Exercise · ZW-DE-MOIVRE-FORMULA-010 · ★★★ Advanced
What is the smallest positive integer \(n\) such that \((1+i)^n\) is a positive real number?
Choose one option.
Show answer
- C. \(8\)
Explanation
\((1+i)^n=2^{n/2}(\cos\frac{n\pi}4+i\sin\frac{n\pi}4)\) is positive real when \(\frac{n\pi}4\) is a multiple of \(2\pi\), i.e. \(n\) a multiple of 8. With \(n=4\) we get \(-4\), real but negative.
Review the concept: De Moivre's formula
De Moivre's formula gives the integer power \(n\) of a complex number \(z\) of modulus \(r\) and argument \(\theta\): it raises the modulus to the \(n\)-th power and multiplies the argument by \(n\).
\[z^n=r^n\left(\cos n\theta+i\sin n\theta\right)\]
More practice problems: de moivre's formula
- ZW-DE-MOIVRE-FORMULA-001 Compute (1+i)^8.
- ZW-DE-MOIVRE-FORMULA-002 Compute (cos π6+isin π6)^6.
- ZW-DE-MOIVRE-FORMULA-003 Use the formula to find (1-i)^4.
- ZW-DE-MOIVRE-FORMULA-004 What is the modulus of (√3+i)^5?
- ZW-DE-MOIVRE-FORMULA-005 Compute (1+i√3)^3.
- ZW-DE-MOIVRE-FORMULA-006 For n=-1, (cos θ+isin θ)^-1 equals:
- ZW-DE-MOIVRE-FORMULA-007 Expanding (cos θ+isin θ)^3 and equating real parts, what expression results for cos 3θ?
- ZW-DE-MOIVRE-FORMULA-008 Compute (√3-i)^6.