89sistem bilangan.pdf
TRANSCRIPT
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SISTEMBILANGAN
NurEdy,PhD.
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SubPokokBahasan
• Bilanganriildansifat-sifatnya• Bilangankompleks
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BILANGANREAL
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SistemBilanganRealBILANGAN
REAL
BILANGAN IRASIONAL
BILANGAN RASIONAL
BILANGAN BULAT BIL RATIONAL
TIDAK BULAT
BIL BULAT NEGATIF
NOL : 0 BIL BULAT POSITIF
BIL BULAT TAK NEGATIF
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OPERASIBILANGANBULAT
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Penjumlahan
• Sifat-sifatkomutaFf: a+b=b+a
– Contoh:2+5=5+2=7
• SifatasosiaFf:(a+b)+c=a+(b+c)– Contoh:(-4)+6=6+(-4)=2
• SifatidenFtas a+0=a=0+a– Contoh:2+0=2=0+2
• Elemeninvers:a+(-a)=0– Contoh:5+(-5)=0
• SifatDistribusi:x(y+z)=xy+xz– Contoh:5(5+5)=5x5+5x5=50
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Operasipadabilanganreal
Pengurangan• a–b–c=a–(b+c)
– Contoh:54–27–10=54–(27+10)=17
• a–(b–c)=a–b+c – Contoh:37–(21–8)=37–21+8=24
• px(a–b)=(pxa)–(pxb)– Contoh:2x(7–3)=(2x7)–(2x3)=8
• (a+b)–c=a+(b–c)– Contoh:(3+4)–2=3+(4–2)
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Perkalian
• SifatkomutaFfaxb=bxa– Contoh:2x3=3x2
• SifatasosiaFf(axb)xc=ax(bxc)– Contoh:(2x3)x4=2x(3x4)
• SifatidenFtasax1=a=1xa– Contoh:5x1=5=1x5
• Elemeninversax(1/a)=1– Contoh:6x(1/6)=1
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Pembagian
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Pembagian
Pembagian
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OPERASIBILANGANPECAHAN
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Penjumlahan
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Penjumlahan
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Pengurangan
3/10
21-2-16
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Perkalian
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Perkalian
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Pembagian
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KONVERSIPECAHAN,PERBANDINGAN,SKALA,PERSEN
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KonversiPecahan
• AgarpengerFankonversidapatdipahamidenganbaikmakauntukmengkonversikanpecahanbiasakebentukpersendapatdilakukandenganmembagipembilangnyaolehpenyebutpadapecahankemudiandikalikan100%.
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KonversiPecahan
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KonversiPecahan
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Perbandingan• Perbandinganduabuahnilaidapatdinyatakansebagai
pembagianataupecahanbiasa.
• Secaraumumperbandinganantarabesaranaterhadapbdituliskansebagai
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JenisPerbandinganPerbandinganSenilai• Disebutsebagaiperbandignansenilaijikaduaperbandinganharganyasama.
Contoh:• 5literminyakmempunyaimassa4kgdan10literminyakmempunyaimassa8kg.Perbandinganantarakualitasminyakdanmassanyadituliskansebagaiberikut:5:10=4:8atau1:2=1:2
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JenisPerbandingan
PerbandinganSenilaiContoh:Mobildengankecepatantetap60km/jam
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JenisPerbandingan
PerbandinganBerbalikNilai• Contoh:Suatupekerjaanjikadikerjakanoleh1orangakanseleaidalam60hari,jika2orang30hari,3orang30haridanseterusnya.
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Skala
• Skalaadalahperbandinganantarajarak(panjangpadagambar)danjarak(panjangsebenarnya).
Contoh:Skalapita=1:200.000Maksudnyajikajarakpadagambar1cmmakajarakpadabumi(sebenarnya)200.000cm.
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Skala
QUIZ• Diketahui:– Skala1:200.000– Jarak2kotapadagambar7,5cm
• BerapaKMjaraksesungguhnya2kotatersebut?
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Skala
Jawab:• Jaraksesungguhnya=7,5x200.000=1500.000cm=15000m=15km
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Skala
QUIZ• Diketahui:– Skala1:200.000– Jarakduakota60km.
• Berapakahjarakpadagambar?
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Skala
Jawab:• 60km=60.000m=6.000.000cm
• Jarakpeta=6.000.000/200.000=60/2=30cm
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Persen
• Persenadalahbentuklaindaripecahanyangpenyebutnyaseratus.
• Simbolyangdigunakanuntukmenyatakanpersenadalah“%”.
• Misalnya2%arFnya2/100• Untukmengubahbentukpersenmenjadipecahandilakukandenganjalanmembagipersentersebutdengan100%.
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Persen
• Contoh
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Persen
• Untukmenyatakanperbandinganantaraduabesaranpersentasedapatditentukandenganpertolonganpernyataanperbandingansehingga
• Besaranpertama:besarankedua=persentase:100%
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Persen
• Contoh
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OperasipadaBilanganBerpangkat
Konsepoperasipadabilanganberpangkat• a3arFnyaaxaxasebanyak3faktor• a3dibacaaberpangkatFga• Secaraumum:anarFnyaaxaxax...asebanyaknfaktor.
• adisebutbilanganberpangkat• adisebutbilangandasarpokok• 3disebutpangkatataueksponenContoh:25=2x2x2x2x2=32
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OperasipadaBilanganBerpangkat
Perkalianbilanganberpangkatyangbilanganpokoknyasama
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OperasipadaBilanganBerpangkat
Pembagianbilanganberpangkat• Siapamaujawab?
• 54:52
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OperasipadaBilanganBerpangkat
Pembagianbilanganberpangkat
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OperasipadaBilanganBerpangkat
Pemangkatanbilanganberpangkat
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OperasipadaBilanganBerpangkat
PemangkatanSuatuPecahan
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OperasipadaBilanganBerpangkat
Bilanganberpangkatnol
50 = 1
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OperasipadaBilanganBerpangkat
PangkatNegaFf
8-2 8-2
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OPERASIPADABILANGANIRRASIONAL(BENTUKAKAR)
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Akar
• Akarmerupakanlawandaripangkatdengantandayangdipunyaiolehsuatubilangan
• adalahuntukmenunjukkanbahwapangkatdaribilangantadidibagiolehindeksyangterdapatpadatandaakar.
• Secaraumumdapatdituliskan:
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Akar
denganmadalahindeksakar
• PenulisanakaryangFdakdisertaidenganindeksberarFindeksdariakartersebutadalah2.Misalnya√3arFnyasamadengan3½
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
Operasiakardilakukansebagaiberikut
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Akar
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Akar
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AkarJawab
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AkarJawab
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BILANGANKOMPLEKS
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BILANGANKOMPLEKS
• BilanganImajiner(j)• BentukBilanganKompleks• OperasiBilanganKompleksBentukBaku• OperasiBilanganKompleksBentukPolar
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KonsepDasar
• Bilanganimajiner(j)merupakanakarkuadratdarisuatubilangannegaFf
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BentukBilanganKompleks
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