Unitate irudikari

Matematikan, unitate irudikaria. ${\displaystyle i}$ izendatzen dena, zenbaki irudikarien multzoko unitatea da. Unitate irudikariaren ${\displaystyle i}$ sinboloa Euleri bururatu zitzaion. Hau betetzen du: ${\displaystyle i\times i=i^{2}=-1}$

${\displaystyle i={\sqrt {-1}}}$

i-ren erro karratua

${\displaystyle i}$-ren erro karratua honela idatz daiteke: ${\displaystyle {\sqrt {i}}=\pm {\frac {\sqrt {2}}{2}}(1+i)}$.
Frogapena:

${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}}(1+i)\right)^{2}\ }$	${\displaystyle =\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\ }$
	${\displaystyle =(\pm 1)^{2}{\frac {2}{4}}(1+i)(1+i)\ }$
	${\displaystyle =1\times {\frac {1}{2}}(1+2i+i^{2})\quad \quad (i^{2}=-1)\ }$
	${\displaystyle ={\frac {1}{2}}(2i)\ }$
	${\displaystyle =i\ }$

i-ren berreketak

${\displaystyle i}$-ren berreturek aurretik jakindako eredu bat jarraitzen dute:

${\displaystyle i^{-3}=i}$

${\displaystyle i^{-2}=-1}$

${\displaystyle i^{-1}=-i}$

${\displaystyle i^{0}=1}$

${\displaystyle i^{1}=i}$

${\displaystyle i^{2}=-1}$

${\displaystyle i^{3}=-i}$

${\displaystyle i^{4}=1}$

${\displaystyle i^{5}=i}$

${\displaystyle i^{6}=-1}$

Hori froga daiteke ondorengoarekin, non n zenbaki osoa den:

${\displaystyle i^{4n}=1}$

${\displaystyle i^{4n+1}=i}$

${\displaystyle i^{4n+2}=-1}$

${\displaystyle i^{4n+3}=-i}$

Ikus, gainera

• Zenbaki konplexua
• Zenbaki erreala
• Euleren identitatea

Kanpo estekak

• Datuak: Q193796