the values for \(a\) and \(b\) are: \[ a = -\frac{13\sqrt{3}}{9}, \] \[ b = \frac{4\sqrt{3}}{9}. \]
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