Square Root of Complex Number in Cartesian Form/Examples/5-12i
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Example of Square Root of Complex Number in Cartesian Form
- $\sqrt {5 - 12 i} = \pm \paren {3 - 2 i}$
Proof
\(\ds \paren {x + i y}^2\) | \(=\) | \(\ds 5 - 12 i\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2\) | \(=\) | \(\ds \dfrac {5 + \sqrt {5^2 + \paren {-12}^2} } 2\) | Square Root of Complex Number in Cartesian Form | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 + \sqrt {169} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {5 + 13} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \pm 3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \pm \dfrac {-12} {2 \times 3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \mp 2\) |
As $2 x y = -12$ it follows that the two solutions are:
- $3 - 2 i$
- $-3 + 2 i$
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Example
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $98 \ \text{(a)}$