3.3 Powers and roots

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== De Moivre's formula==
== De Moivre's formula==
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The computational rules <math>\ \arg (zw) = \arg z + \arg w\ </math> and <math>\ |\,zw\,| = |\,z\,|\cdot|\,w\,|\ </math> mean that
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The computational rules <math>\ \arg (zw) = \arg z + \arg w\ </math> and <math>\ |\,zw\,| = |\,z\,|\,|\,w\,|\ </math> mean that
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{{Fristående formel||<math>\biggl\{\begin{align*}&\arg (z\cdot z) = \arg z + \arg z \\ &|\,z\cdot z\,| = |\,z\,|\cdot|\,z\,|\end{align*}\qquad\biggl\{\begin{align*}&\arg z^3 = 3 \arg z \cr &|\,z^3\,| = |\,z\,|^3\end{align*}\qquad\text{etc.}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*}&\arg (z\times z) = \arg z + \arg z \\ &|\,z\times z\,| = |\,z\,|\times|\,z\,|\end{align*}\qquad\biggl\{\begin{align*}&\arg z^3 = 3 \arg z \cr &|\,z^3\,| = |\,z\,|^3\end{align*}\qquad\text{etc.}</math>}}
For an arbitrary number <math>z=r\,(\cos \alpha +i\,\sin \alpha)</math>, we therefore have the following relationship
For an arbitrary number <math>z=r\,(\cos \alpha +i\,\sin \alpha)</math>, we therefore have the following relationship
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{{Fristående formel||<math>z^n = \bigl(r\,(\cos \alpha +i\sin \alpha)\bigr)^n = r^n\,(\cos n\alpha +i\,\sin n\alpha)\,\mbox{.}</math>}}
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{{Displayed math||<math>z^n = \bigl(r\,(\cos \alpha +i\sin \alpha)\bigr)^n = r^n\,(\cos n\alpha +i\,\sin n\alpha)\,\mbox{.}</math>}}
If <math>|\,z\,|=1</math>, (i.e. <math>z</math> lies on the unit circle) then one has the special relationship
If <math>|\,z\,|=1</math>, (i.e. <math>z</math> lies on the unit circle) then one has the special relationship
<div class="regel">
<div class="regel">
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{{Fristående formel||<math>(\cos \alpha +i\,\sin \alpha)^n = \cos n\alpha +i\,\sin n\alpha\,\mbox{,}</math>}}
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{{Displayed math||<math>(\cos \alpha +i\,\sin \alpha)^n = \cos n\alpha +i\,\sin n\alpha\,\mbox{,}</math>}}
</div>
</div>
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which is usually referred to as ''de Moivres formula''. This relationship is very useful when it comes to deriving trigonometric identities and calculating the roots and powers of complex numbers.
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which is usually referred to as ''de Moivre's formula''. This relationship is very useful when it comes to deriving trigonometric identities and calculating the roots and powers of complex numbers.
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We write <math>z</math> in polar form <math>\ \ z= \frac{1}{\sqrt2} + \frac{i}{\sqrt2} = 1\cdot \Bigl(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\Bigr)\ \ </math> and Moivre's formula gives
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We write <math>z</math> in polar form <math>\ \ z= \frac{1}{\sqrt2} + \frac{i}{\sqrt2} = 1\times \Bigl(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\Bigr)\ \ </math> and de Moivre's formula gives
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{{Fristående formel||<math>\begin{align*}z^3 &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^3 = \cos\frac{3\pi}{4} + i\,\sin\frac{3\pi}{4} = -\frac{1}{\sqrt2} + \frac{1}{\sqrt2}\,i = \frac{-1+i}{\sqrt2}\,\mbox{,}\\[6pt] z^{100} &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^{100} = \cos\frac{100\pi}{4} + i\,\sin\frac{100\pi}{4}\\[4pt] &= \cos 25\pi + i\,\sin 25\pi = \cos \pi + i\,\sin \pi = -1\,\mbox{.}\end{align*}</math>}}
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{{Displayed math||<math>\begin{align*}z^3 &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^3 = \cos\frac{3\pi}{4} + i\,\sin\frac{3\pi}{4} = -\frac{1}{\sqrt2} + \frac{1}{\sqrt2}\,i = \frac{-1+i}{\sqrt2}\,\mbox{,}\\[6pt] z^{100} &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^{100} = \cos\frac{100\pi}{4} + i\,\sin\frac{100\pi}{4}\\[4pt] &= \cos 25\pi + i\,\sin 25\pi = \cos \pi + i\,\sin \pi = -1\,\mbox{.}\end{align*}</math>}}
</div>
</div>
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In the usual way one does an expansion by means of the squaring rules
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In the usual way, one does an expansion of the squared bracketed expression
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{{Fristående formel||<math>\begin{align*} (\cos v + i\,\sin v)^2 &= \cos^2\!v + i^2 \sin^2\!v + 2i \sin v \cos v\\ &= \cos^2\!v - \sin^2\!v + 2i \sin v \cos v\end{align*}</math>}}
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{{Displayed math||<math>\begin{align*} (\cos v + i\,\sin v)^2 &= \cos^2\!v + i^2 \sin^2\!v + 2i \sin v \cos v\\ &= \cos^2\!v - \sin^2\!v + 2i \sin v \cos v\end{align*}</math>}}
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and according to de Moivre's formula one gets
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and according to de Moivre's formula, one gets
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{{Fristående formel||<math>(\cos v + i \sin v)^2 = \cos 2v + i \sin 2v\,\mbox{.}</math>}}
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{{Displayed math||<math>(\cos v + i \sin v)^2 = \cos 2v + i \sin 2v\,\mbox{.}</math>}}
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If one equates the real and imaginary parts of the two expressions one gets the well-known trigonometric formulas
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If one equates the real and imaginary parts of the two expressions one gets the well-known trigonometric formulae
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{{Fristående formel||<math>\biggl\{\begin{align*}\cos 2v &= \cos^2\!v - \sin^2\!v\,\mbox{,}\\[2pt] \sin 2v&= 2 \sin v \cos v\,\mbox{.}\end{align*}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*}\cos 2v &= \cos^2\!v - \sin^2\!v\,\mbox{,}\\[2pt] \sin 2v&= 2 \sin v \cos v\,\mbox{.}\end{align*}</math>}}
</div>
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*<math>\quad 1+i\sqrt{3} = 2\Bigl(\cos\frac{\pi}{3} + i\,\sin\frac{\pi}{3}\,\Bigr)\vphantom{\biggl(}</math>,
*<math>\quad 1+i\sqrt{3} = 2\Bigl(\cos\frac{\pi}{3} + i\,\sin\frac{\pi}{3}\,\Bigr)\vphantom{\biggl(}</math>,
*<math>\quad 1+i = \sqrt2\,\Bigl(\cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)\vphantom{\biggl(}</math>.
*<math>\quad 1+i = \sqrt2\,\Bigl(\cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)\vphantom{\biggl(}</math>.
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Then we get with de Moivre's formula
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Then, with de Moivre's formula, we get
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{{Fristående formel||<math>\frac{(\sqrt3 + i)^{14}}{(1+i\sqrt3\,)^7(1+i)^{10}} = \frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin \frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}}{\displaystyle 2^7\Bigl(\cos \frac{7\pi}{3} + i\,\sin\frac{7\pi}{3}\,\Bigr) \cdot (\sqrt{2}\,)^{10}\Bigl(\cos\frac{10\pi}{4} + i\,\sin\frac{10\pi}{4}\,\Bigr)\vphantom{\biggl(}}</math>}}
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{{Displayed math||<math>\frac{(\sqrt3 + i)^{14}}{(1+i\sqrt3\,)^7(1+i)^{10}} = \frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin \frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}}{\displaystyle 2^7\Bigl(\cos \frac{7\pi}{3} + i\,\sin\frac{7\pi}{3}\,\Bigr) \, (\sqrt{2}\,)^{10}\Bigl(\cos\frac{10\pi}{4} + i\,\sin\frac{10\pi}{4}\,\Bigr)\vphantom{\biggl(}}</math>}}
and this expression can be simplified by performing multiplication and division in polar form
and this expression can be simplified by performing multiplication and division in polar form
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{{Fristående formel||<math>\begin{align*}\frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin\frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}} {\displaystyle 2^{12}\Bigl(\cos\frac{29\pi}{6} + i\,\sin\frac{29\pi}{6}\,\Bigr)\vphantom{\biggl(}} &= 2^2 \Bigl(\cos\Bigl( -\frac{15\pi}{6}\,\Bigr) + i\,\sin\Bigl( -\frac{15\pi}{6}\,\Bigr)\,\Bigr)\\[8pt] &= 4\Bigl(\cos \Bigl( -\frac{\pi}{2}\,\Bigr) + i\,\sin\Bigl( -\frac{\pi}{2}\,\Bigr)\,\Bigr) = -4i\,\mbox{.}\end{align*}</math>}}
