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Waktu sudah berjalan.","start-part-error":"Bagian belum bisa dimulai. Coba lagi ya.","complete-part-cta":"Selesaikan bagian","complete-part-title":"Selesaikan bagian ini?","complete-part-description":"Kalau kamu selesaikan sekarang, jawaban di bagian ini langsung dikunci.","complete-part-success":"Bagian selesai. Sekarang kamu bisa pilih bagian lain atau kembali ke set.","complete-part-error":"Bagian belum bisa diselesaikan. Coba lagi ya.","back-to-set-cta":"Kembali ke set","slug-description":"Kalau sudah siap, mulai try out ini lalu pilih bagian yang ingin kamu kerjakan dulu.","question-unit":"soal"},"Utils":{"search":"Cari","clear-search":"Hapus pencarian","search-bar-placeholder":"Cari konten...","search-placeholder":"Temukan apa yang kamu inginkan...","search-not-found":"Yah, nggak nemu apa-apa nih.","search-loading":"Lagi nyari...","search-help":"Ketik untuk mencari...","search-error":"Ada masalah saat mencari. Coba lagi ya.","search-title":"Cari Materi Belajar","search-description":"Cari materi belajar gratis, artikel edukasi, latihan soal, dan sumber belajar. Temukan topik yang sedang kamu pelajari."},"Home":{"greeting":"Hi, {name}","title":"Mau belajar apa hari ini?","contribute":"Kontribusi","videos":"Video","trending-subjects":"Materi Populer Minggu Ini","views":"{count} views","explore-materials":"Jelajahi Materi","guest":"kamu","continue-learning":"Lanjutkan Belajar","continue-learning-description":"Lanjutkan membaca materi yang belum selesai","no-recent-views":"Belum ada riwayat bacaan","no-recent-views-description":"Yuk, mulai eksplorasi materi pembelajaran yang tersedia!","start-learning":"Mulai Belajar"},"Marketing":{"about":"Tentang","features":"Fitur","pricing":"Harga","sign-in":"Masuk","sign-up":"Daftar","try-nakafa":"Coba Nakafa"},"Weather":{"loading":"Lagi muat cuaca...","error":"Gagal muat data cuaca","unknown-location":"Lokasi nggak diketahui","unknown-country":"Negara nggak diketahui","high":"T","low":"R"},"Videos":{"discover":"Temukan video"},"NotFound":{"title":"Halaman Nggak Ketemu","description":"Waduh! Halaman yang kamu cari nggak ada atau belum tersedia.","home-button":"Beranda","contribute-button":"Kontribusi"},"Error":{"title":"Waduh, ada kesalahan!","description":"Bantu kami dengan melaporkan masalah ini di GitHub ya.","retry":"Coba Lagi","report":"Lapor"},"Contributor":{"title":"Kontributor","description":"Temui orang-orang yang membuat Nakafa menjadi mungkin."},"ComingSoon":{"title":"Segera Hadir","description":"Fitur ini belum tersedia. Silakan periksa kembali nanti atau berkontribusi pada proyek.","contribute":"Berkontribusi"},"Auth":{"continue-with-google":"Masuk dengan Google","logout":"Keluar","login":"Masuk","login-cta-title":"Masuk","login-cta-action":"Akun Nakafa gratis","account":"Akun","settings":"Pengaturan","settings-description":"Kelola pengaturan akun Nakafa, preferensi, dan informasi profil untuk mempersonalisasi pengalaman belajarmu.","profile":"Profil","profile-description":"Jelajahi profil belajar pengguna dengan kontribusi edukasi, komentar bermakna, pelacakan progres belajar, dan keterlibatan komunitas dalam pembelajaran kolaboratif.","chat":"Obrolan","chat-description":"Terhubung dengan asisten AI belajar untuk bimbingan personal, bantuan PR, penjelasan konsep, dan percakapan edukasi interaktif yang disesuaikan dengan kebutuhan belajarmu.","subscriptions":"Langganan","subscriptions-description":"Kelola Nakafa Pro, 3.000 kredit AI bulanan, dan portal pembayaranmu.","get-pro":"Dapatkan Pro","manage":"Kelola","plan-free":"Free","plan-pro":"Pro","name":"Nama","name-description":"Namamu akan muncul di profil dan komentar.","name-footer":"Gunakan 3 sampai 32 karakter ya.","name-placeholder":"Masukkan namamu...","role":"Peran","role-description":"Peranmu membantu kami menyesuaikan pengalaman belajarmu.","role-footer":"Kamu bisa mengubah peran kapan saja.","role-placeholder":"Pilih peran","curriculum":"Kurikulum","curriculum-description":"Preferensi kurikulummu menjaga pintasan mata pelajaran tetap sesuai dengan jalur yang paling sering kamu pakai.","curriculum-footer":"Kamu bisa mengubah kurikulum kapan saja.","curriculum-placeholder":"Pilih kurikulum","delete-account":"Hapus akun","delete-account-description":"Hapus profil dan data pribadimu dari Nakafa secara permanen.","delete-account-footer":"Tindakan ini tidak bisa dibatalkan.","delete-account-dialog-title":"Hapus akun?","delete-account-dialog-description":"Profil dan data pribadimu akan dihapus permanen. Langganan aktif akan dibatalkan, akses Nakafa Pro berakhir, dan sekolah yang kamu buat akan dialihkan ke anggota aktif lain.","delete-account-session-expired":"Sesi ini sudah terlalu lama. Masuk lagi untuk melanjutkan.","delete-account-school-member-required":"Tambahkan satu anggota aktif ke setiap sekolah yang kamu buat, lalu coba lagi.","delete-account-error":"Akunmu belum bisa dihapus. Tunggu sebentar, lalu coba lagi.","delete-account-sign-in-again":"Masuk lagi","teacher":"Guru","student":"Siswa","parent":"Orang tua","administrator":"Administrator","save":"Simpan","developers":"Developer","developers-description":"Akses alat developer, API key, dan sumber integrasi untuk membangun aplikasi dengan konten dan layanan edukasi Nakafa.","api-key":"API Key","api-key-description":"Kelola API keys untuk autentikasi dengan API Nakafa.","api-key-footer":"Buat API key baru di sini.","api-key-confirmation-description":"Simpan API key kamu baik-baik ya, jangan sampai hilang.","new-api-key":"API Key Baru","api-key-created":"Sip, API key berhasil dibuat!","no-api-keys":"Kamu belum punya API key.","create":"Buat","api-key-name":"Nama API Key","api-key-name-placeholder":"Masukkan nama API key kamu...","expires-in":"Kadaluarsa dalam","expires-in-placeholder":"Hari hingga kadaluarsa...","7-days":"7 hari","30-days":"30 hari","90-days":"90 hari","180-days":"180 hari","365-days":"365 hari","never":"Tidak pernah","expires":"Kadaluarsa","api-key-copied":"API key udah disalin!","delete-api-key":"Hapus API Key","delete-api-key-description":"Yakin mau hapus API key ini? Nggak bisa dibalikin lho.","delete":"Hapus","api-key-deleted":"API key udah dihapus."