Opmerkingen

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\multirowsetup }[0]{\raggedright } \newcommand {\tdplotsinandcos }[3]{\pgfmathsetmacro {#1}{sin(#3)}\pgfmathsetmacro {#2}{cos(#3)}} \newcommand {\tdplotdiv }[3]{\pgfmathsetmacro {#1}{#2/#3}} \newcommand {\tdplotcheckdiff }[5]{\par \par \pgfmathparse { abs(#2 - #1)<#3 } \par \ifthenelse {\equal {\pgfmathresult }{1}}{#4}{#5} } \newcommand {\tdplotsetmaincoords }[2]{\pgfmathsetmacro {\tdplotmaintheta }{#1} \pgfmathsetmacro {\tdplotmainphi }{#2} \tdplotcalctransformmainscreen \tikzset {tdplot_main_coords/.style={x={(\raarot cm,\rbarot cm)},y={(\rabrot cm, \rbbrot cm)},z={(\racrot cm, \rbcrot cm)}}}} \newcommand {\tdplotcalctransformmainscreen }[0]{\tdplotsinandcos {\sintheta }{\costheta }{\tdplotmaintheta }\tdplotsinandcos {\sinphi }{\cosphi }{\tdplotmainphi }\tdplotmult {\stsp }{\sintheta }{\sinphi }\tdplotmult {\stcp }{\sintheta 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\pgfmathsetmacro {\tdplotbeta }{#2} \pgfmathsetmacro {\tdplotgamma }{#3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul } \tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } 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Een complex getal kan per definitie op een unieke manier geschreven worden als $a+bi$, met $a,b\in \R $. Dus $1, i, 1+i, 1+3i$ zijn allemaal complexe getallen die we niet eenvoudiger kunnen schrijven. Maar $2i+3+i, i(1+i)$ en $(1+i)(1-i)$ zijn complexe getallen die we wel eenvoudiger kunnen schrijven als $2i+3+i=3+3i, \quad i(1+i)=i+i^2 =-1+i,\quad (1+i)(1-i) = 1 - i^2 = 1+1 =2$.
Als je twee complexe getallen kan optellen, is het ook eenvoudig om ze van elkaar af te trekken: $(a_1+b_1i)-(a_2+b_2i) = (a_1-a_2)+(b_1-b_2)i$. Je kan twee complexe getallen ook delen door elkaar, dit wordt later uitgelegd.
Ingenieurs gebruiken meestal de letter $j$ in plaats van $i$, onder meer om verwarring te voorkomen met de elektrische stroom $i$. Dan geldt dus $j^2=-1$, en $z=a+bj\in \C $.
Men kan aantonen dat de verzameling $\C $ niet totaal kan geordend worden zoals $\Q $ of $\R $. Hiermee wordt bedoeld dat het niet mogelijk is om een definitie te vinden voor $a+bi <c+di$ die zou voldoen aan dezelfde eigenschappen als de ongelijkheid $<$ in $\R $ of $\Q $. Je kan van twee complexe getallen dus niet zeggen welk "het kleinste" is.
In het bijzonder heeft het geen betekenis om te vragen of $2+3i$ positief of negatief is. Zo kan je ook niet zeggen dat $1+i$ positief is en $-1-i$ negatief, maar wel dat ze elkaars tegengestelde zijn. Hetzelfde geldt zelfs voor $i$: het heeft geen betekenis om te beweren dat $i$ positief is, en $-i$ negatief. Het heeft wel betekenis om te zeggen dat $i$ en $-i$ elkaars tegengestelde zijn.