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{{Displayed math||<math>\begin{align*}\frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin\frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}} {\displaystyle 2^{12}\Bigl(\cos\frac{29\pi}{6} + i\,\sin\frac{29\pi}{6}\,\Bigr)\vphantom{\biggl(}} &= 2^2 \Bigl(\cos\Bigl( -\frac{15\pi}{6}\,\Bigr) + i\,\sin\Bigl( -\frac{15\pi}{6}\,\Bigr)\,\Bigr)\\[8pt] &= 4\Bigl(\cos \Bigl( -\frac{\pi}{2}\,\Bigr) + i\,\sin\Bigl( -\frac{\pi}{2}\,\Bigr)\,\Bigr) = -4i\,\mbox{.}\end{align*}</math>}}
</div>
</div>
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== Binomial equations ==
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== ''n''th roots of complex numbers ==
A complex number <math>z</math> is called the ''n''th root of the complex number <math>w</math> if
A complex number <math>z</math> is called the ''n''th root of the complex number <math>w</math> if
<div class="regel">
<div class="regel">
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{{Fristående formel||<math>z^n= w \mbox{.}</math>}}
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{{Displayed math||<math>z^n= w \mbox{.}</math>}}
</div>
</div>
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The above relationship can also be seen as an equation in which <math>z</math> is unknown. This type of equation is called a ''binomial equation''. The solutions are obtained by rewriting both sides in polar form and comparing both the moduli and the arguments.
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The solutions are obtained by rewriting both sides in polar form and comparing both the moduli and the arguments.
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For a given number <math>w=|\,w\,|\,(\cos \theta + i\,\sin \theta)</math> one assumes that <math>z=r\,(\cos \alpha + i\, \sin \alpha)</math> and after insertion, the binomial equation becomes
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For a given number <math>w=|\,w\,|\,(\cos \theta + i\,\sin \theta)</math> one assumes that <math>z=r\,(\cos \alpha + i\, \sin \alpha)</math> and after insertion, the equation becomes
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{{Fristående formel||<math>r^{\,n}\,(\cos n\alpha + i \sin n\alpha) =|w|\,(\cos \theta + i \sin \theta)\,\mbox{,}</math>}}
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{{Displayed math||<math>r^{\,n}\,(\cos n\alpha + i \sin n\alpha) =|w|\,(\cos \theta + i \sin \theta)\,\mbox{,}</math>}}
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where de Moivres formula has been used on the left-hand side. Equating moduli and arguments gives
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where de Moivre's formula has been used on the left-hand side. Equating moduli and arguments gives
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{{Fristående formel||<math>\biggl\{\begin{align*} r^{\,n} &= |w|\,\mbox{,}\\ n\alpha &= \theta + k\cdot 2\pi\,\mbox{.}\end{align*}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*} r^{\,n} &= |w|\,\mbox{,}\\ n\alpha &= \theta + k\times 2\pi\,\mbox{.}\end{align*}</math>}}
Note that we add multiples of <math>2\pi</math> to include all possible values of the argument that have the same direction as <math>\theta</math>. One gets
Note that we add multiples of <math>2\pi</math> to include all possible values of the argument that have the same direction as <math>\theta</math>. One gets
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{{Fristående formel||<math>\biggl\{\begin{align*} r &={\textstyle\sqrt[\scriptstyle n]{|w|}},\\ \alpha &= (\theta + 2k\pi)/n\,, \quad k=0, \pm 1, \pm 2, \ldots\end{align*}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*} r &={\textstyle\sqrt[\scriptstyle n]{|w|}},\\ \alpha &= (\theta + 2k\pi)/n\,, \quad k=0, \pm 1, \pm 2, \ldots\end{align*}</math>}}
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This gives ''one'' value of <math>r</math>, but infinitely many values of <math>\alpha</math>. Despite this, there are not infinitely many solutions. From <math>k = 0</math> to <math>k = n - 1</math> one gets different arguments for <math>z</math> and thus different positions for <math>z</math> in the complex plane. For the other values of <math>k</math> due to the periodicity of the sine and cosine, one returns to these positions and therefore no new solutions are obtained. This reasoning shows that the equation <math>z^n=w</math> has exactly <math>n</math> roots.
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This gives ''one'' value of <math>r</math>, but infinitely many values of <math>\alpha</math>. Despite this, there are not infinitely many solutions. From <math>k = 0</math> to <math>k = n - 1</math> one gets different arguments for <math>z</math> and thus different positions for <math>z</math> in the complex plane. For the other values of <math>k</math>, note that because of the periodicity of sine and cosine, one returns to these positions and therefore no new solutions are obtained. This reasoning shows that the equation <math>z^n=w</math> has exactly <math>n</math> roots.
''Comment''. Note that the arguments of the roots differ from each other by <math>2\pi/n</math> so that the roots are evenly distributed on a circle with radius <math>\sqrt[\scriptstyle n]{|w|}</math> and form corners in a regular ''n-gon'' (an ''n'' sided polygon).
''Comment''. Note that the arguments of the roots differ from each other by <math>2\pi/n</math> so that the roots are evenly distributed on a circle with radius <math>\sqrt[\scriptstyle n]{|w|}</math> and form corners in a regular ''n-gon'' (an ''n'' sided polygon).
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Solve the binomial equation <math>\ z^4= 16\,i\,</math>.
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Solve the equation <math>\ z^4= 16\,i\,</math>.
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This turns the equation <math>\ z^4=16\,i\ </math> into
This turns the equation <math>\ z^4=16\,i\ </math> into
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{{Fristående formel||<math>r^4\,(\cos 4\alpha + i\,\sin 4\alpha) = 16\Bigl(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\,\Bigr)\,\mbox{.}</math>}}
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{{Displayed math||<math>r^4\,(\cos 4\alpha + i\,\sin 4\alpha) = 16\Bigl(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\,\Bigr)\,\mbox{.}</math>}}
Matching the moduli and arguments on both sides gives
Matching the moduli and arguments on both sides gives
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{{Fristående formel||<math>\biggl\{\begin{align*} r^4 &= 16,\\ 4\alpha &= \pi/2 + k\cdot 2\pi,\end{align*}\qquad\text{i.e.}\qquad\biggl\{\begin{align*} r &= \sqrt[\scriptstyle 4]{16}= 2, \\ \alpha &= \pi/8 + k\pi/2\,,\quad k=0,1,2,3.\end{align*}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*} r^4 &= 16,\\ 4\alpha &= \pi/2 + k\times 2\pi,\end{align*}\qquad\text{i.e.}\qquad\biggl\{\begin{align*} r &= \sqrt[\scriptstyle 4]{16}= 2, \\ \alpha &= \pi/8 + k\pi/2\,,\quad k=0,1,2,3.\end{align*}</math>}}
{| width="100%"
{| width="100%"
| width="95%"|
| width="95%"|
The solutions to the equation are thus
The solutions to the equation are thus
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{{Fristående formel||<math>\left\{\begin{align*}\displaystyle z_1&= 2\Bigl(\cos \frac{\pi}{8} + i\,\sin\frac{\pi}{8}\,\Bigr),\\[4pt]
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{{Displayed math||<math>\left\{\begin{align*}\displaystyle z_1&= 2\Bigl(\cos \frac{\pi}{8} + i\,\sin\frac{\pi}{8}\,\Bigr),\\[4pt]
\displaystyle z_2 &= 2\Bigl(\cos\frac{5\pi}{8} + i\,\sin\frac{5\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt]
\displaystyle z_2 &= 2\Bigl(\cos\frac{5\pi}{8} + i\,\sin\frac{5\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt]
\displaystyle z_3 &= 2\Bigl(\cos\frac{9\pi}{8} + i\,\sin\frac{9\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt]
\displaystyle z_3 &= 2\Bigl(\cos\frac{9\pi}{8} + i\,\sin\frac{9\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt]
\displaystyle z_4 &= 2\Bigl(\cos\frac{13\pi}{8} + i\,\sin\frac{13\pi}{8}\,\Bigr).\end{align*}\right.</math>}}
\displaystyle z_4 &= 2\Bigl(\cos\frac{13\pi}{8} + i\,\sin\frac{13\pi}{8}\,\Bigr).\end{align*}\right.</math>}}
| width="5%" |
| width="5%" |
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||{{:3.3 - Figur - Komplexa talen z₁, z₂, z₃ och z₄}}
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||{{:3.3 - Figure - The complex numbers z₁, z₂, z₃ and z₄}}
|}
|}
</div>
</div>
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If we manipulate <math>i</math> as if it were a real number and treat a complex number <math>z</math> as a function of just <math>\alpha</math> ( where <math>r</math> is a constant),