},"EventAccess":{"title":"Akses event partner","unavailable-invalid-code":"Link event ini nggak valid atau sudah nggak tersedia.","unavailable-disabled":"Akses event ini sedang dimatikan untuk sementara.","unavailable-not-started":"Akses event ini belum dibuka. Coba lagi nanti ya.","unavailable-ended":"Akses event ini sudah berakhir.","sign-in-description":"Masuk untuk klaim akses.","ready-description":"Aktifkan untuk mulai.","active-until":"Aktif sampai {date}","ended-at":"Berakhir {date}","sign-in-cta":"Masuk","redeem-cta":"Aktifkan","already-active-cta":"Sudah aktif","view-tryout-cta":"Lihat","redeem-success":"Akses event aktif sampai {date}.","redeem-error":"Akses event belum bisa diaktifkan. Coba lagi ya."},"Ai":{"ask-nina":"Tanya Nina","text-placeholder":"Tanya Nina apa saja...","error-message":"Waduh, ada masalah nih. Coba ganti model AI-nya ya.","insufficient-credits":"Kredit AI kamu habis. Tunggu reset kredit berikutnya atau upgrade ke Pro untuk 3.000 kredit bulanan.","retry":"Coba lagi","rate-limit-message":"Kamu terlalu cepat! Coba lagi nanti ya.","nakafa":"Nakafa","nakafa-search":"hasil Nakafa","nakafa-search-results":"Konten ditemukan","nakafa-search-articles":"Artikel ditemukan","nakafa-search-subject":"Materi ditemukan","nakafa-search-tryout":"Try out ditemukan","nakafa-search-quran":"Ayat ditemukan","nakafa-search-empty":"Tidak ada konten yang cocok.","nakafa-content":"Materi","nakafa-exercise":"Latihan","nakafa-quran":"Quran","nakafa-quran-verse-count":"{count, plural, other {# ayat}}","question-number":"Soal {number}","nakafa-loading":"Lagi membaca {kind}...","nakafa-error":"Gagal membaca {kind}.","math-apart":"Mengurai pecahan","math-cancel":"Membatalkan faktor","math-circle":"Mencari lingkaran","math-combination":"Menghitung kombinasi","math-compare":"Membandingkan","math-cumulative_probability":"Mencari peluang kumulatif","math-determinant":"Mencari determinan","math-differentiate":"Menurunkan","math-distance":"Menghitung jarak","math-distribution":"Mengecek distribusi","math-domain":"Mencari domain","math-eigen_analysis":"Menganalisis struktur eigen","math-eigenvalues":"Mencari nilai eigen","math-eigenvectors":"Mencari vektor eigen","math-evaluate":"Menghitung","math-expected_value":"Menghitung ekspektasi","math-expand":"Mengembangkan","math-factor":"Memfaktorkan","math-gcd":"Mencari FPB","math-integrate":"Mengintegralkan","math-intersection":"Mencari irisan","math-inverse":"Mencari invers","math-interval_probability":"Mencari peluang rentang","math-is_prime":"Mengecek prima","math-lcm":"Mencari KPK","math-limit":"Mencari limit","math-line":"Mencari garis","math-linear_system":"Menyelesaikan sistem","math-matrix_multiply":"Mengalikan matriks","math-mean":"Mencari rata-rata","math-median":"Mencari median","math-midpoint":"Mencari titik tengah","math-mode":"Mencari modus","math-modular":"Menghitung sisa","math-permutation":"Menghitung permutasi","math-point_probability":"Mencari peluang","math-prime_factorization":"Memfaktorkan prima","math-product":"Mengalikan deret","math-quartiles":"Mencari kuartil","math-rank":"Mencari rank","math-rationalize":"Merasionalkan","math-roots":"Mencari akar","math-rref":"Mereduksi matriks","math-series":"Menguraikan deret","math-simplify":"Menyederhanakan","math-slope":"Mencari gradien","math-solve":"Menyelesaikan","math-standard_deviation":"Mencari simpangan","math-summation":"Menjumlahkan deret","math-tail_probability":"Mencari peluang ekor","math-together":"Menggabungkan pecahan","math-variance":"Mencari varians","math-variance_probability":"Mencari varians","math-z_score":"Mencari z-score","math-error":"Bagian ini belum bisa dicek. Tulis ekspresi atau data matematikanya lebih lengkap.","math-loading":"Mengolah perhitungan...","math-step":"Langkah {number}","math-relation-differentiate":"turunannya","math-relation-limit":"limitnya","math-relation-solve":"solusinya","math-compare-verified":"sama dengan","math-compare-contradicted":"berbeda dengan","math-compare-inconclusive":"belum terbukti sama dengan","math-input":"Diketahui","math-condition":"Syarat","math-item-algebraic-multiplicity":"Multiplisitas aljabar","math-item-counterexample":"Contoh pembeda","math-item-diagonalizable":"Dapat didiagonalkan","math-item-domain":"Domain","math-item-eigenbasis":"Basis eigenspace","math-item-eigenvalue":"Nilai eigen","math-item-eigenvector":"Vektor eigen","math-item-factor":"Faktor","math-item-geometric-multiplicity":"Multiplisitas geometri","math-item-mode":"Modus","math-item-approximation":"Perkiraan","math-item-q1":"Kuartil bawah","math-item-q2":"Median","math-item-q3":"Kuartil atas","math-item-result":"Hasil","math-item-root":"Akar","math-item-singularity":"Titik bermasalah","math-item-solution":"Solusi","math-item-status":"Kesimpulan","math-value-divergent":"Tidak konvergen","math-value-no":"Tidak","math-value-yes":"Ya","thinking":"Berpikir...","thought-for":"Berpikir selama {duration} detik","thought-for-a-few-seconds":"Berpikir selama beberapa detik","follow-up":"Lanjutkan","copy-message":"Salin pesan","retry-message":"Ulangi