Materi Pertemuan 2 Matematika Dasar untuk Biologi
❇✐❧❛♥❣❛♥
■♥❞❡❦s
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥✱ ■♥❞❡❦s ❞❛♥ ▲♦❣❛r✐t♠❛
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
❋▼■P❆ ❯◆❨
❙❡♣t❡♠❜❡r ✶✾✱ ✷✵✶✻
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❍✐♠♣✉♥❛♥ ❇✐❧❛♥❣❛♥
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❇✐❧❛♥❣❛♥ ❆s❧✐ ✭◆✮
❇✐❧❛♥❣❛♥ ❇✉❧❛t ✭❩✮
❇✐❧❛♥❣❛♥ ❘❛s✐♦♥❛❧
✭◗✮
❇✐❧❛♥❣❛♥ ❘✐✐❧ ✭❘✮
❇✐❧❛♥❣❛♥ ❑♦♠♣❧❡❦s
✭❈✮
❋✐❣✉r❡✿ ❍✐♠♣✉♥❛♥✲❤✐♠♣✉♥❛♥
❇✐❧❛♥❣❛♥
❘❡❢ ✿
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
●❛r✐s ❇✐❧❛♥❣❛♥
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❋✐❣✉r❡✿ ●❛r✐s ❜✐❧❛♥❣❛♥ r❡❛❧
❘❡❢ ✿ ❤tt♣s✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❘❡❛❧❴♥✉♠❜❡r
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❋❛❦t♦r✐❛❧
❉❡✜♥✐t✐♦♥
❋❛❦t♦r✐❛❧ ❞❛r✐ s✉❛t✉ ❜✐❧❛♥❣❛♥ ❜✉❧❛t ♣♦s✐t✐❢ ♥✱ ♥✦✱ ❞✐❞❡✜♥✐s✐❦❛♥
s❡❜❛❣❛✐ ❤❛s✐❧ ♣❡r❦❛❧✐❛♥ ❞❛r✐ ♥ ❜✐❧❛♥❣❛♥ ❛s❧✐ ♣❡rt❛♠❛✳
n! = n × (n − ✶) × · · · × ✸ × ✷ × ✶
❊①❛♠♣❧❡
❝♦♥t♦❤ ❢❛❦t♦r✐❛❧
✹! = ✹ × ✸ × ✷ × ✶
✼! = ✼ × ✻ × ✺ × ✹ × ✸ × ✷ × ✶
✶! = ✶
✵! = ✶
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇❡♥t✉❦ ❡❦s♣♦♥❡♥s✐❛❧
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❉❡✜♥✐t✐♦♥
ab = a × a × · · · × a
a❞✐s❡❜✉t ❜❛s✐s✱ b ❞✐s❡❜✉t ♣❛♥❣❦❛t✳
❙✐❢❛t ✶
✉♥t✉❦ s❡❜❛r❛♥❣ ❜❛s✐s t❛❦ ♥♦❧ a ❞❛♥ ❜✐❧❛♥❣❛♥ ❛s❧✐ b ❞❛♥ c ✱
ab × ac = ab+c
c
ab = ab×c
ab /ac = ab−c
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
a−b = ✶▼❛t❡♠❛t✐❦❛
/ab
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
✹✹ × ✷✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
✹✹ × ✷✸ = ...
✸✼ //✸✹ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❙❝✐❡♥t✐✜❝ ◆♦t❛t✐♦♥
❉❡✜♥✐t✐♦♥
a × ✶✵b
❞❡♥❣❛♥ ✶ ≤ a < ✶✵
❈♦♥t♦❤
❇✐❧❛♥❣❛♥ ❛✈♦❣❛❞r♦ ✭✻, ✵✷✷ × ✶✵✷✸ ✮
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❙✐♥❣❦❛t❛♥ ✉♥t✉❦ s❛t✉❛♥
❋✐❣✉r❡✿ s❛t✉❛♥
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❘❡❢❡r❡♥s✐
✶
❈❛✉st♦♥✱ ❉❛✈✐❞✳❘✳✱ ▼❛t❡♠❛t✐❦❛ ❉❛s❛r ❯♥t✉❦ ❇✐♦❧♦❣✐✇❛♥✳
●❛❞❥❛❤ ▼❛❞❛ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❞✐t❡r❥❡♠❛❤❦❛♥ ♦❧❡❤ ❉r✳
❙✉❜❛♥❛r ❞❛♥ ❉rs✳ ▼♦❝❤❛♠♠❛❞ ◆❛s✐r✱ ▼✳❙❝
✷
❋♦st❡r✱ P❡t❡r ❈✳✱ ❊❛s② ▼❛t❤❡♠❛t✐❝s ❢♦r ❇✐♦❧♦❣✐sts✳ ❍❛r✇♦♦❞
❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡r✳
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
■♥❞❡❦s