If we manipulate <math>i</math> as if it were a real number and treat a complex number <math>z</math> as a function of just <math>\alpha</math> ( where <math>r</math> is a constant),
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{{Fristående formel||<math>f(\alpha) = r\,(\cos \alpha + i\,\sin \alpha)</math>}}
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{{Displayed math||<math>f(\alpha) = r\,(\cos \alpha + i\,\sin \alpha)</math>}}
we get after differentiation
we get after differentiation
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{{Fristående formel||<math>\begin{align*} f^{\,\prime}(\alpha) &= -r\sin \alpha + r\,i\,\cos \alpha = r\,i^2 \sin \alpha + r\,i\,\cos \alpha = i\,r\,(\cos \alpha + i\,\sin \alpha) = i\,f(\alpha)\\ f^{\,\prime\prime} (\alpha) &= - r\,\cos \alpha - r\,i\,\sin \alpha = i^2\,r\,(\cos \alpha + i\,\sin \alpha) = i^2\, f(\alpha)\cr &\text{etc.}\end{align*}</math>}}
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{{Displayed math||<math>\begin{align*} f^{\,\prime}(\alpha) &= -r\sin \alpha + r\,i\,\cos \alpha = r\,i^2 \sin \alpha + r\,i\,\cos \alpha = i\,r\,(\cos \alpha + i\,\sin \alpha) = i\,f(\alpha)\\ f^{\,\prime\prime} (\alpha) &= - r\,\cos \alpha - r\,i\,\sin \alpha = i^2\,r\,(\cos \alpha + i\,\sin \alpha) = i^2\, f(\alpha)\cr &\text{etc.}\end{align*}</math>}}
The only real-valued functions which behave like this are <math>f(x)= e^{\,kx}</math>, which justifies the definition
The only real-valued functions which behave like this are <math>f(x)= e^{\,kx}</math>, which justifies the definition
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{{Fristående formel||<math>e^{\,i\alpha} = \cos \alpha + i\,\sin \alpha\,\mbox{.}</math>}}
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{{Displayed math||<math>e^{\,i\alpha} = \cos \alpha + i\,\sin \alpha\,\mbox{.}</math>}}
This definition turns out to be a completely natural generalisation of the exponential function for the real numbers. Putting <math>z=a+ib</math> one gets
This definition turns out to be a completely natural generalisation of the exponential function for the real numbers. Putting <math>z=a+ib</math> one gets
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{{Fristående formel||<math>e^{\,z} = e^{\,a+ib} = e^{\,a} \cdot e^{\,ib} = e^{\,a}(\cos b + i\,\sin b)\,\mbox{.}</math>}}
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{{Displayed math||<math>e^{\,z} = e^{\,a+ib} = e^{\,a} \, e^{\,ib} = e^{\,a}(\cos b + i\,\sin b)\,\mbox{.}</math>}}
The definition of <math>e^{\,z}</math> may be regarded as a convenient notation for the polar form of a complex number, as <math>z=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{\,i\alpha}\,</math>.
The definition of <math>e^{\,z}</math> may be regarded as a convenient notation for the polar form of a complex number, as <math>z=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{\,i\alpha}\,</math>.
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For a real number <math>z</math> the definition is consistent with the case when the exponent is real, as <math>z=a+0\cdot i</math> which gives
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For a real number <math>z</math> the definition is consistent with the case when the exponent is real, as <math>z=a+0\times i</math> which gives
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{{Fristående formel||<math>e^{\,z} = e^{\,a+0\cdot i} = e^a (\cos 0 + i \sin 0) = e^a \cdot 1 = e^a\,\mbox{.}</math>}}
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{{Displayed math||<math>e^{\,z} = e^{\,a+0\times i} = e^a (\cos 0 + i \sin 0) = e^a \times 1 = e^a\,\mbox{.}</math>}}
</div>
</div>
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A further indication of why the above definition is so natural is given by the relationship
A further indication of why the above definition is so natural is given by the relationship
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{{Fristående formel||<math>\bigl(e^{\,i\alpha}\bigr)^n = (\cos \alpha + i \sin \alpha)^n = \cos n\alpha + i \sin n \alpha = e^{\,in\alpha}\,\mbox{,}</math>}}
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{{Displayed math||<math>\bigl(e^{\,i\alpha}\bigr)^n = (\cos \alpha + i \sin \alpha)^n = \cos n\alpha + i \sin n \alpha = e^{\,in\alpha}\,\mbox{,}</math>}}
which demonstrates that de Moivre's formula is actually identical to the well-known law of exponents,
which demonstrates that de Moivre's formula is actually identical to the well-known law of exponents,
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{{Fristående formel||<math>\left(a^x\right)^y = a^{x\,y}\,\mbox{.}</math>}}
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{{Displayed math||<math>\left(a^x\right)^y = a^{x\,y}\,\mbox{.}</math>}}
</div>
</div>
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From the above definition, one can obtain the relationship
From the above definition, one can obtain the relationship
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{{Fristående formel||<math>e^{\pi\,i} = \cos \pi + i \sin \pi = -1</math>}}
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{{Displayed math||<math>e^{\pi\,i} = \cos \pi + i \sin \pi = -1</math>}}
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which connects together the, generally regarded, most basic numbers in mathematics: <math>e</math>, <math>\pi</math>, <math>i</math> and 1.
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which connects together what are generall regarded as the most basic numbers in mathematics: <math>e</math>, <math>\pi</math>, <math>i</math> and 1.
This relationship is seen by many as the most beautiful in mathematics and was discovered by Euler in the early 1700's.
This relationship is seen by many as the most beautiful in mathematics and was discovered by Euler in the early 1700's.
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Put <math>w = z + i</math>. We then get the binomial equation <math>\ w^3=-8i\,</math>. To begin with, we rewrite <math>w</math> and <math>-8i</math> in polar form
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Put <math>w = z + i</math>. We then get the equation <math>\ w^3=-8i\,</math>. To begin with, we rewrite <math>w</math> and <math>-8i</math> in polar form
*<math>\quad w=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{i\alpha}\,\mbox{,}</math>
*<math>\quad w=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{i\alpha}\,\mbox{,}</math>
*<math>\quad -8i = 8\Bigl(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\,\Bigr) = 8\,e^{3\pi i/2}\vphantom{\biggl(}\,\mbox{.}</math>
*<math>\quad -8i = 8\Bigl(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\,\Bigr) = 8\,e^{3\pi i/2}\vphantom{\biggl(}\,\mbox{.}</math>
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The equation in polar form is <math>\ r^3e^{3\alpha i}=8\,e^{3\pi i/2}\ </math> and matching the moduli and arguments on both sides gives,
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In polar form, the equation is <math>\ r^3e^{3\alpha i}=8\,e^{3\pi i/2}\ </math>; matching the moduli and arguments on both sides gives
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{{Fristående formel||<math>\biggl\{\begin{align*} r^3 &= 8\,\mbox{,}\\ 3\alpha &= 3\pi/2+2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r&=\sqrt[\scriptstyle 3]{8}\,\mbox{,}\\ \alpha&= \pi/2+2k\pi/3\,,\quad k=0,1,2\,\mbox{.}\end{align*}</math>}}
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{{Displayed math||<math>\biggl\{\begin{align*} r^3 &= 8\,\mbox{,}\\ 3\alpha &= 3\pi/2+2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r&=\sqrt[\scriptstyle 3]{8}\,\mbox{,}\\ \alpha&= \pi/2+2k\pi/3\,,\quad k=0,1,2\,\mbox{.}\end{align*}</math>}}
The roots of the equation are thus
The roots of the equation are thus
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If for <math>z=a+ib</math> one has <math>|\,z\,|=r</math> and <math>\arg z = \alpha</math> then for <math>\overline{z}= a-ib</math> one gets <math>|\,\overline{z}\,|=r</math> and <math>\arg \overline{z} = - \alpha</math>.This means that <math>z=r\,e^{i\alpha}</math> and <math>\overline{z} = r\,e^{-i\alpha}</math>. The equation can be written
If for <math>z=a+ib</math> one has <math>|\,z\,|=r</math> and <math>\arg z = \alpha</math> then for <math>\overline{z}= a-ib</math> one gets <math>|\,\overline{z}\,|=r</math> and <math>\arg \overline{z} = - \alpha</math>.This means that <math>z=r\,e^{i\alpha}</math> and <math>\overline{z} = r\,e^{-i\alpha}</math>. The equation can be written
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{{Fristående formel||<math>(r\,e^{i\alpha})^2 = r\,e^{-i\alpha}\qquad\text{or}\qquad r^2 e^{2i\alpha}= r\,e^{-i\alpha}\,\mbox{,}</math>}}