pesan","see":"Lihat","scrape":"Scrape","scrape-loading":"Lagi membaca sumber...","scrape-done":"Sumber dibaca","scrape-error":"Gagal membaca sumber.","web-search":"Pencarian web","web-search-loading":"Lagi nyari info terkait...","web-search-error":"Gagal nemu info terkait nih.","web-search-empty":"Tidak ada sumber yang didapatkan.","model":"Model","model-subtitle-nakafa-lite":"Belajar harian","model-subtitle-nakafa-pro":"Penalaran mendalam","show-more":"Lebih banyak","show-less":"Lebih sedikit","new-chat-title":"Mau belajar apa hari ini?","new-chat-description":"Tanya aku aja ya, jangan ragu-ragu!","suggestion-current-material":"materi ini","suggestion-explain":"Jelasin ini","suggestion-explain-prompt":"Jelaskan {title} dengan bahasa yang mudah dipahami siswa.","suggestion-example":"Buat contoh soal","suggestion-example-prompt":"Buatkan contoh soal tentang {title} beserta pembahasan langkah demi langkah.","suggestion-summary":"Ambil intinya","suggestion-summary-prompt":"Ringkas {title} menjadi ide penting yang perlu aku ingat.","new-chat":"Obrolan baru","chats":"Obrolan","search-chats":"Cari obrolan kamu...","rename-chat":"Ubah nama obrolan","delete-chat":"Hapus obrolan","delete-chat-description":"Yakin ingin menghapus obrolan ini? Nggak bisa dikembalikan lho.","confirm":"Konfirmasi","share-chat":"Bagikan obrolan","share-chat-description":"Hati-hati saat membagikan link ke orang asing ya.","private":"Pribadi","private-description":"Obrolan hanya dapat dilihat oleh kamu.","public":"Publik","public-description":"Obrolan dapat dilihat oleh siapa saja.","create-link":"Buat link","copy-link":"Salin link","link-copied":"Link berhasil disalin!","private-chat-error":"Akses ditolak","private-chat-error-description":"Ini bukan obrolanmu. Minta akses ke pemiliknya dulu ya.","credits-label":"Kredit","token-usage":"Penggunaan Token","input-tokens":"Input","output-tokens":"Output","total-tokens":"Total","recent-chats":"Obrolan Terbaru","speed":"Kecepatan pemutaran"},"Pricing":{"headline":"Kita juga pernah jadi pelajar. Ngerti rasanya milih antara belajar atau makan. Akses dasar gratis. Selamanya.","free-title":"Gratis","free-description":"Mulai belajar sesuai kecepatanmu","free-feature-1":"Akses ke 5.000+ materi","free-feature-2":"2.000+ latihan soal dengan pembahasan","free-feature-3":"Tutor AI dasar (10 pesan per hari)","free-feature-4":"1 try out lengkap gratis","free-feature-5":"Akses komunitas di Discord","free-cta":"Mulai Belajar","pro-title":"Pro","pro-description":"Belajar lebih serius dengan Nina, 3.000 kredit AI bulanan, dan akses try out berulang.","pro-period":"/bulan","pro-feature-1":"Semua fitur Gratis, plus:","pro-feature-2":"3.000 kredit AI setiap bulan untuk model premium Nina","pro-feature-3":"Mulai set try out lain atau ulangi try out, dengan skor IRT pada set yang mendukung dan pembahasan detail","pro-feature-4":"Download materi sebagai PDF dan dengarkan versi audio","pro-feature-5":"Bantuan prioritas saat kamu butuh arahan","pro-cta":"Dapatkan Pro"},"Faq":{"badge":"Bantuan","headline":"Masih ada pertanyaan?","description":"Tidak menemukan yang kamu cari? Hubungi kami langsung.","cta-contact":"Hubungi founder","q1":"Apakah Nakafa benar-benar gratis?","a1":"Ya. Konten inti dan 1 try out lengkap gratis. Pro diperlukan untuk mulai set try out lain atau mengulang try out, memakai model AI premium, mendapat 3.000 kredit AI bulanan, dan mengunduh PDF. Materi pembelajaran dasar tetap tanpa paywall.","q2":"Model AI apa yang bisa saya gunakan?","a2":"Semua pengguna bisa memakai Nakafa Lite dan Nakafa Pro. Kredit menentukan seberapa banyak tiap model bisa digunakan.","q3":"Bisakah saya mengunduh pelajaran untuk offline?","a3":"Unduhan PDF adalah fitur Pro. Kamu bisa menyimpan pelajaran apa pun sebagai PDF untuk belajar offline atau mencetak. Pengguna gratis selalu bisa mengakses konten online.","q4":"Mata pelajaran dan kelas apa yang tercakup?","a4":"Kami mencakup SD hingga universitas. Matematika, Fisika, Kimia, Biologi, Ilmu Komputer, dan lainnya. Dari konsep dasar SD hingga kursus universitas tingkat lanjut.","q5":"Apakah data saya pribadi?","a5":"Ya. Kami tidak menjual data kamu. Percakapan AI diproses dengan aman. Kamu bisa menghapus riwayat chat kapan saja. Kami mengutamakan privasi.","q6":"Bisakah sekolah menggunakan Nakafa?","a6":"Tentu saja. Kami menawarkan solusi kustom untuk kelas dan institusi pendidikan. Hubungi founder untuk integrasi LMS, manajemen siswa, dan branding kustom.","q7":"Bagaimana Nakafa memastikan akurasi konten?","a7":"Semua konten dibuat oleh kontributor yang berkualifikasi dan ditinjau oleh komunitas. Pendekatan source-available kami membuat siapa pun bisa meninjau materi dan mengusulkan perbaikan. Latihan soal punya pembahasan supaya kamu belajar dari kesalahan, bukan hanya skor."},"Features":{"story":"Belajar tidak berhenti setelah kamu selesai membaca. Coba pemahamanmu lewat Tryout, lihat yang masih bikin ragu, lalu tanya Nina tanpa mulai dari awal.","subjects-title":"Mulai dari pelajaran yang kamu butuhkan","subjects-navigation":"Pelajaran yang tersedia","tryout-title":"Setelah belajar, coba lihat apa yang benar-benar kamu pahami","nina-title":"Kalau masih ragu, Nina sudah tahu bagian mana yang kamu maksud","nina-prompt":"Pada operasi $$\\triangle(a,b)=ka+\\frac{b}{m}$$, diketahui $$\\triangle(2,4)=8$$ dan $$\\triangle(3,5)=11$$. Bagaimana cara mencari $$\\triangle(5,9)$$?","nina-reasoning":"Bandingkan dua persamaan yang diketahui. Selisihnya menunjukkan nilai bulat positif $$k$$ dan $$m$$ sebelum perhitungan terakhir.","nina-answer":"Kurangi persamaan pertama dari persamaan kedua:\n\n$$k+\\frac{1}{m}=3$$\n\nKarena $$k$$ dan $$m$$ bilangan bulat positif, $$\\frac{1}{m}$$ hanya bisa bulat saat $$m=1$$. Jadi $$k=2$$.