Square Root of Complex Number in Cartesian Form
Theorem
Let $z \in \C$ be a complex number.
Let $z = x + i y$ where $x, y \in \R$ are real numbers.
Let $z$ not be wholly real, that is, such that $y \ne 0$.
Then the square root of $z$ is given by:
- $z^{1/2} = \pm \paren {a + i b}$
where:
\(\ds a\) | \(=\) | \(\ds \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}\) | ||||||||||||
\(\ds b\) | \(=\) | \(\ds \frac y {\cmod y} \sqrt {\frac {-x + \sqrt {x^2 + y^2} } 2}\) |
Proof
Let $a + i b \in z^{1/2}$.
Then:
\(\ds \paren {a + i b}^2\) | \(=\) | \(\ds x + i y\) | Definition 4 of Square Root of Complex Number | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds a^2 + 2 i a b - b^2\) | \(=\) | \(\ds x + i y\) | Square of Sum and $i^2 = -1$ |
Equating imaginary parts in $(1)$:
\(\ds 2 a b\) | \(=\) | \(\ds y\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds \frac y {2 a}\) | rearranging |
Equating real parts in $(1)$:
\(\ds a^2 - b^2\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^2 - \paren {\frac y {2 a} }^2\) | \(=\) | \(\ds x\) | substituting for $b$ from $(2)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 4 a^4 - 4 a^2 x - y^2\) | \(=\) | \(\ds 0\) | multiplying by $4 a^2$ and rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^2\) | \(=\) | \(\ds \frac {4 x \pm \sqrt {16 x^2 + 16 y^2} } 8\) | Quadratic Formula | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^2\) | \(=\) | \(\ds \frac {x \pm \sqrt {x^2 + y^2} } 2\) | dividing top and bottom by $4$ | ||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds \pm \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}\) | taking the square root |
Note that in $(3)$, only the positive square root of the discriminant $x^2 + y^2$ is used.
This is because the negative square root of $x^2 + y^2$ would yield $\dfrac {x - \sqrt {x^2 + y^2} } 2 < 0$.
As $a \in \R$, it is necessary that $\dfrac {x + \sqrt {x^2 + y^2} } 2 > 0$.
Hence $\sqrt {x^2 + y^2} > 0$.
Then:
\(\ds b\) | \(=\) | \(\ds \frac y {2 a}\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac y {2 \paren {\pm \sqrt {\dfrac {x + \sqrt {x^2 + y^2} } 2} } }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {y^2} {2 \paren {x + \sqrt {x^2 + y^2} } }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {y^2 \paren {x - \sqrt {x^2 + y^2} } } {2 \paren {x + \sqrt {x^2 + y^2} } \paren {x - \sqrt {x^2 + y^2} } }\) | multiplying top and bottom by $x - \sqrt {x^2 + y^2}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {y^2 \paren {x - \sqrt {x^2 + y^2} } } {2 \paren {x^2 - \paren {x^2 + y^2} } }\) | Difference of Two Squares | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {y^2 \paren {x - \sqrt {x^2 + y^2} } } {- 2 y^2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b^2\) | \(=\) | \(\ds \frac {-x + \sqrt {x^2 + y^2} } 2\) |
But from $(2)$ we have:
- $b = \dfrac y {2 a}$
and so having picked either the positive square root or negative square root of either $a^2$ or $b^2$, the root of the other is forced.
So:
- if $y > 0$, then $a$ and $b$ are both of the same sign.
Thus:
- $b = 1 \times \sqrt {\dfrac {-x + \sqrt {x^2 + y^2} } 2}$
- if $y < 0$, then $a$ and $b$ are of opposite sign.
Thus:
- $b = \paren {-1} \times \sqrt {\dfrac {-x + \sqrt {x^2 + y^2} } 2}$
Hence:
- $b = \dfrac y {\cmod y} \sqrt {\dfrac {-x + \sqrt {x^2 + y^2} } 2}$
$\blacksquare$
Examples
Example: $i$
- $\sqrt i = \pm \left({\dfrac {\sqrt 2} 2 + \dfrac {\sqrt 2} 2 i}\right)$
Example: $2 + 2 i$
- $\sqrt {2 + 2 i} = \pm \left({\sqrt {1 + \sqrt 2} + i \sqrt {\sqrt 2 - 1}}\right)$
Example: $3 + 4 i$
- $\sqrt {3 + 4 i} = \pm \left({2 + i}\right)$
Example: $-8 + 6 i$
- $\sqrt {-8 + 6 i} = \pm \paren {1 + 3 i}$
Example: $5 - 12 i$
- $\sqrt {5 - 12 i} = \pm \paren {3 - 2 i}$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $80 \ \text{(b)}$