+
{{Displayed math||<math>(r\,e^{i\alpha})^2 = r\,e^{-i\alpha}\qquad\text{or}\qquad r^2 e^{2i\alpha}= r\,e^{-i\alpha}\,\mbox{,}</math>}}
which directly gives that <math>r=0</math> is a solution, i.e. <math>z=0</math>. If we assume that <math>r\not=0</math> then the equation can be written as <math>\ r\,e^{3i\alpha} = 1\,</math>, which gives after matching moduli and arguments
which directly gives that <math>r=0</math> is a solution, i.e. <math>z=0</math>. If we assume that <math>r\not=0</math> then the equation can be written as <math>\ r\,e^{3i\alpha} = 1\,</math>, which gives after matching moduli and arguments
-
{{Fristående formel||<math>\biggl\{\begin{align*} r &= 1\,\mbox{,}\\ 3\alpha &= 0 + 2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r &= 1\,\mbox{,}\\ \alpha &= 2k\pi/3\,\mbox{,}\quad k=0,1,2\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\biggl\{\begin{align*} r &= 1\,\mbox{,}\\ 3\alpha &= 0 + 2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r &= 1\,\mbox{,}\\ \alpha &= 2k\pi/3\,\mbox{,}\quad k=0,1,2\,\mbox{.}\end{align*}</math>}}
The solutions are
The solutions are
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== Completing the square ==
== Completing the square ==
-
The squaring rules,
+
The well-known rules
-
{{Fristående formel||<math>\left\{\begin{align*} (a+b)^2 &= a^2+2ab+b^2\\ (a-b)^2 &= a^2-2ab+b^2\end{align*}\right.</math>}}
+
{{Displayed math||<math>\left\{\begin{align*} (a+b)^2 &= a^2+2ab+b^2\\ (a-b)^2 &= a^2-2ab+b^2\end{align*}\right.</math>}}
-
which are usually used to expand parenthesis can also be used in reverse to obtain quadratic expressions. For example,
+
which are usually used to expand brackets can also be used in reverse to obtain quadratic expressions. For example,
-
{{Fristående formel||<math>\begin{align*} x^2+4x+4 &= (x+2)^2\,\mbox{,}\\ x^2-10x+25 &= (x-5)^2\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2+4x+4 &= (x+2)^2\,\mbox{,}\\ x^2-10x+25 &= (x-5)^2\,\mbox{.}\end{align*}</math>}}
This can be used to solve quadratic equations, for example,
This can be used to solve quadratic equations, for example,
-
{{Fristående formel||<math>\begin{align*} x^2+4x+4 &= 9\,\mbox{,}\\ (x+2)^2 &= 9\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2+4x+4 &= 9\,\mbox{,}\\ (x+2)^2 &= 9\,\mbox{.}\end{align*}</math>}}
Taking roots then gives that <math>x+2=\pm\sqrt{9}</math> and thus that <math>x=-2\pm 3</math>, i.e. <math>x=1</math> or <math>x=-5</math>.
Taking roots then gives that <math>x+2=\pm\sqrt{9}</math> and thus that <math>x=-2\pm 3</math>, i.e. <math>x=1</math> or <math>x=-5</math>.
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Sometimes it is necessary to add or subtract an appropriate number to obtain a suitable expression. The above equation, for example, could just as easily been presented to us as
Sometimes it is necessary to add or subtract an appropriate number to obtain a suitable expression. The above equation, for example, could just as easily been presented to us as
-
{{Fristående formel||<math>x^2+4x-5=0\,\mbox{.}</math>}}
+
{{Displayed math||<math>x^2+4x-5=0\,\mbox{.}</math>}}
By adding 9 to both sides, we get a suitable expression on the left side:
By adding 9 to both sides, we get a suitable expression on the left side:
-
{{Fristående formel||<math>\begin{align*} x^2+4x-5+9 &= 0+9\,\mbox{,}\\ x^2+4x+4\phantom{{}+9} &= 9\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2+4x-5+9 &= 0+9\,\mbox{,}\\ x^2+4x+4\phantom{{}+9} &= 9\,\mbox{.}\end{align*}</math>}}
This method is called ''completing the square''.
This method is called ''completing the square''.
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The coefficient in front of <math>x</math> is <math>-6</math> and it shows that we must have the number <math>(-3)^2=9</math> as the constant term on the left-hand side to make a complete square. By adding <math>2</math> to both sides we achieve this:
The coefficient in front of <math>x</math> is <math>-6</math> and it shows that we must have the number <math>(-3)^2=9</math> as the constant term on the left-hand side to make a complete square. By adding <math>2</math> to both sides we achieve this:
-
{{Fristående formel||<math>\begin{align*} x^2-6x+7+2 &= 2+2\,\mbox{,}\\ x^2-6x+9\phantom{{}+2} &= 4\,\mbox{,}\\ \rlap{(x-3)^2}\phantom{x^2-6x+7+2}{} &= 4\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2-6x+7+2 &= 2+2\,\mbox{,}\\ x^2-6x+9\phantom{{}+2} &= 4\,\mbox{,}\\ \rlap{(x-3)^2}\phantom{x^2-6x+7+2}{} &= 4\,\mbox{.}\end{align*}</math>}}
Taking roots then gives <math>x-3=\pm 2</math>, which means that <math>x=1</math> or <math>x=5</math>.
Taking roots then gives <math>x-3=\pm 2</math>, which means that <math>x=1</math> or <math>x=5</math>.
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The equation can be written as <math>z^2+8z+17=0</math>. By subtracting 1 on both sides, we get a complete square on the left-hand side:
The equation can be written as <math>z^2+8z+17=0</math>. By subtracting 1 on both sides, we get a complete square on the left-hand side:
-
{{Fristående formel||<math>\begin{align*} z^2+8z+17-1 &= 0-1\,\mbox{,}\\ z^2+8z+16\phantom{{}-1} &= -1\,\mbox{,}\\ \rlap{(z+4)^2}\phantom{z^2+8z+17-1}{} &= -1\,\mbox{,}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} z^2+8z+17-1 &= 0-1\,\mbox{,}\\ z^2+8z+16\phantom{{}-1} &= -1\,\mbox{,}\\ \rlap{(z+4)^2}\phantom{z^2+8z+17-1}{} &= -1\,\mbox{,}\end{align*}</math>}}
and thus <math>z+4=\pm\sqrt{-1}</math>. In other words, the solutions are <math>z=-4-i</math> and <math>z=-4+i</math>.
and thus <math>z+4=\pm\sqrt{-1}</math>. In other words, the solutions are <math>z=-4-i</math> and <math>z=-4+i</math>.
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Half the coefficient of <math>x</math> is <math>-\tfrac{4}{3}</math>. We thus add <math>\bigl(-\tfrac{4}{3}\bigr)^2=\tfrac{16}{9}</math> to both sides
Half the coefficient of <math>x</math> is <math>-\tfrac{4}{3}</math>. We thus add <math>\bigl(-\tfrac{4}{3}\bigr)^2=\tfrac{16}{9}</math> to both sides
-
{{Fristående formel||<math>\begin{align*} x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1 &= 2+\tfrac{16}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}}{}+1 &= \tfrac{34}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1} &= \tfrac{25}{9}\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1 &= 2+\tfrac{16}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}}{}+1 &= \tfrac{34}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1} &= \tfrac{25}{9}\,\mbox{.}\end{align*}</math>}}
-
Now it's easy to get to <math>x-\tfrac{4}{3}=\pm\tfrac{5}{3}</math> and thus to get that <math>x=\tfrac{4}{3}\pm\tfrac{5}{3}</math>, i.e. <math>x=-\tfrac{1}{3}</math> or <math>x=3</math>.
+
Now it is easy to get to <math>x-\tfrac{4}{3}=\pm\tfrac{5}{3}</math> and thus to get that <math>x=\tfrac{4}{3}\pm\tfrac{5}{3}</math>, i.e. <math>x=-\tfrac{1}{3}</math> or <math>x=3</math>.
</div>
</div>
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Completing the square gives
Completing the square gives
-
{{Fristående formel||<math>\begin{align*} x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q &= \Bigl(\frac{p}{2}\Bigr)^2\,\mbox{,}\\ \rlap{\Bigl(x+\frac{p}{2}\Bigr)^2}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \Bigl(\frac{p}{2}\Bigr)^2-q\,\mbox{,}\\ \rlap{x+\frac{p}{2}}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\ \mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q &= \Bigl(\frac{p}{2}\Bigr)^2\,\mbox{,}\\ \rlap{\Bigl(x+\frac{p}{2}\Bigr)^2}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \Bigl(\frac{p}{2}\Bigr)^2-q\,\mbox{,}\\ \rlap{x+\frac{p}{2}}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\ \mbox{.}\end{align*}</math>}}
-
This gives the usual formula, ''pq-formula'', for solutions to quadratic equations
+
This gives the usual formula for solutions of quadratic equations
-
{{Fristående formel||<math>x=-\frac{p}{2}\pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\,\mbox{.}</math>}}
+
{{Displayed math||<math>x=-\frac{p}{2}\pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\,\mbox{.}</math>}}
</div>
</div>
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-
{{Fristående formel||<math>z^2-(12+4i)z+(-(6+2i))^2-4+24i=(-(6+2i))^2\,\mbox{.}</math>}}
+
{{Displayed math||<math>z^2-(12+4i)z+(-(6+2i))^2-4+24i=(-(6+2i))^2\,\mbox{.}</math>}}
Expanding the square on the right-hand side <math>\ (-(6+2i))^2=36+24i+4i^2=32+24i\ </math> and completing the square on the left-hand side gives
Expanding the square on the right-hand side <math>\ (-(6+2i))^2=36+24i+4i^2=32+24i\ </math> and completing the square on the left-hand side gives