\n\nMasukkan keduanya ke operasi terakhir:\n\n$$\\triangle(5,9)=5(2)+\\frac{9}{1}=19$$\n\nJadi $$\\triangle(5,9)=19$$.","projectile-title":"Gerak yang sulit dibayangkan jadi lebih mudah saat bisa kamu lihat sendiri","projectile-controls":"Pilih lintasan bola meriam","projectile-sixty-degree":"Enam Puluh Derajat","projectile-long-drive":"Jarak Jauh","projectile-high-arc":"Lengkung Tinggi","projectile-view-label":"Visual analisis gerak parabola bola meriam","projectile-horizontal-component":"Komponen mendatar","projectile-vertical-component":"Komponen vertikal","projectile-peak-time":"Waktu puncak","projectile-flight-time":"Waktu terbang","projectile-range":"Jangkauan","projectile-instantaneous-velocity":"Kecepatan pada dua detik"},"CurriculaSection":{"headline":"Kurikulummu boleh berbeda. Alur belajarmu tetap menyatu.","description":"Mulai dari jalur sekolahmu. Materi, Tryout, dan Nina mengikuti kurikulum yang kamu gunakan.","cta":"Lihat semua kurikulum","navigation":"Kurikulum yang tersedia"},"SchoolsSection":{"badge":"Untuk Sekolah","headline":"Setiap sekolah punya kebutuhan berbeda. Kami mengerti.","description":"Butuh LMS, integrasi konten, atau alur kelas khusus? Pilih yang cocok untuk sekolahmu.","cta":"Diskusikan kebutuhanmu"},"TrustSection":{"headline":"Kamu tidak harus percaya begitu saja. Ikuti pelajarannya sampai ke sumber.","description":"Baca penjelasannya, ikuti rujukannya, atau buka sumber materinya saat kamu ingin memeriksa lebih jauh.","open-lesson":"Buka materi ini","open-lesson-short":"Buka materi","open-source":"Buka .md","human-label":"Manusia","agent-label":"Agen","comparison-slider-label":"Atur lebar tampilan manusia dan agen","lesson-preview-label":"Tampilan materi Nakafa","source-preview-label":"Sumber Markdown Nakafa","lesson-heading":"Dari Kertas hingga Pandemi","lesson-opening-before-folds":"Pernahkah kamu membayangkan melipat selembar kertas sebanyak","lesson-folds-math":"42 \\text{ kali}","lesson-opening-after-folds":"? Jika mungkin dilakukan, ketebalannya akan melebihi jarak bumi ke bulan! Ini karena setiap lipatan menghasilkan penggandaan ketebalan kertas, inilah yang disebut","lesson-growth-term":"pertumbuhan eksponensial","lesson-growth-before-year":"Pertumbuhan eksponensial terjadi ketika sesuatu bertambah dengan faktor pengali tetap dalam setiap interval waktu. Pada awal","lesson-year-math":"2020","lesson-growth-after-year":", dunia mengalami contoh nyata pertumbuhan eksponensial melalui penyebaran virus COVID-19. Satu orang terinfeksi dapat menularkan ke dua orang, kemudian empat, delapan, dan seterusnya.","lesson-definition-heading":"Definisi Eksponen","lesson-definition":"Eksponen adalah cara singkat untuk menuliskan perkalian berulang. Bayangkan kamu sedang menghitung berapa banyak orang yang tertular virus dalam kasus seperti COVID-19. Pada setiap fase penularan, jumlah orang yang tertular akan bertambah dengan pola yang menarik:","lesson-sequence-math":"1 = 2^0 \\quad 2 = 2^1 \\quad 4 = 2 \\times 2 = 2^2 \\quad 8 = 2 \\times 2 \\times 2 = 2^3 \\quad 16 = 2 \\times 2 \\times 2 \\times 2 = 2^4"},"About":{"meta-title":"Materi Belajar Terbuka Gratis dari SD sampai Universitas","title":"Belajar sampai paham.","description":"Pilih pelajaran, uji pemahaman lewat Tryout, lalu tanya Nina saat kamu buntu.","explore-subjects":"Jelajahi pelajaran","try-tryout":"Coba Tryout","contribute":"Kontribusi","start-learning":"Mulai belajar","join-community":"Bergabung komunitas","community-title":"Ketemu orang yang ngerti apa yang kamu pelajari","community-description":"Di balik tiap materi ada orang yang pernah bingung juga. 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(x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Pelajari ekspansi binomial dengan contoh i²=-1.", authors: [{ name: "Nabil Akbarazzima Fatih" }], date: "2025-05-01", subject: "Bilangan Kompleks", }; ## Mengalikan Dua Bilangan Kompleks Mengalikan dua bilangan kompleks mirip seperti mengalikan dua bentuk aljabar binomial. Kita bisa menggunakan sifat distributif perkalian terhadap penjumlahan. Mari kita lihat bagaimana cara mengalikan dengan . Ingat bahwa , jadi kita bisa substitusi: Sekarang, kita kelompokkan bagian real dan bagian imajiner: Jadi, rumus umum untuk perkalian bilangan kompleks adalah: ## Contoh Perhitungan Misalkan dan . Tentukan . **Penyelesaian:** Menggunakan sifat distributif: Atau menggunakan rumus umum dengan : Hasilnya sama! Visualisasi dari ,{" "} , dan hasil perkaliannya{" "} . } cameraPosition={[0, 0, 15]} showZAxis={false} data={[ { points: [ { x: 0, y: 0, z: 0 }, { x: 2, y: 1, z: 0 }, ], color: "#0284c7", labels: [{ text: "z₁ = 2+i", at: 1, offset: [0.5, 0.5, 0] }], cone: { position: "end" }, }, { points: [ { x: 0, y: 0, z: 0 }, { x: 1, y: -2, z: 0 }, ], color: "#059669", labels: [{ text: "z₂ = 1-2i", at: 1, offset: [0.5, -0.5, 0] }], cone: { position: "end" }, }, { points: [ { x: 0, y: 0, z: 0 }, { x: 4, y: -3, z: 0 }, ], color: "#e11d48", labels: [{ text: "z₁ × z₂ = 4-3i", at: 1, offset: [0.5, -0.5, 0] }], cone: { position: "end" }, }, ]} /> ## Latihan Misalkan dan . Tentukan . ### Kunci Jawaban Menggunakan sifat distributif: Menggunakan rumus umum dengan : 36:o13c,5c:["$","$L62",null,{"content":"$63","slug":"/id/materi/matematika/bilangan-kompleks/perkalian-bilangan-kompleks","sourceUrl":"https://github.com/nakafaai/aksara/blob/a48d644c809419c93dc247239eabba9ced519051/packages/corpus/material/lesson/mathematics/complex-number/multiplication-complex-numbers/id.mdx"}] 64:Tab0,36:oab0,

