Exercise · ZW-POLAR-FORM-010 · ★★★ Advanced
Multiplying by \(1+i\) is geometrically equivalent to:
Choose one option.
Show answer
- D. Rotating by \(\pi/4\) and scaling by \(\sqrt2\)
Explanation
\(1+i=\sqrt2\left(\cos\frac\pi4+i\sin\frac\pi4\right)\): when multiplying, moduli multiply (factor \(\sqrt2\)) and arguments add (rotation by \(\pi/4\)).
Review the concept: Polar form
The polar form of a complex number \(z\neq0\) expresses it with its modulus \(r\) and an argument \(\theta\), an angle that \(z\) makes with the positive real axis.
\[z=r(\cos\theta+i\sin\theta),\qquad r=|z|\]
More practice problems: polar form
- ZW-POLAR-FORM-001 What is the polar form of 1+i?
- ZW-POLAR-FORM-002 Write -1 in polar form with θ∈(-π,π].
- ZW-POLAR-FORM-003 What is the principal argument of -√3+i?
- ZW-POLAR-FORM-004 Write 2(cos π3+isin π3) in the form a+bi.
- ZW-POLAR-FORM-005 Compute 2(cos π6+isin π6)·3(cos π3+isin π3).
- ZW-POLAR-FORM-006 Let z=4(cos (5π)/(6)+isin (5π)/(6)) and w=2(cos π3+isin π3). Compute z/w.
- ZW-POLAR-FORM-007 What is the principal argument, in (-π,π], of -2-2i?
- ZW-POLAR-FORM-008 A complex number has argument π/4. Which of these is also an argument of it?