-
{{Fristående formel||<math>\begin{align*} (z-(6+2i))^2-4+24i &= 32+24i\,\mbox{,}\\ \rlap{(z-(6+2i))^2}\phantom{(z-(6+2i))^2-4+24i}{} &= 36\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} (z-(6+2i))^2-4+24i &= 32+24i\,\mbox{,}\\ \rlap{(z-(6+2i))^2}\phantom{(z-(6+2i))^2-4+24i}{} &= 36\,\mbox{.}\end{align*}</math>}}
-
After a taking roots, we have that <math>\ z-(6+2i)=\pm 6\ </math> and therefore the solutions are <math>z=12+2i</math> and <math>z=2i</math>.
+
After taking roots, we have that <math>\ z-(6+2i)=\pm 6\ </math> and therefore the solutions are <math>z=12+2i</math> and <math>z=2i</math>.
</div>
</div>
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If one wants to bring about a square in an expression one can use the same technique. In order not to change the value of the expression one both adds and subtracts the missing constant term, such as in the following,
If one wants to bring about a square in an expression one can use the same technique. In order not to change the value of the expression one both adds and subtracts the missing constant term, such as in the following,
-
{{Fristående formel||<math>\begin{align*} x^2+10x+3 &= x^2+10x+25+3-25\\ &= (x+5)^2-22\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} x^2+10x+3 &= x^2+10x+25+3-25\\ &= (x+5)^2-22\,\mbox{.}\end{align*}</math>}}
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Add and subtract the term <math>\bigl(\frac{1}{2}(2-4i)\bigr)^2=(1-2i)^2=-3-4i\,</math>,
Add and subtract the term <math>\bigl(\frac{1}{2}(2-4i)\bigr)^2=(1-2i)^2=-3-4i\,</math>,
-
{{Fristående formel||<math>\begin{align*} z^2+(2-4i)z+1-3i &= z^2+(2-4i)z+(1-2i)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(-3-4i)+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2+4+i\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} z^2+(2-4i)z+1-3i &= z^2+(2-4i)z+(1-2i)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(-3-4i)+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2+4+i\,\mbox{.}\end{align*}</math>}}
</div>
</div>
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==Solving using a formula==
==Solving using a formula==
-
To solve quadratic equations sometimes the simplest method is to use the usual formula for quadratic equations. However, this may lead to that one ends up with terms of the type <math>\sqrt{a+ib}</math>. One can then assume
+
To solve quadratic equations, sometimes the simplest method is to use the usual formula for quadratic equations. However, this may leave one with terms of the type <math>\sqrt{a+ib}</math>. One can then assume
-
{{Fristående formel||<math>z=x+iy=\sqrt{a+ib}\,\mbox{.}</math>}}
+
{{Displayed math||<math>z=x+iy=\sqrt{a+ib}\,\mbox{.}</math>}}
By squaring both sides we get
By squaring both sides we get
-
{{Fristående formel||<math>\begin{align*} (x+iy)^2 &= a+ib\,\mbox{,}\\ x^2 - y^2 + 2xy\,i &= a+ib\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} (x+iy)^2 &= a+ib\,\mbox{,}\\ x^2 - y^2 + 2xy\,i &= a+ib\,\mbox{.}\end{align*}</math>}}
Matching the real and imaginary parts gives
Matching the real and imaginary parts gives
-
{{Fristående formel||<math>\left\{\begin{align*} &x^2 - y^2 = a\,\mbox{,}\\ &2xy=b\,\mbox{.}\end{align*}\right.</math>}}
+
{{Displayed math||<math>\left\{\begin{align*} &x^2 - y^2 = a\,\mbox{,}\\ &2xy=b\,\mbox{.}\end{align*}\right.</math>}}
These equations can be solved by substitution, for example, <math>y= b/(2x)</math> can be inserted in the first equation.
These equations can be solved by substitution, for example, <math>y= b/(2x)</math> can be inserted in the first equation.
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Put <math>\ x+iy=\sqrt{-3-4i}\ </math> where <math>x</math> and <math>y</math> are real numbers. Squaring both sides gives
Put <math>\ x+iy=\sqrt{-3-4i}\ </math> where <math>x</math> and <math>y</math> are real numbers. Squaring both sides gives
-
{{Fristående formel||<math>\begin{align*} (x+iy)^2 &= -3-4i\,\mbox{,}\\ x^2 - y^2 + 2xyi &= -3-4i\,\mbox{,}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} (x+iy)^2 &= -3-4i\,\mbox{,}\\ x^2 - y^2 + 2xyi &= -3-4i\,\mbox{,}\end{align*}</math>}}
which leads to the system of equations
which leads to the system of equations
-
{{Fristående formel||<math>\Bigl\{\begin{align*} x^2 - y^2 &= -3\,\mbox{,}\\ 2xy&= -4\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\Bigl\{\begin{align*} x^2 - y^2 &= -3\,\mbox{,}\\ 2xy&= -4\,\mbox{.}\end{align*}</math>}}
From the second equation, we can solve for <math>\ y=-4/(2x) = -2/x\ </math> and put it into the first equation to get
From the second equation, we can solve for <math>\ y=-4/(2x) = -2/x\ </math> and put it into the first equation to get
-
{{Fristående formel||<math>x^2-\frac{4}{x^2} = -3 \quad \Leftrightarrow \quad x^4 +3x^2 - 4=0\,\mbox{.}</math>}}
+
{{Displayed math||<math>x^2-\frac{4}{x^2} = -3 \quad \Leftrightarrow \quad x^4 +3x^2 - 4=0\,\mbox{.}</math>}}
This is a quadratic equation in <math>x^2</math> which can be seen more easily by putting <math>t=x^2</math>,
This is a quadratic equation in <math>x^2</math> which can be seen more easily by putting <math>t=x^2</math>,
-
{{Fristående formel||<math>t^2 +3t -4=0\,\mbox{.}</math>}}
+
{{Displayed math||<math>t^2 +3t -4=0\,\mbox{.}</math>}}
The solutions are <math>t = 1</math> and <math>t = -4</math>. The latter solution must be rejected, as <math>x</math> and <math>y</math> have been assumed to be real numbers, and thus <math>x^2=-4</math> cannot be true. We get <math>x=\pm\sqrt{1}</math>, which gives us two possible solutions
The solutions are <math>t = 1</math> and <math>t = -4</math>. The latter solution must be rejected, as <math>x</math> and <math>y</math> have been assumed to be real numbers, and thus <math>x^2=-4</math> cannot be true. We get <math>x=\pm\sqrt{1}</math>, which gives us two possible solutions
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So we can conclude that
So we can conclude that
-
{{Fristående formel||<math>\sqrt{-3-4i} = \biggl\{\begin{align*} &\phantom{-}1-2i\,\mbox{,}\\ &-1+2i\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\sqrt{-3-4i} = \biggl\{\begin{align*} &\phantom{-}1-2i\,\mbox{,}\\ &-1+2i\,\mbox{.}\end{align*}</math>}}
</div>
</div>
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<br>
<br>
<br>
<br>
-
The formula for solutions to a quadratic equations (see example 3) gives that
+
The formula for solutions of a quadratic equation (see example 3) gives that
-
{{Fristående formel||<math>z= 1\pm \sqrt{1-10} = 1\pm \sqrt{-9}= 1\pm 3i\,\mbox{.}</math>}}
+
{{Displayed math||<math>z= 1\pm \sqrt{1-10} = 1\pm \sqrt{-9}= 1\pm 3i\,\mbox{.}</math>}}
</li>
</li>
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<br>
<br>
<br>
<br>
-
Here, once again , the ''pq''-formula may be used giving the solutions directly
+
Here, once again , the formula may be used, giving the solutions directly
-
{{Fristående formel||<math>\begin{align*} z &= -2+i\pm\sqrt{\smash{(-2+i)^2+4i}\vphantom{i^2}} = -2+i\pm\sqrt{4-4i+i^{\,2}+4i}\\ &=-2+i\pm\sqrt{3} = -2\pm\sqrt{3}+i\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} z &= -2+i\pm\sqrt{\smash{(-2+i)^2+4i}\vphantom{i^2}} = -2+i\pm\sqrt{4-4i+i^{\,2}+4i}\\ &=-2+i\pm\sqrt{3} = -2\pm\sqrt{3}+i\,\mbox{.}\end{align*}</math>}}
</li>
</li>
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<br>
<br>
<br>
<br>
-
Division of both sides with <math>i</math> gives
+
Division of both sides by <math>i</math> gives
-
{{Fristående formel||<math>\begin{align*} z^2 + \frac{2+6i}{i}z +\frac{2+11i}{i} &= 0\,\mbox{,}\\ z^2+ (6-2i)z + 11-2i &= 0\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} z^2 + \frac{2+6i}{i}z +\frac{2+11i}{i} &= 0\,\mbox{,}\\ z^2+ (6-2i)z + 11-2i &= 0\,\mbox{.}\end{align*}</math>}}
-
Applying the ''pq''- formula gives
+
Applying the formula gives
-
{{Fristående formel||<math>\begin{align*} z &= -3+i \pm \sqrt{\smash{(-3+i)^2 -(11-2i)}\vphantom{i^2}}\\ &= -3+i \pm \sqrt{-3-4i}\\ &= -3+i\pm(1-2i)\end{align*}</math>}}
+
{{Displayed math||<math>\begin{align*} z &= -3+i \pm \sqrt{\smash{(-3+i)^2 -(11-2i)}\vphantom{i^2}}\\ &= -3+i \pm \sqrt{-3-4i}\\ &= -3+i\pm(1-2i)\end{align*}</math>}}
-
where we used the resulting value of<math>\ \sqrt{-3-4i}\ </math> which we obtained in example 15. The solutions are therefore
+
where we used the resulting value of <math>\ \sqrt{-3-4i}\ </math> that we obtained in example 15. The solutions are therefore
-
{{Fristående formel||<math>z=\biggl\{\begin{align*} &-2-i\,\mbox{,}\\ &-4+3i\,\mbox{.}\end{align*}</math>}}
+
{{Displayed math||<math>z=\biggl\{\begin{align*} &-2-i\,\mbox{,}\\ &-4+3i\,\mbox{.}\end{align*}</math>}}
</li>
</li>
</ol>
</ol>
</div>
</div>