z_1 = x_1 + iy_1

36:o87d,5d:["$","article",null,{"className":"mx-auto max-w-3xl px-6","children":[false,[["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"mengalikan-dua-bilangan-kompleks","children":["$","a",null,{"aria-label":"Link to Mengalikan Dua Bilangan Kompleks","className":"group/heading inline-flex items-center gap-4","href":"#mengalikan-dua-bilangan-kompleks","title":"Mengalikan Dua Bilangan Kompleks","children":[["$","span",null,{"className":"text-pretty","children":"Mengalikan Dua Bilangan Kompleks"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}],"\n",["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Mengalikan dua bilangan kompleks mirip seperti mengalikan dua bentuk aljabar binomial. Kita bisa menggunakan sifat distributif perkalian terhadap penjumlahan."}],"\n",["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Mari kita lihat bagaimana cara mengalikan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$64"}}]}]," dengan ","$L65","."]}],"\n","$L66","\n","$L67","\n","$L68","\n","$L69","\n","$L6a","\n","$L6b","\n","$L6c","\n","$L6d","\n","$L6e","\n","$L6f","\n","$L70","\n","$L71","\n","$L72","\n","$L73","\n","$L74","\n","$L75","\n","$L76","\n","$L77","\n","$L78","\n","$L79","\n","$L7a","\n","$L7b","\n","$L7c"]]}] 36:o1b8,5f:["$","footer",null,{"className":"relative py-20","children":["$","div",null,{"className":"z-10 mx-auto max-w-3xl px-6","children":["$","$L7d",null,{"slug":"material/lesson/mathematics/complex-number/multiplication-complex-numbers"}]}]}] 60:["$","$L7e",null,{"audio":{"contentType":"material","locale":"id","slug":"material/lesson/mathematics/complex-number/multiplication-complex-numbers"},"contextTitle":"Perkalian Bilangan Kompleks"}] 36:o7a7,61:["$","aside",null,{"children":["$","$L7f",null,{"cookieName":"sidebar_state:right","keyboardShortcut":"x","sidebarDesktop":1280,"children":[["$","$L80",null,{"className":"fixed top-20 right-6 size-9 bg-background/80 backdrop-blur-xs xl:hidden","icon":[["path",{"d":"M4 5L16 5","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":"1.5","key":"0"}],["path",{"d":"M4 12L20 12","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":"1.5","key":"1"}],["path",{"d":"M4 19L12 19","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":"1.5","key":"2"}]],"size":"icon","variant":"outline"}],["$","$L81",null,{"containerClassName":"lg:hidden xl:block","side":"right","children":[["$","$L82",null,{"children":["$","$L83",null,{"children":["$","$L84",null,{"children":["$","$L85",null,{"render":["$","$L46",null,{"href":"/id/materi/matematika/bilangan-kompleks/perkalian-bilangan-kompleks","title":"Perkalian Bilangan Kompleks"}],"size":"lg","children":["$","div",null,{"className":"grid flex-1 text-left text-sm leading-tight","children":[["$","span",null,{"className":"truncate font-medium","children":"Perkalian Bilangan Kompleks"}],["$","$L86",null,{"children":"Kalikan bilangan kompleks dengan sifat distributif dan rumus (x₁x₂-y₁y₂)+i(x₁y₂+x₂y₁). Pelajari ekspansi binomial dengan contoh i²=-1."}]]}]}]}]}]}],["$","$L87",null,{"children":["$","$L88",null,{"data":[{"label":"Mengalikan Dua Bilangan Kompleks","href":"#mengalikan-dua-bilangan-kompleks","children":[]},{"label":"Contoh Perhitungan","href":"#contoh-perhitungan","children":[]},{"label":"Latihan","href":"#latihan","children":[{"label":"Kunci Jawaban","href":"#kunci-jawaban","children":[]}]}],"title":"Pada halaman ini"}]}],["$","$L89",null,{"className":"border-t","children":["$","$L83",null,{"children":[["$","$L8a",null,{}],false,"$L8b","$L8c",["$","$L8d",null,{}]]}]}]]}]]}]}] 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36:o8e2,c7:I[776218,["/_next/static/immutable/chunks/2ozbzbkbjrs5g.js","/_next/static/immutable/chunks/1u94c24vqx26q.js","/_next/static/immutable/chunks/44psnuid4hcf5.js","/_next/static/immutable/chunks/2b2z9fhnbh5e0.js","/_next/static/immutable/chunks/0ni87mxribyly.js","/_next/static/immutable/chunks/0g7ln_b7f132w.js","/_next/static/immutable/chunks/1bapa2m2ksez8.js","/_next/static/immutable/chunks/0ugejllqhnrwj.js","/_next/static/immutable/chunks/1gze7y7ht-33b.js","/_next/static/immutable/chunks/393_w_-eb8uv-.js","/_next/static/immutable/chunks/0chv-zggdaw_f.js","/_next/static/immutable/chunks/287vic71a6h_7.js","/_next/static/immutable/chunks/2tmwfq8jsbwhd.js","/_next/static/immutable/chunks/136a_3pwwx3sd.js","/_next/static/immutable/chunks/1j7t38u2zaqis.js","/_next/static/immutable/chunks/02shfbk-zzfca.js","/_next/static/immutable/chunks/2d0hfm5dv_pzb.js","/_next/static/immutable/chunks/1k-yr9yysem7x.js","/_next/static/immutable/chunks/06f0haa4q81t6.js","/_next/static/immutable/chunks/0kbnvbl7eptep.js","/_next/static/immutable/chunks/3brmn4mcym1z7.js","/_next/static/immutable/chunks/0_ln2_6lkmx32.js","/_next/static/immutable/chunks/2tvds67b366ea.js","/_next/static/immutable/chunks/442lhk44e6ykq.js","/_next/static/immutable/chunks/3b6wkamawtv3r.js","/_next/static/immutable/chunks/2rdc3wpvx-0y9.js","/_next/static/immutable/chunks/3b0fdgj_zftzq.js","/_next/static/immutable/chunks/27ei9vnzeiq85.js","/_next/static/immutable/chunks/014aoh0qn81eo.js","/_next/static/immutable/chunks/1c00lzvzs706d.js","/_next/static/immutable/chunks/3ywgyv40llk3m.js","/_next/static/immutable/chunks/058g38eq00elg.js","/_next/static/immutable/chunks/1rug2kwvfis1u.js","/_next/static/immutable/chunks/21b3i3nowssv-.js","/_next/static/immutable/chunks/2qkbg9ig3vnaf.js","/_next/static/immutable/chunks/2zves0p1ggtcw.js","/_next/static/immutable/chunks/3cpp7c7yc-8su.js","/_next/static/immutable/chunks/1l-5amv9gm5ec.js","/_next/static/immutable/chunks/2yr4tqc4fd3s6.js","/_next/static/immutable/chunks/36ofm1pm6eeme.js","/_next/static/immutable/chunks/05s1116x2xxgz.js","/_next/static/immutable/chunks/34f45g3k3ibf_.js","/_next/static/immutable/chunks/1uw8afofehcs7.js","/_next/static/immutable/chunks/3wms78zzv6zr5.js","/_next/static/immutable/chunks/3s93rqpoi-ptk.js"],"Button"] 36:o8,8e:Tab0,36:oab0,

