Bilangan dari bentuk \(a + bi\) dengan \(a\) dan \(b\) adalah bilangan riil disebut dengan bilangan kompleks.
\(i\) = imajiner
\(\color{blue} i = \sqrt{-1}\)
\(\color{blue} i^2 = -1\)
Cara Penulisan
Contoh 1:
\(3 + \sqrt{-4}\)
\(3 + \sqrt{4 \times -1}\)
\(3 + 2\sqrt{-1}\)
\(3 + 2i\)
Contoh 2:
\(7 \:-\: \sqrt{-25}\)
\(7 \:-\: \sqrt{25 \times -1}\)
\(7 \:-\: 5\sqrt{-1}\)
\(7 \:-\: 5i\)
Penjumlahan dan Pengurangan
\(\color{blue} a + bi + c + di = (a + c) + (b + d)i\)
\(\color{blue} a + bi\:-\:(c + di) = (a\:-\:c) + (b\:-\:d)i\)
Contoh 1
Nyatakan dalam bentuk \(a + bi\)
\(12 + 5i + 3 + 2i \:-\: (10 + 4i)\)
\(12 + 5i + 3 + 2i \:-\:10 \:-\:4i\)
\(12 + 3 \:-\:10 + (5 + 2\:-\:4)i\)
\(5 + 3i\)
Contoh 2
Nyatakan dalam bentuk \(a + bi\)
\(6i\:-\:(2\:-\:3i) + 10 + 5i\)
\(6i\:-\:2 + 3i + 10 + 5i\)
\(-2 + 10 + (6 + 3 + 5)i\)
\(8 + 14i\)
Perkalian
\(\color{blue} (a + bi)(c + di)\)
\(\color{blue} ac + (ad + bc)i + bdi^2\)
\(\color{blue} ac + (ad + bc)i \:-\:bd\)
Catatan: \(\color{blue} i^2 = -1\)
Contoh 1
Nyatakan dalam bentuk \(a + bi\)
\((3 + 5i)^2\)
\(3^2 + 2\cdot 3 \cdot 5i + (5i)^2\)
\(9 + 30i + 25i^2\)
\(9 + 30i \:-\:25\)
\(-16 + 30i\)
Contoh 2
Nyatakan dalam bentuk \(a + bi\)
\((3\:-\:6i)(8 + 2i)\)
\(24 + 6i\:-\:48i\:-\:12i^2\)
\(24 \:-\:42i + 12\)
\(36\:-\:42i\)
Pembagian
\(\color{blue} \dfrac{a + bi}{c} = \dfrac{a}{c} + \dfrac{b}{c}i\)
Contoh 1
\(\dfrac{12 + 4i}{2}\)
\(\dfrac{12}{2} + \dfrac{4}{2}i\)
\(6 + 2i\)
\(\color{blue} \dfrac{a + bi}{ci}\)
Kalikan pembilang dan penyebut dengan \(i\)
\(\color{blue} \dfrac{(a + bi)\cdot \color{red}i}{ci\cdot \color{red}i}\)
\(\color{blue} \dfrac{ai + bi^2}{ci^2}\)
\(\color{blue} \dfrac{ai \:-\:b}{-c}\)
\(\color{blue} \dfrac{ai}{-c} + \dfrac{-b}{-c}\)
\(\color{blue} \dfrac{b}{c}\:-\:\dfrac{a}{c}i\)
Contoh 2
\(\dfrac{6 + 9i}{3i}\)
\(\dfrac{(6 + 9i)\cdot \color{red}i}{3i\cdot \color{red}i}\)
\(\dfrac{6i + 9i^2}{3i^2}\)
\(\dfrac{6i\:-\:9}{-3}\)
\(\dfrac{6i}{-3} + \dfrac{-9}{-3}\)
\(-2i + 3\)
\(3\:-\:2i\)
\(\color{blue} \dfrac{a + bi}{c + di}\)
Kalikan pembilang dan penyebut dengan \(c\:-\:di\)
\(\color{blue} \dfrac{a + bi}{c + di}\times \color{red} \dfrac{c\:-\:di}{c\:-\:di}\)
\(\color{blue} \dfrac{(a + bi)(c\:-\:di)}{(c + di)(c\:-\:di)}\)
\(\color{blue} \dfrac{ac\:-\:adi + bci\:-\:bdi^2}{c^2\:-\:cdi + cdi \:-\:d^2i^2}\)
\(\color{blue} \dfrac{ac + (bc\:-\:ad)i+ bd}{c^2 + d^2}\)
\(\color{blue} \dfrac{(ac + bd) + (bc\:-\:ad)i}{c^2 + d^2}\)
Contoh 3a
\(\dfrac{10 + 4i}{5 + 2i}\)
\(\dfrac{10 + 4i}{5 + 2i}\times \color{red} \dfrac{5\:-\:2i}{5\:-\:2i}\)
\(\dfrac{(10 + 4i)(5\:-\:2i)}{(5 + 2i)(5\:-\:2i)}\)
\(\dfrac{50\:-\:20i + 20i\:-\:8i^2}{25\:-\:4i^2}\)
\(\dfrac{50 + 8}{25 + 4}\)
\(\dfrac{58}{29}\)
\(2\)
Contoh 3b
\(\dfrac{8 + 3i}{2\:-\:5i}\)
\(\dfrac{8 + 3i}{2\:-\:5i}\times \color{red} \dfrac{2 + 5i}{2 + 5i}\)
\(\dfrac{(8 + 3i)(2 + 5i)}{(2\:-\:5i)(2 + 5i)}\)
\(\dfrac{16 + 40i + 6i + 15i^2}{4\:-\:25i^2}\)
\(\dfrac{16 + 46i \:-\:15}{4 + 25}\)
\(\dfrac{1 + 46i}{29}\)
\(\dfrac{1}{29} + \dfrac{46}{29}i\)