Current revision

       Theory          Exercises      

Contents:

  • De Moivre's formula
  • Binomial equations
  • Exponential function
  • Euler's formula
  • Completing the square
  • Quadratic equations

Learning outcomes:

After this section, you will have learned how to:

  • Calculate the powers of complex numbers with de Moivre's formula.
  • Calculate the roots of certain complex numbers by rewriting to polar form.
  • Solve binomial equations.
  • Complete the square for complex quadratic expressions.
  • Solve complex quadratic equations.

De Moivre's formula

The computational rules \displaystyle \ \arg (zw) = \arg z + \arg w\ and \displaystyle \ |\,zw\,| = |\,z\,|\,|\,w\,|\ mean that

\displaystyle \biggl\{\begin{align*}&\arg (z\times z) = \arg z + \arg z \\ &|\,z\times z\,| = |\,z\,|\times|\,z\,|\end{align*}\qquad\biggl\{\begin{align*}&\arg z^3 = 3 \arg z \cr &|\,z^3\,| = |\,z\,|^3\end{align*}\qquad\text{etc.}

For an arbitrary number \displaystyle z=r\,(\cos \alpha +i\,\sin \alpha), we therefore have the following relationship

\displaystyle z^n = \bigl(r\,(\cos \alpha +i\sin \alpha)\bigr)^n = r^n\,(\cos n\alpha +i\,\sin n\alpha)\,\mbox{.}

If \displaystyle |\,z\,|=1, (i.e. \displaystyle z lies on the unit circle) then one has the special relationship

\displaystyle (\cos \alpha +i\,\sin \alpha)^n = \cos n\alpha +i\,\sin n\alpha\,\mbox{,}

which is usually referred to as de Moivre's formula. This relationship is very useful when it comes to deriving trigonometric identities and calculating the roots and powers of complex numbers.


Example 1


If \displaystyle z = \frac{1+i}{\sqrt2}, determine \displaystyle z^3 and \displaystyle z^{100}.


We write \displaystyle z in polar form \displaystyle \ \ z= \frac{1}{\sqrt2} + \frac{i}{\sqrt2} = 1\times \Bigl(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\Bigr)\ \ and de Moivre's formula gives

\displaystyle \begin{align*}z^3 &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^3 = \cos\frac{3\pi}{4} + i\,\sin\frac{3\pi}{4} = -\frac{1}{\sqrt2} + \frac{1}{\sqrt2}\,i = \frac{-1+i}{\sqrt2}\,\mbox{,}\\[6pt] z^{100} &= \Bigl( \cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)^{100} = \cos\frac{100\pi}{4} + i\,\sin\frac{100\pi}{4}\\[4pt] &= \cos 25\pi + i\,\sin 25\pi = \cos \pi + i\,\sin \pi = -1\,\mbox{.}\end{align*}

Example 2


In the usual way, one does an expansion of the squared bracketed expression

\displaystyle \begin{align*} (\cos v + i\,\sin v)^2 &= \cos^2\!v + i^2 \sin^2\!v + 2i \sin v \cos v\\ &= \cos^2\!v - \sin^2\!v + 2i \sin v \cos v\end{align*}

and according to de Moivre's formula, one gets

\displaystyle (\cos v + i \sin v)^2 = \cos 2v + i \sin 2v\,\mbox{.}

If one equates the real and imaginary parts of the two expressions one gets the well-known trigonometric formulae


\displaystyle \biggl\{\begin{align*}\cos 2v &= \cos^2\!v - \sin^2\!v\,\mbox{,}\\[2pt] \sin 2v&= 2 \sin v \cos v\,\mbox{.}\end{align*}

Example 3


Simplify \displaystyle \ \ \frac{(\sqrt3 + i)^{14}}{(1+i\sqrt3\,)^7(1+i)^{10}}\,.


We write the numbers \displaystyle \sqrt{3}+i, \displaystyle 1+i\sqrt{3} and \displaystyle 1+i in polar form

  • \displaystyle \quad\sqrt{3} + i = 2\Bigl(\cos\frac{\pi}{6} + i\,\sin\frac{\pi}{6}\,\Bigr)\vphantom{\biggl(},
  • \displaystyle \quad 1+i\sqrt{3} = 2\Bigl(\cos\frac{\pi}{3} + i\,\sin\frac{\pi}{3}\,\Bigr)\vphantom{\biggl(},
  • \displaystyle \quad 1+i = \sqrt2\,\Bigl(\cos\frac{\pi}{4} + i\,\sin\frac{\pi}{4}\,\Bigr)\vphantom{\biggl(}.

Then, with de Moivre's formula, we get

\displaystyle \frac{(\sqrt3 + i)^{14}}{(1+i\sqrt3\,)^7(1+i)^{10}} = \frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin \frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}}{\displaystyle 2^7\Bigl(\cos \frac{7\pi}{3} + i\,\sin\frac{7\pi}{3}\,\Bigr) \, (\sqrt{2}\,)^{10}\Bigl(\cos\frac{10\pi}{4} + i\,\sin\frac{10\pi}{4}\,\Bigr)\vphantom{\biggl(}}

and this expression can be simplified by performing multiplication and division in polar form

\displaystyle \begin{align*}\frac{\displaystyle 2^{14}\Bigl(\cos\frac{14\pi}{6} + i\,\sin\frac{14\pi}{6}\,\Bigr)\vphantom{\biggl(}} {\displaystyle 2^{12}\Bigl(\cos\frac{29\pi}{6} + i\,\sin\frac{29\pi}{6}\,\Bigr)\vphantom{\biggl(}} &= 2^2 \Bigl(\cos\Bigl( -\frac{15\pi}{6}\,\Bigr) + i\,\sin\Bigl( -\frac{15\pi}{6}\,\Bigr)\,\Bigr)\\[8pt] &= 4\Bigl(\cos \Bigl( -\frac{\pi}{2}\,\Bigr) + i\,\sin\Bigl( -\frac{\pi}{2}\,\Bigr)\,\Bigr) = -4i\,\mbox{.}\end{align*}


nth roots of complex numbers

A complex number \displaystyle z is called the nth root of the complex number \displaystyle w if

\displaystyle z^n= w \mbox{.}

The solutions are obtained by rewriting both sides in polar form and comparing both the moduli and the arguments.

For a given number \displaystyle w=|\,w\,|\,(\cos \theta + i\,\sin \theta) one assumes that \displaystyle z=r\,(\cos \alpha + i\, \sin \alpha) and after insertion, the equation becomes

\displaystyle r^{\,n}\,(\cos n\alpha + i \sin n\alpha) =|w|\,(\cos \theta + i \sin \theta)\,\mbox{,}

where de Moivre's formula has been used on the left-hand side. Equating moduli and arguments gives

\displaystyle \biggl\{\begin{align*} r^{\,n} &= |w|\,\mbox{,}\\ n\alpha &= \theta + k\times 2\pi\,\mbox{.}\end{align*}

Note that we add multiples of \displaystyle 2\pi to include all possible values of the argument that have the same direction as \displaystyle \theta. One gets

\displaystyle \biggl\{\begin{align*} r &={\textstyle\sqrt[\scriptstyle n]{|w|}},\\ \alpha &= (\theta + 2k\pi)/n\,, \quad k=0, \pm 1, \pm 2, \ldots\end{align*}

This gives one value of \displaystyle r, but infinitely many values of \displaystyle \alpha. Despite this, there are not infinitely many solutions. From \displaystyle k = 0 to \displaystyle k = n - 1 one gets different arguments for \displaystyle z and thus different positions for \displaystyle z in the complex plane. For the other values of \displaystyle k, note that because of the periodicity of sine and cosine, one returns to these positions and therefore no new solutions are obtained. This reasoning shows that the equation \displaystyle z^n=w has exactly \displaystyle n roots.

Comment. Note that the arguments of the roots differ from each other by \displaystyle 2\pi/n so that the roots are evenly distributed on a circle with radius \displaystyle \sqrt[\scriptstyle n]{|w|} and form corners in a regular n-gon (an n sided polygon).


Exempel 4


Solve the equation \displaystyle \ z^4= 16\,i\,.


Write \displaystyle z and \displaystyle 16\,i in polar form

  • \displaystyle \quad z=r\,(\cos \alpha + i\,\sin \alpha)\,,
  • \displaystyle \quad 16\,i= 16\Bigl(\cos\frac{\pi}{2} + i\,\sin\frac{\pi}{2}\,\Bigr)\vphantom{\biggl(}.

This turns the equation \displaystyle \ z^4=16\,i\ into

\displaystyle r^4\,(\cos 4\alpha + i\,\sin 4\alpha) = 16\Bigl(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\,\Bigr)\,\mbox{.}

Matching the moduli and arguments on both sides gives

\displaystyle \biggl\{\begin{align*} r^4 &= 16,\\ 4\alpha &= \pi/2 + k\times 2\pi,\end{align*}\qquad\text{i.e.}\qquad\biggl\{\begin{align*} r &= \sqrt[\scriptstyle 4]{16}= 2, \\ \alpha &= \pi/8 + k\pi/2\,,\quad k=0,1,2,3.\end{align*}

The solutions to the equation are thus

\displaystyle \left\{\begin{align*}\displaystyle z_1&= 2\Bigl(\cos \frac{\pi}{8} + i\,\sin\frac{\pi}{8}\,\Bigr),\\[4pt]

\displaystyle z_2 &= 2\Bigl(\cos\frac{5\pi}{8} + i\,\sin\frac{5\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt] \displaystyle z_3 &= 2\Bigl(\cos\frac{9\pi}{8} + i\,\sin\frac{9\pi}{8}\,\Bigr),\vphantom{\biggl(}\\[4pt] \displaystyle z_4 &= 2\Bigl(\cos\frac{13\pi}{8} + i\,\sin\frac{13\pi}{8}\,\Bigr).\end{align*}\right.