z_2 = x_2 + iy_2

36:oa0,65:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$8e"}}]}] 90:T1504,36:o1504,

z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)

36:o841,66:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$90"}}]}],"$L91"]}]}],"$L92","$L93"]}] 94:T454,

i^2 = -1

67:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Ingat bahwa ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$94"}}]}],", jadi kita bisa substitusi:"]}] 95:T19f0,36:o19f0,

= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)

36:o360,68:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-compact","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$95"}}]}],"$L96"]}]}],"$L97"]}] 69:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Sekarang, kita kelompokkan bagian real dan bagian imajiner:"}] 98:T1933,36:o1933,

= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)

36:o35f,6a:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-compact","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$98"}}]}],"$L99"]}]}],"$L9a"]}] 6b:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Jadi, rumus umum untuk perkalian bilangan kompleks adalah:"}] 9b:T1f2b,36:o1f2b,

z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

36:o5f4,6c:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9b"}}]}],"$L9c"]}]}] 6d:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"contoh-perhitungan","children":["$","a",null,{"aria-label":"Link to Contoh Perhitungan","className":"group/heading inline-flex items-center gap-4","href":"#contoh-perhitungan","title":"Contoh Perhitungan","children":[["$","span",null,{"className":"text-pretty","children":"Contoh Perhitungan"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] 9d:T613,36:o613,

z_1 = 2+i

36:o8,9e:T63d,36:o63d,

z_2 = 1-2i

36:o31b,6e:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Misalkan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9d"}}]}]," dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9e"}}]}],". Tentukan ","$L9f","."]}] 6f:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["$","strong",null,{"className":"font-medium","children":"Penyelesaian:"}]}] 70:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Menggunakan sifat distributif:"}] a0:Tc20,36:oc20,

z_1 \times z_2 = (2+i)(1-2i)

36:o2d6,71:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a0"}}]}],"$La1"]}]}],"$La2","$La3","$La4","$La5","$La6","$La7"]}] a8:T10f2,36:o10f2,

x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2

36:o139,72:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Atau menggunakan rumus umum dengan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a8"}}]}],":"]}] a9:T1f2b,36:o1f2b,

z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

36:o348,73:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a9"}}]}],"$Laa"]}]}],"$Lab","$Lac","$Lad","$Lae"]}] 74:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Hasilnya sama!"}] af:T613,36:o613,

z_1 = 2+i

36:o8,b0:T63d,36:o63d,

z_2 = 1-2i

36:o837,75:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":"Visualisasi Perkalian Bilangan Kompleks"}],["$","div",null,{"className":"text-muted-foreground text-sm","data-slot":"card-description","children":["Visualisasi dari ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$af"}}]}],","," ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$b0"}}]}],", dan hasil perkaliannya"," ","$Lb1","."]}]]}],"$Lb2"]}] 76:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"latihan","children":["$","a",null,{"aria-label":"Link to Latihan","className":"group/heading inline-flex items-center gap-4","href":"#latihan","title":"Latihan","children":[["$","span",null,{"className":"text-pretty","children":"Latihan"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] b3:T613,36:o613,

z_1 = 1+i

36:o8,b4:T9ed,36:o9ed,

z_2 = \frac{1}{2} - 2i

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z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)

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x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2

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z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

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36:o3b,91:["$","$Lc8",null,{"orientation":"horizontal"}] c9:T14b5,36:o14b5,

= x_1(x_2 + iy_2) + iy_1(x_2 + iy_2)

36:o181,92:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$c9"}}]}],"$Lca"]}]}] cb:T1ae3,36:o1ae3,

= (x_1x_2 + ix_1y_2) + (ix_2y_1 + i^2y_1y_2)

36:o1b3,93:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$cb"}}]}],"$Lcc"]}]}] 96:["$","$Lc8",null,{"orientation":"horizontal"}] cd:T1868,36:o1868,

= x_1x_2 + ix_1y_2 + ix_2y_1 - y_1y_2

36:o1b3,97:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$cd"}}]}],"$Lce"]}]}] 99:["$","$Lc8",null,{"orientation":"horizontal"}] cf:T1902,36:o1902,

= (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

36:o1b2,9a:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$cf"}}]}],"$Ld0"]}]}] 9c:["$","$Lc8",null,{"orientation":"horizontal"}] d1:T73d,36:o73d,

z_1 \times z_2

36:od1,9f:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d1"}}]}] a1:["$","$Lc8",null,{"orientation":"horizontal"}] d2:T776,36:o776,

= 2(1-2i) + i(1-2i)

36:o1a8,a2:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d2"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] d3:T8b1,36:o8b1,

= (2 - 4i) + (i - 2i^2)

36:o1a8,a3:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d3"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] d4:Tb99,36:ob99,

= (2 - 4i) + (i - 2(-1)) \quad \text{(karena } i^2 = -1)

36:o180,a4:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d4"}}]}],"$Ld5"]}]}] d6:T6f9,36:o6f9,

= (2 - 4i) + (i + 2)

36:o1a8,a5:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d6"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] d7:T71d,36:o71d,

= (2+2) + (-4i + i)

36:o1a0,a6:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d7"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] 36:o58e,a7:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

= 4 - 3i

"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] 36:o3a,aa:["$","$Lc8",null,{"orientation":"horizontal"}] d8:Taba,36:oaba,

= (2)(1) - (1)(-2) + i((2)(-2) + (1)(1))

36:o1a8,ab:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d8"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] d9:T797,36:o797,

= (2 - (-2)) + i(-4 + 1)

36:o1a8,ac:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$d9"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] da:T5ee,36:o5ee,

