z = \textcolor{red}{x} + \textcolor{blue}{y}i- $i = \sqrt{-1}$ er imaginær enhet
- $\textnormal{Re}(z) = \textcolor{red}{x}$ er realdelen til $z$
- $\textnormal{Im}(z) = \textcolor{blue}{y}$ er imaginærdelen til $z$
+ Eksempel: $z=3+4i$
Gitt ett kompleks tall:
z = \textcolor{red}{3} + \textcolor{blue}{4} iRealdelen til $z$ er:
\textcolor{red}{x} = \Re(z) = \textnormal{Re}(z) = \textcolor{red}{3}Imaginærdelen til $z$ er:
\textcolor{blue}{y} = \Im(z) = \textnormal{Im}(z) = \textcolor{blue}{4}Merk at både realdel og imaginærdel til $z$ kan skrives med morsomme krøllbokstaver og med vanlige stive bokstaver.
+ Hva er den kompleks konjugerte?
Hvis du vil finne den konjugerte til $z = x + yi$, bytter du bare fortegn foran $i$:
\overline{z} = \overline{\textcolor{red}{x} \textcolor{blue}{+ y} i} = \textcolor{red}{x} \textcolor{blue}{- y}iStreken over betyr bare at vi vil ha den kompleks konjugerte av det som står under.
Eksempler:
\begin{aligned}
\overline{\textcolor{red}{1} \textcolor{blue}{+} i} = \textcolor{red}{1} \textcolor{bue}{-} i \\
\overline{\textcolor{red}{3} \textcolor{blue}{- 2} i} = \textcolor{red}{3} \textcolor{blue}{+ 2} i \\
\end{aligned}En kul greie:
z\overline{z} = (\textcolor{red}{x} \textcolor{blue}{+ y}i) (\textcolor{red}{x} \textcolor{blue}{- y} i) = x^2 - xyi + xyi - y^2i^2 = \textcolor{red}{x^2} + \textcolor{blue}{y^2}der vi kan bruke tredje kvadratsetning.
Eksempel hvis $z = \textcolor{red}{3} \textcolor{blue}{+ 4} i$:
z \overline{z} = (\textcolor{red}{3} \textcolor{blue}{+ 4} i)(\textcolor{red}{3} \textcolor{blue}{- 4} i)= \textcolor{red}{3^2} + \textcolor{blue}{4^2} = 25+ Addisjon av komplekse tall
Gitt to komplekse tall:
\textcolor{green}{z_1} = \textcolor{green}{x_1 + y_1i} \\
\textcolor{purple}{z_2} = \textcolor{purple}{x_2 + y_2i}Addisjon:
\textcolor{green}{z_1} + \textcolor{purple}{z_2} = (\textcolor{green}{x_1} + \textcolor{purple}{x_2}) + (\textcolor{green}{y_1} + \textcolor{purple}{y_2})i Å legge sammen komplekse tall ligner på hvordan du legger sammen vektorer.
Eksempel:
\textcolor{green}{z_1} = \textcolor{green}{1 + i} \quad \textnormal{ og } \quad
\textcolor{purple}{z_2} = \textcolor{purple}{3 - 2i}\Rightarrow \quad \textcolor{green}{z_1} + \textcolor{purple}{z_2} = (\textcolor{green}{1}+\textcolor{purple}{3}) + (\textcolor{green}{1} - \textcolor{purple}{2})i = 4 - iOg, vips, er vi ferdige!
+ Multiplikasjon av komplekse tall
Gitt to komplekse tall:
z_1 = \textcolor{red}{x_1} + \textcolor{blue}{y_1}i \\
z_2 = \textcolor{red}{x_2} + \textcolor{blue}{y_2}iMultiplikasjon:
\begin{aligned}
z_1 \cdot z_2 & = (\textcolor{red}{x_1} + \textcolor{blue}{y_1}i) \cdot (\textcolor{red}{x_2} + \textcolor{blue}{y_2}i) \\
\Rightarrow \quad z_1 \cdot z_2 & = \textcolor{red}{x_1x_2} + \textcolor{red}{x_1} \textcolor{blue}{y_2}i + \textcolor{red}{x_2} \textcolor{blue}{y_1}i + \textcolor{blue}{y_1y_2} i^2 \\
\end{aligned}Husk at $i^2 = -1$:
\begin{aligned}
\Rightarrow \quad z_1 \cdot z_2 & = \textcolor{red}{x_1x_2} + \textcolor{red}{x_1} \textcolor{blue}{y_2}i + \textcolor{red}{x_2} \textcolor{blue}{y_1}i - \textcolor{blue}{y_1y_2} \\
\Rightarrow \quad z_1 \cdot z_2 & = (\textcolor{red}{x_1x_2} - \textcolor{blue}{y_1y_2}) + (\textcolor{red}{x_1} \textcolor{blue}{y_2} + \textcolor{red}{x_2} \textcolor{blue}{y_1})i
\end{aligned}Eksempel:
z_1 = \textcolor{red}{1} + i \quad \textnormal{ og } \quad
z_2 = \textcolor{red}{3} \textcolor{blue}{- 2}i \\
\begin{aligned} \\
z_1 \cdot z_2 & = (\textcolor{red}{1}+i) \cdot (\textcolor{red}{3} \textcolor{blue}{- 2}i) \\
\Rightarrow \quad z_1 \cdot z_2 & = \textcolor{red}{1} \cdot \textcolor{red}{3} + \textcolor{red}{1} \cdot (\textcolor{blue}{-2}i) + \textcolor{red}{3} \cdot i + i \cdot (\textcolor{blue}{-2}i) \\
\Rightarrow \quad z_1 \cdot z_2 & = 3 - 2i + 3i + 2 \\
\Rightarrow \quad z_1 \cdot z_2 & = 5 + i
\end{aligned}Og, vips, er vi ferdige!
+ Divisjon av komplekse tall
Når vi vil dele to komplekse tall på hverandre, multipliserer vi både teller og nevner med den kompleks konjugerte av nevneren:
\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}}Eksempel:
\frac{\textcolor{red}{2} \textcolor{blue}{+} i}{\textcolor{red}{1} \textcolor{blue}{+ 3} i}
= \frac{(\textcolor{red}{2} \textcolor{blue}{+} i)(\textcolor{red}{1} \textcolor{blue}{- 3} i)}{(\textcolor{red}{1} \textcolor{blue}{+ 3} i)(\textcolor{red}{1} \textcolor{blue}{- 3}i)} = \frac{2 - 6i + i - 3i^2}{\textcolor{red}{1^2} + \textcolor{blue}{3^2}} = \frac{5 - 5i}{10} = \frac{1}{2} - \frac{1}{2}i Og, vips, er vi ferdige!