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Can someone please show me how to express these two quantities in polar and Cartesian form?

$$ \lvert z_1z_2\rvert$$ and $$ arg(z_1z_2)$$

I understand for the Cartesian form of the first, I just get $$(x^2+y^2)^{1/2} + (x^2+y^2)^{1/2} $$ But then how can I do the same for the polar form?

Many thanks

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    It depends on which form the complex number was originally written in. – 2017-01-31
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    Good job on the mathiax - Keep it up! – 2017-01-31
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    This is the only information given in the question so I assumed the answer would take a generic form. Thanks everyone, I understand now. – 2017-01-31

3 Answers 3

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It's straightforward to remember what the polar form of a complex number is. To this end, just know that

$$z = x+iy$$

$$|z| = \sqrt{x^2 + y^2}$$

$$\theta = \text{arg}(z) = \arctan\frac{y}{x}$$

If you have two complex numbers, then...

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    Careful with the arctan ofc. – 2017-01-31
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Let $z_1=r_1e^{i\theta_1}$ and likewise for $z_2$. We then know that

$$|z_1z_2|=|z_1|\cdot|z_2|=r_1r_2$$

$$\arg(z_1z_2)=\arg(z_1)+\arg(z_2)=\theta_1+\theta_2$$

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For $z_1=x_1+iy_1=r_1e^{i\theta}$ and $z_2=x_2+iy_2=r_2e^{i\phi}$ then $$|z_1z_2|=\sqrt{x_1^2+y_1^2}\sqrt{x_2^2+y_2^2}=r_1r_2$$ and $$\arg(z_1z_2)=\arctan\frac{x_1y_2+x_2y_1}{x_1x_2-y_1y_2}=\theta+\phi$$