= (2+2) + i(-3)

36:o1a0,ad:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$da"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] 36:o58e,ae:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

= 4 - 3i

"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] 36:o8,db:T9ad,36:o9ad,

z_1 \times z_2 = 4-3i

36:o385,b1:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$db"}}]}] b2:["$","div",null,{"className":"px-6","data-slot":"card-content","children":["$","$Ldc",null,{"cameraPosition":[0,0,15],"showZAxis":false,"children":[["$","$Ldd","line-0,0,0;2,1,0",{"points":[{"x":0,"y":0,"z":0},{"x":2,"y":1,"z":0}],"color":"#0284c7","labels":[{"text":"z₁ = 2+i","at":1,"offset":[0.5,0.5,0]}],"cone":{"position":"end"}}],["$","$Ldd","line-0,0,0;1,-2,0",{"points":[{"x":0,"y":0,"z":0},{"x":1,"y":-2,"z":0}],"color":"#059669","labels":[{"text":"z₂ = 1-2i","at":1,"offset":[0.5,-0.5,0]}],"cone":{"position":"end"}}],["$","$Ldd","line-0,0,0;4,-3,0",{"points":[{"x":0,"y":0,"z":0},{"x":4,"y":-3,"z":0}],"color":"#e11d48","labels":[{"text":"z₁ × z₂ = 4-3i","at":1,"offset":[0.5,-0.5,0]}],"cone":{"position":"end"}}]]}]}] de:T73d,36:o73d,

z_1 \times z_2

36:od1,b5:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$de"}}]}] b7:["$","$Lc8",null,{"orientation":"horizontal"}] df:Tdd8,36:odd8,

= 1(\frac{1}{2} - 2i) + i(\frac{1}{2} - 2i)

36:o180,b8:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$df"}}]}],"$Le0"]}]}] e1:Tf35,36:of35,

= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2i^2)

36:o180,b9:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$e1"}}]}],"$Le2"]}]}] e3:Te26,36:oe26,

= (\frac{1}{2} - 2i) + (\frac{1}{2}i - 2(-1))

36:o180,ba:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$e3"}}]}],"$Le4"]}]}] e5:Td7d,36:od7d,

= (\frac{1}{2} - 2i) + (\frac{1}{2}i + 2)

36:o180,bb:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$e5"}}]}],"$Le6"]}]}] e7:Tda3,36:oda3,

= (\frac{1}{2} + 2) + (-2i + \frac{1}{2}i)

36:o181,bc:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$e7"}}]}],"$Le8"]}]}] e9:T1401,36:o1401,

= (\frac{1}{2} + \frac{4}{2}) + (-\frac{4}{2}i + \frac{1}{2}i)

36:o180,bd:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$e9"}}]}],"$Lea"]}]}] eb:Ta22,36:oa22,

= \frac{5}{2} - \frac{3}{2}i

36:o1db,be:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$eb"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] c1:["$","$Lc8",null,{"orientation":"horizontal"}] ec:T1152,36:o1152,

= ((1)(\frac{1}{2}) - (1)(-2)) + i((1)(-2) + (\frac{1}{2})(1))

36:o180,c2:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$ec"}}]}],"$Led"]}]}] ee:Tdf5,36:odf5,

= (\frac{1}{2} - (-2)) + i(-2 + \frac{1}{2})

36:o181,c3:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$ee"}}]}],"$Lef"]}]}] f0:T10a1,36:o10a1,

= (\frac{1}{2} + 2) + i(-\frac{4}{2} + \frac{1}{2})

36:o180,c4:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$f0"}}]}],"$Lf1"]}]}] f2:Tf7d,36:of7d,

= (\frac{1}{2} + \frac{4}{2}) + i(-\frac{3}{2})

36:o180,c5:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$f2"}}]}],"$Lf3"]}]}] f4:Ta22,36:oa22,

= \frac{5}{2} - \frac{3}{2}i

36:o1a0,c6:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L8f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$f4"}}]}],["$","$Lc8",null,{"orientation":"horizontal"}]]}]}] 36:o2ee,ca:["$","$Lc8",null,{"orientation":"horizontal"}] cc:["$","$Lc8",null,{"orientation":"horizontal"}] ce:["$","$Lc8",null,{"orientation":"horizontal"}] d0:["$","$Lc8",null,{"orientation":"horizontal"}] d5:["$","$Lc8",null,{"orientation":"horizontal"}] e0:["$","$Lc8",null,{"orientation":"horizontal"}] e2:["$","$Lc8",null,{"orientation":"horizontal"}] e4:["$","$Lc8",null,{"orientation":"horizontal"}] e6:["$","$Lc8",null,{"orientation":"horizontal"}] e8:["$","$Lc8",null,{"orientation":"horizontal"}] ea:["$","$Lc8",null,{"orientation":"horizontal"}] ed:["$","$Lc8",null,{"orientation":"horizontal"}] ef:["$","$Lc8",null,{"orientation":"horizontal"}] f1:["$","$Lc8",null,{"orientation":"horizontal"}] f3:["$","$Lc8",null,{"orientation":"horizontal"}] 36:o3a,2c:C 2a:C 2b:C 23:C 25:C 26:C 27:C 28:C 2e:false 2d:77722 36:C 9b:Tab0,

z_2 = x_2 + iy_2

72:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9b"}}]}] 9d:T1504,

z_1 \times z_2 = (x_1 + iy_1)(x_2 + iy_2)

73:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9d"}}]}],"$L9e"]}]}],"$L9f","$La0"]}] a1:T454,

i^2 = -1

74:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Ingat bahwa ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a1"}}]}],", jadi kita bisa substitusi:"]}] a2:T19f0,

= (x_1x_2 + ix_1y_2) + (ix_2y_1 + (-1)y_1y_2)

75:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-compact","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a2"}}]}],"$La3"]}]}],"$La4"]}] 76:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Sekarang, kita kelompokkan bagian real dan bagian imajiner:"}] a5:T1933,

= (x_1x_2 - y_1y_2) + (ix_1y_2 + ix_2y_1)

77:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-compact","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a5"}}]}],"$La6"]}]}],"$La7"]}] 78:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Jadi, rumus umum untuk perkalian bilangan kompleks adalah:"}] a8:T1f2b,

z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

79:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a8"}}]}],"$La9"]}]}] 7a:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"contoh-perhitungan","children":["$","a",null,{"aria-label":"Link to Contoh Perhitungan","className":"group/heading inline-flex items-center gap-4","href":"#contoh-perhitungan","title":"Contoh Perhitungan","children":[["$","span",null,{"className":"text-pretty","children":"Contoh Perhitungan"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] aa:T613,

z_1 = 2+i

ab:T63d,

z_2 = 1-2i

7b:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Misalkan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$aa"}}]}]," dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$ab"}}]}],". Tentukan ","$Lac","."]}] 7c:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["$","strong",null,{"className":"font-medium","children":"Penyelesaian:"}]}] 7d:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Menggunakan sifat distributif:"}] ad:Tc20,

z_1 \times z_2 = (2+i)(1-2i)

