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Nombor khayalan

Sistem nombor matematik
Asas
$\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C}$ Nombor asli $\mathbb {N}$ Nombor negatif $\mathbb {Z} ^{-}$ Integer $\mathbb {Z}$ Nombor nisbah $\mathbb {Q}$ Nombor bukan nisbah $\mathbb {Q} '$ Nombor nyata $\mathbb {R}$ Nombor khayalan $\mathbb {I}$ Nombor kompleks $\mathbb {C}$ Nombor algebra ${\overline {\mathbb {Q} }}$ Nombor transenden $\mathbb {T}$
Perluasan kompleks
Nombor dwikompleks Nombor hiperkompleks Kuaternion $\mathbb {H}$ Kokuaternion Bikuaternion Oktonion $\mathbb {O}$ Sedenion $\mathbb {S}$ Tesarina Hipernombor Nombor supernyata Nombor hipernyata Nombor sureal
Lain-lain
Nombor kompleks belah $\mathbb {R} ^{1,1}$ Nombor bersiri Nombor melampaui terhingga Nombor ordinal Nombor kardinal $\aleph _{n}$ Nombor perdana $\mathbb {P}$ Nombor p-adik Nombor boleh bina Nombor boleh kira Jujukan integer Pemalar matematik $x$ Nombor besar Pi $\pi$ Nombor Euler $e$ Unit khayalan $i$ Ketakterhinggaan $\infty$

Dalam matematik, nombor khayalan merupakan nombor nyata yang didarabi dengan unit khayalan $i$ , dimana $i^{2}=-1$ dan $i={\sqrt {-1}}$ .

Seperti, $21i$ , $-7i$ dan $(2/5)i$ . Apabila nombor khayalan digabungkan dengan nombor nyata, ia terhasil nombor yang dikenali sebagi nombor kompleks. Semua nombor khayalan dalam satah Cartes dipeta pada paksi khayalan yang berserenjang dengan paksi nyata.

Sifat

Nombor khayalan berkongsi hanya sikit persamaan dengan nombor nyata biasa.

Penambahan dan Penolakan

$ai+bi=(a+b)i$
$ai-bi=(a-b)i$

dimana $a,b\in \mathbb {R}$

Pendaraban

$ai\times bi=-ab$

dimana $a,b\in \mathbb {R}$

Pembahagian

${\frac {ai}{bi}}={\frac {a}{b}}$
${\frac {a}{bi}}=-{\frac {a}{b}}i$

dimana $a,b\in \mathbb {R}$

Nilai Mutlak

Nilai mutlak untuk suatu nilai $x$ , ditakrifkan sebagai jarak daripada 0 ke $x$ atas satah Cartes. Mengikut definisi itu, nilai mutlak bagi $xi$ adalah;

$|xi|=x$

Logaritma Asli

Rumusan logaritma asli bagi nombor kompleks boleh didapati melalui bentuk polar;

$z=re^{i\theta }$

dimana $e$ adalah nombor Euler, $z\in \mathbb {C}$ , $r=|z|$ , $\theta$ adalah argumen dan $i$ ialah unit khayalan.

$z=re^{i\theta }$
$\mathrm {ln} (z)=\mathrm {ln} (re^{i\theta })$
$\mathrm {ln} (z)=\mathrm {ln} (r)+\mathrm {ln} (e^{i\theta })$
$\mathrm {ln} (z)=\mathrm {ln} (r)+i\theta +2\pi n$
$\mathrm {ln} (z)=\mathrm {ln} (|z|)+i\theta +2\pi n$

Jadi,

$\mathrm {ln} (xi)=\mathrm {ln} (x)+{\frac {\pi }{2}}i+2\pi n$

Sejarah

Nombor khayalan telah ditakrifkan pada tahun 1572 oleh Rafael Bombelli. Pada masa itu, nombor khayalan tidak difikirkan wujud, sama seperti nombor sifar dan negatif yang difikirkan tidak wujud dan tidak berguna.

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