Exercise · ZW-COMPLEX-MODULUS-010 · ★★★ Advanced
Which of these equalities holds for all complex \(z,w\)?
Choose one option.
Show answer
- C. \(|zw|=|z|\,|w|\)
Explanation
Only the multiplicative one always holds. The sum satisfies \(|z+w|\le|z|+|w|\); “\(|z|=|\operatorname{Re}z|+|\operatorname{Im}z|\)” fails for \(z=1+i\) (\(\sqrt2\ne2\)); and \(z=i\) gives \(|z|^2=1\ne-1=z^2\).
Review the concept: Modulus of a complex number
The modulus \(|z|\) of a complex number \(z=a+bi\) is the distance from the point \((a,b)\) to the origin of the complex plane.
\[|z|=\sqrt{a^2+b^2}=\sqrt{z\bar z}\]
More practice problems: modulus of a complex number
- ZW-COMPLEX-MODULUS-001 Compute |3+4i|.
- ZW-COMPLEX-MODULUS-002 What is |-5+12i|?
- ZW-COMPLEX-MODULUS-003 Compute |1-i|.
- ZW-COMPLEX-MODULUS-004 For a complex z, what does zbar z equal?
- ZW-COMPLEX-MODULUS-005 Which set does |z-(1+i)|=2 describe?
- ZW-COMPLEX-MODULUS-006 Compute |(1+i)(3-4i)|.
- ZW-COMPLEX-MODULUS-007 Compute |(6+8i)/(3-4i)|.
- ZW-COMPLEX-MODULUS-008 For which real t does |t+3i|=5 hold?