7e:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$ad"}}]}],"$Lae"]}]}],"$Laf","$Lb0","$Lb1","$Lb2","$Lb3","$Lb4"]}] b5:T10f2,

x_1 = 2, y_1 = 1, x_2 = 1, y_2 = -2

7f:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Atau menggunakan rumus umum dengan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$b5"}}]}],":"]}] b6:T1f2b,

z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

80:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$b6"}}]}],"$Lb7"]}]}],"$Lb8","$Lb9","$Lba","$Lbb"]}] 81:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Hasilnya sama!"}] bc:T613,

z_1 = 2+i

bd:T63d,

z_2 = 1-2i

82:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":"Visualisasi Perkalian Bilangan Kompleks"}],["$","div",null,{"className":"text-muted-foreground text-sm","data-slot":"card-description","children":["Visualisasi dari ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$bc"}}]}],","," ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$bd"}}]}],", dan hasil perkaliannya"," ","$Lbe","."]}]]}],"$Lbf"]}] 83:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"latihan","children":["$","a",null,{"aria-label":"Link to Latihan","className":"group/heading inline-flex items-center gap-4","href":"#latihan","title":"Latihan","children":[["$","span",null,{"className":"text-pretty","children":"Latihan"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] c0:T613,

z_1 = 1+i

c1:T9ed,

z_2 = \frac{1}{2} - 2i

84:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Misalkan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$c0"}}]}]," dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$c1"}}]}],". Tentukan ","$Lc2","."]}] 85:["$","h3",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-xl","id":"kunci-jawaban","children":["$","a",null,{"aria-label":"Link to Kunci Jawaban","className":"group/heading inline-flex items-center gap-4","href":"#kunci-jawaban","title":"Kunci Jawaban","children":[["$","span",null,{"className":"text-pretty","children":"Kunci Jawaban"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] 86:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Menggunakan sifat distributif:"}] c3:Tf51,

z_1 \times z_2 = (1+i)(\frac{1}{2} - 2i)

87:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$c3"}}]}],"$Lc4"]}]}],"$Lc5","$Lc6","$Lc7","$Lc8","$Lc9","$Lca","$Lcb"]}] cc:T14a0,

x_1 = 1, y_1 = 1, x_2 = \frac{1}{2}, y_2 = -2

88:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Menggunakan rumus umum dengan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$cc"}}]}],":"]}] cd:T1f2b,

z_1 \times z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)

89:["$","div",null,{"className":"my-4 space-y-0 last:mb-0 *:data-math-block:rounded-none *:data-math-block:border-b-0 [&>[data-math-block]:first-child]:rounded-t-xl [&>[data-math-block]:last-child]:rounded-b-xl [&>[data-math-block]:last-child]:border-b content-auto-math-stack-spacious","children":[["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9c",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$cd"}}]}],"$Lce"]}]}],"$Lcf","$Ld0","$Ld1","$Ld2","$Ld3"]}] 6b:["$","nav",null,{"aria-label":"Pagination navigation","className":"mt-10 pt-10","children":["$","div",null,{"className":"mx-auto grid max-w-3xl gap-6 px-6 sm:grid-cols-2","children":[["$","$Ld4",null,{"className":"group flex h-auto flex-col whitespace-normal py-3 shadow-xs items-start","nativeButton":false,"render":["$","$L4e",null,{"href":"/id/materi/matematika/bilangan-kompleks/modulus-dan-argumen-bilangan-kompleks?ctx=united-states~high-school-mathematics-number-quantity","title":"Modulus dan Argumen Bilangan Kompleks","children":[["$","div",null,{"className":"flex items-center gap-2 font-normal text-muted-foreground text-sm transition-colors group-hover:text-accent-foreground","children":[["$","svg",null,{"ref":"$undefined","xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M5.5 12.002H19","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}],["$","path","1",{"d":"M10.9999 18.002C10.9999 18.002 4.99998 13.583 4.99997 12.0019C4.99996 10.4208 11 6.00195 11 6.00195","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}]]}],"Sebelumnya",false]}],["$","p",null,{"className":"w-full text-foreground transition-colors group-hover:text-accent-foreground","children":"Modulus dan Argumen Bilangan Kompleks"}]]}],"variant":"outline"}],["$","$Ld4",null,{"className":"group flex h-auto flex-col whitespace-normal py-3 shadow-xs items-end","nativeButton":false,"render":["$","$L4e",null,{"href":"/id/materi/matematika/bilangan-kompleks/argumen-utama-bilangan-kompleks?ctx=united-states~high-school-mathematics-number-quantity","title":"Argumen Utama Bilangan Kompleks","children":[["$","div",null,{"className":"flex items-center gap-2 font-normal text-muted-foreground text-sm transition-colors group-hover:text-accent-foreground","children":[false,"Selanjutnya",["$","svg",null,{"ref":"$undefined","xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M18.5 12L4.99997 12","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}],["$","path","1",{"d":"M13 18C13 18 19 13.5811 19 12C19 10.4188 13 6 13 6","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}]]}]]}],["$","p",null,{"className":"w-full text-foreground transition-colors group-hover:text-accent-foreground text-right","children":"Argumen Utama Bilangan Kompleks"}]]}],"variant":"outline"}]]}]}] 98:["$","$L91",null,{"children":["$","$L92",null,{"render":["$","a",null,{"aria-label":"Lapor","href":"https://github.com/nakafaai/nakafa.com/issues/new/choose","rel":"noopener noreferrer","target":"_blank","title":"Lapor","children":[["$","svg",null,{"ref":"$undefined","xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M3.01206 6.59744C2.89214 7.65674 3.6584 10.1951 5.87693 10.255","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2,"opacity":"$undefined"}],["$","path","1",{"d":"M18.2891 10.195C19.3684 10.255 20.9873 8.99582 20.9873 6.59741","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2,"opacity":"$undefined"}],["$","path","2",{"d":"M20.0876 20.9879C20.1476 19.6088 19.2482 17.4742 17.5093 17.4142","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2,"opacity":"$undefined"}],["$","path","3",{"d":"M3.90138 20.9881C3.84009 19.5998 4.7594 17.4509 6.53675 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