Complex Numbers Calculator

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Example

Created on 2024-06-20Asked by Aiden Roberts (Solvelet student)

Find the modulus and argument of the complex number

z = 3 - 4i

Solution

Step 1: Identify the complex number

z = 3 - 4i

. Step 2: Calculate the modulus of

z

|z| = \sqrt{x^2 + y^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.

Step 3: Calculate the argument of

z

\arg(z) = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-4}{3}\right) = -\tan^{-1}\left(\frac{4}{3}\right).

Step 4: Conclusion. The modulus of

z = 3 - 4i

is 5, and the argument is

-\tan^{-1}\left(\frac{4}{3}\right)

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Luna Wilson on Solvelet

1. Find the argument and modulus of the complex number

z = -3 + 4i

2. Perform the division:

\frac{2 + i}{1 - 3i}

DefinitionSuch tools are used beyond the academic sphere and can be met in physics, mathematics, engineering, and complex analysis. In this regard, the characteristics of electric current are usually traced with the help of complex valued functions, in addition to the behavior of mechanical systems. As real numbers fail to bring to life some of the models, the application of complex numbers appeared to be the only solution to the problem. Taking into account the context, complex exponential function z can be described as f z = ez with ex means is a real function, as the cosine and the sign. A complex number can be portrayed in the way of a + bi, where a and b stand for real numbers and i is the unit of the imaginary. This measurement maintains the equation i = −1. The real value of the imaginary and its multiplication to be is equally 3 and 4.

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