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Snowfall
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When is it true that [itex]\arg\left(z_1 z_2\right) = \arg(z_1) + \arg(z_2) [/itex]?
Snowfall said:When is it true that [itex]\arg\left(z_1 z_2\right) = \arg(z_1) + \arg(z_2) [/itex]?
You mean like defining, for real numbers a, b, c, d,chiro said:Why don't you just write out the equations for argument and solve them?
Snowfall said:You mean like defining, for real numbers a, b, c, d,
[tex]z_{1} := ai+b, ~ z_{2} := ci+d[/tex]
then solving the following daunting looking equation:
[tex]\tan^{-1}\left(\frac{ad+bc}{bd-ac}\right) = \tan^{-1}\frac{b}{a}+\tan^{-1}\frac{d}{c}.[/tex]
Please by all means show me how to solve that one!
The argument of a complex number is the angle formed between the positive real axis and the line joining the origin to the complex number on the complex plane. It is also referred to as the phase angle or the polar angle.
The argument of a complex number can be calculated using the inverse tangent function (arctan) of the imaginary part divided by the real part of the complex number. This can also be represented using the formula arg(z) = tan-1(b/a), where z is the complex number a + bi.
The argument of a complex number lies between -π (exclusive) and π (inclusive). However, it is often represented as an angle between 0 and 2π in the polar coordinate system.
The argument of a complex number and its modulus (or absolute value) are related through the formula z = |z|cosθ + i|z|sinθ, where θ is the argument of the complex number. This formula is known as the polar form of a complex number.
The argument of a complex number is important in understanding the behavior and properties of complex numbers. It helps in visualizing complex numbers on the complex plane, finding roots of complex numbers, and solving equations involving complex numbers.