[Image]


Exponential form of complex numbers

If we manipulate \displaystyle i as if it were a real number and treat a complex number \displaystyle z as a function of just \displaystyle \alpha ( where \displaystyle r is a constant),

\displaystyle f(\alpha) = r\,(\cos \alpha + i\,\sin \alpha)

we get after differentiation

\displaystyle \begin{align*} f^{\,\prime}(\alpha) &= -r\sin \alpha + r\,i\,\cos \alpha = r\,i^2 \sin \alpha + r\,i\,\cos \alpha = i\,r\,(\cos \alpha + i\,\sin \alpha) = i\,f(\alpha)\\ f^{\,\prime\prime} (\alpha) &= - r\,\cos \alpha - r\,i\,\sin \alpha = i^2\,r\,(\cos \alpha + i\,\sin \alpha) = i^2\, f(\alpha)\cr &\text{etc.}\end{align*}

The only real-valued functions which behave like this are \displaystyle f(x)= e^{\,kx}, which justifies the definition

\displaystyle e^{\,i\alpha} = \cos \alpha + i\,\sin \alpha\,\mbox{.}

This definition turns out to be a completely natural generalisation of the exponential function for the real numbers. Putting \displaystyle z=a+ib one gets

\displaystyle e^{\,z} = e^{\,a+ib} = e^{\,a} \, e^{\,ib} = e^{\,a}(\cos b + i\,\sin b)\,\mbox{.}

The definition of \displaystyle e^{\,z} may be regarded as a convenient notation for the polar form of a complex number, as \displaystyle z=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{\,i\alpha}\,.


Example 5


For a real number \displaystyle z the definition is consistent with the case when the exponent is real, as \displaystyle z=a+0\times i which gives

\displaystyle e^{\,z} = e^{\,a+0\times i} = e^a (\cos 0 + i \sin 0) = e^a \times 1 = e^a\,\mbox{.}

Example 6


A further indication of why the above definition is so natural is given by the relationship

\displaystyle \bigl(e^{\,i\alpha}\bigr)^n = (\cos \alpha + i \sin \alpha)^n = \cos n\alpha + i \sin n \alpha = e^{\,in\alpha}\,\mbox{,}

which demonstrates that de Moivre's formula is actually identical to the well-known law of exponents,

\displaystyle \left(a^x\right)^y = a^{x\,y}\,\mbox{.}

Example 7


From the above definition, one can obtain the relationship

\displaystyle e^{\pi\,i} = \cos \pi + i \sin \pi = -1

which connects together what are generall regarded as the most basic numbers in mathematics: \displaystyle e, \displaystyle \pi, \displaystyle i and 1. This relationship is seen by many as the most beautiful in mathematics and was discovered by Euler in the early 1700's.

Example 8


Solve the equation \displaystyle \ (z+i)^3 = -8i.


Put \displaystyle w = z + i. We then get the equation \displaystyle \ w^3=-8i\,. To begin with, we rewrite \displaystyle w and \displaystyle -8i in polar form

  • \displaystyle \quad w=r\,(\cos \alpha + i\,\sin \alpha) = r\,e^{i\alpha}\,\mbox{,}
  • \displaystyle \quad -8i = 8\Bigl(\cos \frac{3\pi}{2} + i\,\sin\frac{3\pi}{2}\,\Bigr) = 8\,e^{3\pi i/2}\vphantom{\biggl(}\,\mbox{.}

In polar form, the equation is \displaystyle \ r^3e^{3\alpha i}=8\,e^{3\pi i/2}\ ; matching the moduli and arguments on both sides gives

\displaystyle \biggl\{\begin{align*} r^3 &= 8\,\mbox{,}\\ 3\alpha &= 3\pi/2+2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r&=\sqrt[\scriptstyle 3]{8}\,\mbox{,}\\ \alpha&= \pi/2+2k\pi/3\,,\quad k=0,1,2\,\mbox{.}\end{align*}

The roots of the equation are thus

  • \displaystyle \quad w_1 = 2\,e^{\pi i/2} = 2\Bigl(\cos \frac{\pi}{2} + i\,\sin\frac{\pi}{2}\,\Bigr) = 2i\,\mbox{,}\quad\vphantom{\biggl(}
  • \displaystyle \quad w_2 = 2\,e^{7\pi i/6} = 2\Bigl(\cos\frac{7\pi}{6} + i\,\sin\frac{7\pi}{6}\,\Bigr) = -\sqrt{3}-i\,\mbox{,}\quad\vphantom{\Biggl(}
  • \displaystyle \quad w_3 = 2\,e^{11\pi i/6} = 2\Bigl(\cos\frac{11\pi}{6} + i\,\sin\frac{11\pi}{6}\,\Bigr) = \sqrt{3}-i\,\mbox{,}\quad\vphantom{\biggl(}

i.e. \displaystyle z_1 = 2i-i=i, \displaystyle z_2 = - \sqrt{3}-2i and \displaystyle z_3 = \sqrt{3}-2i.

Example 9


Solve \displaystyle \ z^2 = \overline{z}\,.


If for \displaystyle z=a+ib one has \displaystyle |\,z\,|=r and \displaystyle \arg z = \alpha then for \displaystyle \overline{z}= a-ib one gets \displaystyle |\,\overline{z}\,|=r and \displaystyle \arg \overline{z} = - \alpha.This means that \displaystyle z=r\,e^{i\alpha} and \displaystyle \overline{z} = r\,e^{-i\alpha}. The equation can be written

\displaystyle (r\,e^{i\alpha})^2 = r\,e^{-i\alpha}\qquad\text{or}\qquad r^2 e^{2i\alpha}= r\,e^{-i\alpha}\,\mbox{,}

which directly gives that \displaystyle r=0 is a solution, i.e. \displaystyle z=0. If we assume that \displaystyle r\not=0 then the equation can be written as \displaystyle \ r\,e^{3i\alpha} = 1\,, which gives after matching moduli and arguments

\displaystyle \biggl\{\begin{align*} r &= 1\,\mbox{,}\\ 3\alpha &= 0 + 2k\pi\,\mbox{,}\end{align*}\qquad\Leftrightarrow\qquad\biggl\{\begin{align*} r &= 1\,\mbox{,}\\ \alpha &= 2k\pi/3\,\mbox{,}\quad k=0,1,2\,\mbox{.}\end{align*}

The solutions are

  • \displaystyle \quad z_1 = e^0 = 1\,\mbox{,}
  • \displaystyle \quad z_2 = e^{2\pi i/ 3} = \cos\frac{2\pi}{3} + i\,\sin\frac{2\pi}{3} = -\frac{1}{2} + \frac{\sqrt3}{2}\,i\,\mbox{,}\vphantom{\Biggl(}
  • \displaystyle \quad z_3 = e^{4\pi i/ 3} = \cos\frac{4\pi}{3} + i\,\sin\frac{4\pi}{3} = -\frac{1}{2} - \frac{\sqrt3}{2}\,i\,\mbox{,}
  • \displaystyle \quad z_4 = 0\,\mbox{.}


Completing the square

The well-known rules

\displaystyle \left\{\begin{align*} (a+b)^2 &= a^2+2ab+b^2\\ (a-b)^2 &= a^2-2ab+b^2\end{align*}\right.

which are usually used to expand brackets can also be used in reverse to obtain quadratic expressions. For example,

\displaystyle \begin{align*} x^2+4x+4 &= (x+2)^2\,\mbox{,}\\ x^2-10x+25 &= (x-5)^2\,\mbox{.}\end{align*}

This can be used to solve quadratic equations, for example,

\displaystyle \begin{align*} x^2+4x+4 &= 9\,\mbox{,}\\ (x+2)^2 &= 9\,\mbox{.}\end{align*}

Taking roots then gives that \displaystyle x+2=\pm\sqrt{9} and thus that \displaystyle x=-2\pm 3, i.e. \displaystyle x=1 or \displaystyle x=-5.


Sometimes it is necessary to add or subtract an appropriate number to obtain a suitable expression. The above equation, for example, could just as easily been presented to us as

\displaystyle x^2+4x-5=0\,\mbox{.}

By adding 9 to both sides, we get a suitable expression on the left side:

\displaystyle \begin{align*} x^2+4x-5+9 &= 0+9\,\mbox{,}\\ x^2+4x+4\phantom{{}+9} &= 9\,\mbox{.}\end{align*}

This method is called completing the square.


Example 10

  1. Solve the equation \displaystyle \ x^2-6x+7=2\,.

    The coefficient in front of \displaystyle x is \displaystyle -6 and it shows that we must have the number \displaystyle (-3)^2=9 as the constant term on the left-hand side to make a complete square. By adding \displaystyle 2 to both sides we achieve this:
    \displaystyle \begin{align*} x^2-6x+7+2 &= 2+2\,\mbox{,}\\ x^2-6x+9\phantom{{}+2} &= 4\,\mbox{,}\\ \rlap{(x-3)^2}\phantom{x^2-6x+7+2}{} &= 4\,\mbox{.}\end{align*}

    Taking roots then gives \displaystyle x-3=\pm 2, which means that \displaystyle x=1 or \displaystyle x=5.