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥✱ ■♥❞❡❦s ❞❛♥ ▲♦❣❛r✐t♠❛
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
❋▼■P❆ ❯◆❨
❙❡♣t❡♠❜❡r ✶✾✱ ✷✵✶✻
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❍✐♠♣✉♥❛♥ ❇✐❧❛♥❣❛♥
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❇✐❧❛♥❣❛♥ ❆s❧✐ ✭◆✮
❇✐❧❛♥❣❛♥ ❇✉❧❛t ✭❩✮
❇✐❧❛♥❣❛♥ ❘❛s✐♦♥❛❧
✭◗✮
❇✐❧❛♥❣❛♥ ❘✐✐❧ ✭❘✮
❇✐❧❛♥❣❛♥ ❑♦♠♣❧❡❦s
✭❈✮
❋✐❣✉r❡✿ ❍✐♠♣✉♥❛♥✲❤✐♠♣✉♥❛♥
❇✐❧❛♥❣❛♥
❘❡❢ ✿
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
●❛r✐s ❇✐❧❛♥❣❛♥
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❋✐❣✉r❡✿ ●❛r✐s ❜✐❧❛♥❣❛♥ r❡❛❧
❘❡❢ ✿ ❤tt♣s✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴❘❡❛❧❴♥✉♠❜❡r
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❋❛❦t♦r✐❛❧
❉❡✜♥✐t✐♦♥
❋❛❦t♦r✐❛❧ ❞❛r✐ s✉❛t✉ ❜✐❧❛♥❣❛♥ ❜✉❧❛t ♣♦s✐t✐❢ ♥✱ ♥✦✱ ❞✐❞❡✜♥✐s✐❦❛♥
s❡❜❛❣❛✐ ❤❛s✐❧ ♣❡r❦❛❧✐❛♥ ❞❛r✐ ♥ ❜✐❧❛♥❣❛♥ ❛s❧✐ ♣❡rt❛♠❛✳
n! = n × (n − ✶) × · · · × ✸ × ✷ × ✶
❊①❛♠♣❧❡
❝♦♥t♦❤ ❢❛❦t♦r✐❛❧
✹! = ✹ × ✸ × ✷ × ✶
✼! = ✼ × ✻ × ✺ × ✹ × ✸ × ✷ × ✶
✶! = ✶
✵! = ✶
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇❡♥t✉❦ ❡❦s♣♦♥❡♥s✐❛❧
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❉❡✜♥✐t✐♦♥
ab = a × a × · · · × a
a❞✐s❡❜✉t ❜❛s✐s✱ b ❞✐s❡❜✉t ♣❛♥❣❦❛t✳
❙✐❢❛t ✶
✉♥t✉❦ s❡❜❛r❛♥❣ ❜❛s✐s t❛❦ ♥♦❧ a ❞❛♥ ❜✐❧❛♥❣❛♥ ❛s❧✐ b ❞❛♥ c ✱
ab × ac = ab+c
c
ab = ab×c
ab /ac = ab−c
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
a−b = ✶▼❛t❡♠❛t✐❦❛
/ab
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
✹✹ × ✷✸ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❈♦♥t♦❤
❇✐❧❛♥❣❛♥
■♥❞❡❦s
✶✵✷ × ✶✵✸ = ...
✷✷ × ✷✸ = ...
✹✹ × ✷✸ = ...
✸✼ //✸✹ = ...
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❙❝✐❡♥t✐✜❝ ◆♦t❛t✐♦♥
❉❡✜♥✐t✐♦♥
a × ✶✵b
❞❡♥❣❛♥ ✶ ≤ a < ✶✵
❈♦♥t♦❤
❇✐❧❛♥❣❛♥ ❛✈♦❣❛❞r♦ ✭✻, ✵✷✷ × ✶✵✷✸ ✮
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❙✐♥❣❦❛t❛♥ ✉♥t✉❦ s❛t✉❛♥
❋✐❣✉r❡✿ s❛t✉❛♥
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐
❇✐❧❛♥❣❛♥
■♥❞❡❦s
❘❡❢❡r❡♥s✐
✶
❈❛✉st♦♥✱ ❉❛✈✐❞✳❘✳✱ ▼❛t❡♠❛t✐❦❛ ❉❛s❛r ❯♥t✉❦ ❇✐♦❧♦❣✐✇❛♥✳
●❛❞❥❛❤ ▼❛❞❛ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❞✐t❡r❥❡♠❛❤❦❛♥ ♦❧❡❤ ❉r✳
❙✉❜❛♥❛r ❞❛♥ ❉rs✳ ▼♦❝❤❛♠♠❛❞ ◆❛s✐r✱ ▼✳❙❝
✷
❋♦st❡r✱ P❡t❡r ❈✳✱ ❊❛s② ▼❛t❤❡♠❛t✐❝s ❢♦r ❇✐♦❧♦❣✐sts✳ ❍❛r✇♦♦❞
❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡r✳
◆✐❦❡♥❛s✐❤ ❇✐♥❛t❛r✐
▼❛t❡♠❛t✐❦❛ ❉❛s❛r ✉♥t✉❦ ❇✐♦❧♦❣✐