In Problems 35-42, find z w and (z)/(w). Leave your answers in polar form. z=1-i w=1-

Step 1: Convert u005C( z u005C) and u005C( w u005C) to polar form.u000Au002D For u005C( z u003D 1 u002D i u005C), calculate the modulus

In Problems 35-42, find z w and (z)/(w). Leave your answers...

Key Concepts

Complex Numbers and Representations

Complex numbers can be expressed both in rectangular form (a + bi) and in polar form (r(cos? + i sin?) or re^(i?)). This duality is fundamental because while the rectangular form is often more intuitive for addition and subtraction, the polar form greatly simplifies multiplication, division, and the computation of powers and roots.

Conversion from Rectangular to Polar Form

Converting a complex number from its rectangular representation (a + bi) to its polar form involves calculating its modulus (r = ?(a² + b²)) and its argument (? = arctan(b/a)). This conversion is crucial because it allows one to represent and operate on complex numbers in a way that leverages trigonometric identities and properties.

Multiplication in Polar Form

When multiplying two complex numbers in polar form, you simply multiply their moduli and add their arguments. This method contrasts with the multiplication of complex numbers in rectangular form and is much more straightforward when the numbers are already expressed in terms of magnitude and angle.

Division in Polar Form

Dividing complex numbers in polar form is analogous to multiplication: you divide their moduli and subtract the arguments. This property makes division of complex numbers much simpler when they are expressed in polar form compared to when they are in rectangular form.

In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form. $$ \begin{aligned} & z=1-i \\ & w=1-\sqrt{3} i \end{aligned} $$

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