  2. Solve the equation \displaystyle \ z^2+21=4-8z\,.

    The equation can be written as \displaystyle z^2+8z+17=0. By subtracting 1 on both sides, we get a complete square on the left-hand side:
    \displaystyle \begin{align*} z^2+8z+17-1 &= 0-1\,\mbox{,}\\ z^2+8z+16\phantom{{}-1} &= -1\,\mbox{,}\\ \rlap{(z+4)^2}\phantom{z^2+8z+17-1}{} &= -1\,\mbox{,}\end{align*}

    and thus \displaystyle z+4=\pm\sqrt{-1}. In other words, the solutions are \displaystyle z=-4-i and \displaystyle z=-4+i.

Generally, completing the square may be regarded as arranging that "the square of half the coefficient of the x-term" is the constant term in the quadratic expression. This term can always be added to the two sides without worrying about the other terms and then manipulating the equation. If the coefficients of the expression are complex numbers, one still can go about it in the same way.


Example 11


Solve the equation \displaystyle \ x^2-\frac{8}{3}x+1=2\,.


Half the coefficient of \displaystyle x is \displaystyle -\tfrac{4}{3}. We thus add \displaystyle \bigl(-\tfrac{4}{3}\bigr)^2=\tfrac{16}{9} to both sides

\displaystyle \begin{align*} x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1 &= 2+\tfrac{16}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}}{}+1 &= \tfrac{34}{9}\,\mbox{,}\\ \rlap{\bigl(x-\tfrac{4}{3}\bigr)^2}\phantom{x^2-\tfrac{8}{3}x+\tfrac{16}{9}+1} &= \tfrac{25}{9}\,\mbox{.}\end{align*}

Now it is easy to get to \displaystyle x-\tfrac{4}{3}=\pm\tfrac{5}{3} and thus to get that \displaystyle x=\tfrac{4}{3}\pm\tfrac{5}{3}, i.e. \displaystyle x=-\tfrac{1}{3} or \displaystyle x=3.

Example 12


Solve the equation \displaystyle \ x^2+px+q=0\,.


Completing the square gives

\displaystyle \begin{align*} x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q &= \Bigl(\frac{p}{2}\Bigr)^2\,\mbox{,}\\ \rlap{\Bigl(x+\frac{p}{2}\Bigr)^2}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \Bigl(\frac{p}{2}\Bigr)^2-q\,\mbox{,}\\ \rlap{x+\frac{p}{2}}\phantom{x^2+px+\Bigl(\frac{p}{2}\Bigr)^2+q}{} &= \pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\ \mbox{.}\end{align*}

This gives the usual formula for solutions of quadratic equations

\displaystyle x=-\frac{p}{2}\pm \sqrt{\Bigl(\frac{p}{2}\Bigr)^2-q}\,\mbox{.}

Example 13


Solve the equation \displaystyle \ z^2-(12+4i)z-4+24i=0\,.


Half the coefficient of \displaystyle z is \displaystyle -(6+2i) so we add the square of this expression to both sides


\displaystyle z^2-(12+4i)z+(-(6+2i))^2-4+24i=(-(6+2i))^2\,\mbox{.}

Expanding the square on the right-hand side \displaystyle \ (-(6+2i))^2=36+24i+4i^2=32+24i\ and completing the square on the left-hand side gives

\displaystyle \begin{align*} (z-(6+2i))^2-4+24i &= 32+24i\,\mbox{,}\\ \rlap{(z-(6+2i))^2}\phantom{(z-(6+2i))^2-4+24i}{} &= 36\,\mbox{.}\end{align*}

After taking roots, we have that \displaystyle \ z-(6+2i)=\pm 6\ and therefore the solutions are \displaystyle z=12+2i and \displaystyle z=2i.

If one wants to bring about a square in an expression one can use the same technique. In order not to change the value of the expression one both adds and subtracts the missing constant term, such as in the following,

\displaystyle \begin{align*} x^2+10x+3 &= x^2+10x+25+3-25\\ &= (x+5)^2-22\,\mbox{.}\end{align*}


Example 14


Complete the square in the expression \displaystyle \ z^2+(2-4i)z+1-3i\,.


Add and subtract the term \displaystyle \bigl(\frac{1}{2}(2-4i)\bigr)^2=(1-2i)^2=-3-4i\,,

\displaystyle \begin{align*} z^2+(2-4i)z+1-3i &= z^2+(2-4i)z+(1-2i)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(1-2i)^2+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2-(-3-4i)+1-3i\\ &= \bigl(z+(1-2i)\bigr)^2+4+i\,\mbox{.}\end{align*}


Solving using a formula

To solve quadratic equations, sometimes the simplest method is to use the usual formula for quadratic equations. However, this may leave one with terms of the type \displaystyle \sqrt{a+ib}. One can then assume

\displaystyle z=x+iy=\sqrt{a+ib}\,\mbox{.}

By squaring both sides we get

\displaystyle \begin{align*} (x+iy)^2 &= a+ib\,\mbox{,}\\ x^2 - y^2 + 2xy\,i &= a+ib\,\mbox{.}\end{align*}

Matching the real and imaginary parts gives

\displaystyle \left\{\begin{align*} &x^2 - y^2 = a\,\mbox{,}\\ &2xy=b\,\mbox{.}\end{align*}\right.

These equations can be solved by substitution, for example, \displaystyle y= b/(2x) can be inserted in the first equation.


Example 15


Calculate \displaystyle \ \sqrt{-3-4i}\,.


Put \displaystyle \ x+iy=\sqrt{-3-4i}\ where \displaystyle x and \displaystyle y are real numbers. Squaring both sides gives

\displaystyle \begin{align*} (x+iy)^2 &= -3-4i\,\mbox{,}\\ x^2 - y^2 + 2xyi &= -3-4i\,\mbox{,}\end{align*}

which leads to the system of equations

\displaystyle \Bigl\{\begin{align*} x^2 - y^2 &= -3\,\mbox{,}\\ 2xy&= -4\,\mbox{.}\end{align*}

From the second equation, we can solve for \displaystyle \ y=-4/(2x) = -2/x\ and put it into the first equation to get

\displaystyle x^2-\frac{4}{x^2} = -3 \quad \Leftrightarrow \quad x^4 +3x^2 - 4=0\,\mbox{.}

This is a quadratic equation in \displaystyle x^2 which can be seen more easily by putting \displaystyle t=x^2,

\displaystyle t^2 +3t -4=0\,\mbox{.}

The solutions are \displaystyle t = 1 and \displaystyle t = -4. The latter solution must be rejected, as \displaystyle x and \displaystyle y have been assumed to be real numbers, and thus \displaystyle x^2=-4 cannot be true. We get \displaystyle x=\pm\sqrt{1}, which gives us two possible solutions

  • \displaystyle \ x=-1\ which gives \displaystyle \ y=-2/(-1)=2\,,
  • \displaystyle \ x=1\ which gives \displaystyle \ y=-2/1=-2\,.

So we can conclude that

\displaystyle \sqrt{-3-4i} = \biggl\{\begin{align*} &\phantom{-}1-2i\,\mbox{,}\\ &-1+2i\,\mbox{.}\end{align*}

Example 16


  1. Solve the equation \displaystyle \ z^2-2z+10=0\,.

    The formula for solutions of a quadratic equation (see example 3) gives that
    \displaystyle z= 1\pm \sqrt{1-10} = 1\pm \sqrt{-9}= 1\pm 3i\,\mbox{.}
  2. Solve the equation \displaystyle \ z^2 + (4-2i)z -4i=0\,\mbox{.}

    Here, once again , the formula may be used, giving the solutions directly
    \displaystyle \begin{align*} z &= -2+i\pm\sqrt{\smash{(-2+i)^2+4i}\vphantom{i^2}} = -2+i\pm\sqrt{4-4i+i^{\,2}+4i}\\ &=-2+i\pm\sqrt{3} = -2\pm\sqrt{3}+i\,\mbox{.}\end{align*}
  3. Solve the equation \displaystyle \ iz^2+(2+6i)z+2+11i=0\,\mbox{.}

    Division of both sides by \displaystyle i gives
    \displaystyle \begin{align*} z^2 + \frac{2+6i}{i}z +\frac{2+11i}{i} &= 0\,\mbox{,}\\ z^2+ (6-2i)z + 11-2i &= 0\,\mbox{.}\end{align*}

    Applying the formula gives

    \displaystyle \begin{align*} z &= -3+i \pm \sqrt{\smash{(-3+i)^2 -(11-2i)}\vphantom{i^2}}\\ &= -3+i \pm \sqrt{-3-4i}\\ &= -3+i\pm(1-2i)\end{align*}

    where we used the resulting value of \displaystyle \ \sqrt{-3-4i}\ that we obtained in example 15. The solutions are therefore

    \displaystyle z=\biggl\{\begin{align*} &-2-i\,\mbox{,}\\ &-4+3i\,\mbox{.}\end{align*}
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