Akrab bersama sandi matematika
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    Akrab bersama sandi matematika Akrab bersama sandi matematika Document Transcript

    • AKRAB BERSAMA SANDI MATEMATIKA Oleh : Fithri Angelia Permana, S.Si (WI LPMP NAD) lpmp-aceh.com/download/download.php?fileId=46Apakah matematika ilmu yang sulit?Secara umum, semakin kompleks suatu fenomena, semakin kompleks pula alat (dalam hal inijenis matematika) yang melalui berbagai perumusan (model matematikanya) diharapkan mampuuntuk mendapatkan atau sekedar mendekati solusi eksak seakurat-akuratnya.Jadi tingkatkesulitan suatu jenis atau cabang matematika bukan disebabkan oleh jenis atau cabangmatematika itu sendiri, tetapi disebabkan oleh sulit dan kompleksnya fenomena yang solusinyadiusahakan dicari atau didekati oleh perumusan (model matematikanya) dengan menggunakanjenis atau cabang matematika tersebut.Sebaliknya berbagai fenomena fisik yg mudah di amati, misalnya jumlah penduduk di seluruhIndonesia, tak memerlukan jenis atau cabang matematika yang canggih. Kemampuan aritmatikasudah cukup untuk mencari solusi (jumlah penduduk) dengan keakuratan yang cukuptinggi.Dalam matematika sering digunakan simbol-simbol yang umum dikenal olehmatematikawan. Sering kali pengertian simbol ini tidak dijelaskan, karena dianggap maknanyatelah diketahui. Hal ini kadang menyulitkan bagi mereka yang awam. Daftar berikut ini berisibanyak simbol beserta artinya.Matematika sebagai bahasaDi manakah letak semua konsep-konsep matematika, misalnya letak bilangan 1? Banyak parapakar matematika, misalnya para pakar Teori Model (lihat model matematika) yg jugamendalami filosofi di balik konsep-konsep matematika bersepakat bahwa semua konsep-konsepmatematika secara universal terdapat di dalam pikiran setiap manusia.Jadi yang dipelajari dalammatematika adalah berbagai simbol dan ekspresi untuk mengkomunikasikannya. Misalnya orangJawa secara lisan memberi simbol bilangan 3 dengan mengatakan "Telu", sedangkan dalam
    • bahasa Indonesia, bilangan tersebut disimbolkan melalui ucapan "Tiga". Inilah sebabnya, banyakpakar mengkelompokkan matematika dalam kelompok bahasa, atau lebih umum lagi dalamkelompok (alat) komunikasi, bukan sains.Dalam pandangan formalis, matematika adalah penelaahan struktur abstrak yang didefinisikansecara aksioma dengan menggunakan logika simbolik dan notasi matematika; ada pulapandangan lain, misalnya yang dibahas dalam filosofi matematika.Struktur spesifik yang diselidiki oleh matematikawan sering kali berasal dari ilmu pengetahuanalam, dan sangat umum di fisika, tetapi matematikawan juga mendefinisikan dan menyelidikistruktur internal dalam matematika itu sendiri, misalnya, untuk menggeneralisasikan teori bagibeberapa sub-bidang, atau alat membantu untuk perhitungan biasa. Akhirnya, banyakmatematikawan belajar bidang yang dilakukan mereka untuk sebab estetis saja, melihat ilmupasti sebagai bentuk seni daripada sebagai ilmu praktis atau terapan.Matematika tingkat lanjutdigunakan sebagai alat untuk mempelajari berbagai fenomena fisik yg kompleks, khususnyaberbagai fenomena alam yang teramati, agar pola struktur, perubahan, ruang dan sifat-sifatfenomena bisa didekati atau dinyatakan dalam sebuah bentuk perumusan yg sistematis dan penuhdengan berbagai konvensi, simbol dan notasi. Hasil perumusan yang menggambarkan prilakuatau proses fenomena fisik tersebut biasa disebut model matematika dari fenomena.Faktanya, kita melihat bahwa siswa kelas rendah (umumnya siswa kelas 1) lebih menyenangipelajaran matematika dibanding pelajaran bahasa Indonesia. Hal ini, ”mungkin” karenamatematika merupakan ilmu yang mudah dimengerti dengan lambang bilangan yang sederhanaserta dekat dengan kehidupan si siswa. Fakta juga menunjukkan bahwa setelah siswa berada dikelas tinggi, mereka malah membenci matematika sampai kepada guru yang mengajarkannya.Banyak faktor yang mendukung terjadinya hal ini, guru yang kurang mampu mentransferilmunya, materi yang terlalu abstrak dan kode atau sandi di matematika yang tidak begitufamiliar. Seseorang yang tidak paham sandi/kode dari lambang matematika akan mengalamikesulitan untuk memahami maksud permasalahan matematika yang ada. Ada beberapasandi/kode matematika yang ditampilkan untuk menambah wawasan kita bersama.Simbol matematika dasar
    • Namambol Dibaca sebagai Penjelasan Contoh Kategori kesamaan x = y berarti x and y mewakili hal = sama dengan atau nilai yang sama. 1+1=2 Umum Ketidaksamaan x ≠ y berarti x dan y tidak mewakili ≠ tidak sama dengan hal atau nilai yang sama. 1≠2 Umum ketidaksamaan < lebih kecil dari; lebih x < y berarti x lebih kecil dari y. 3<4 besar dari 5>4 x > y means x lebih besar dari y. > order theory inequality ≤ lebih kecil dari atau x ≤ y berarti x lebih kecil dari atau sama dengan y. sama dengan, lebih 3 ≤ 4 and 5 ≤ 5 besar dari atau sama 5 ≥ 4 and 5 ≥ 5 x ≥ y means x lebih besar dari atau dengan ≥ order theory sama dengan y. tambah 4 + 6 berarti jumlah antara 4 dan tambah 2+7=9 6. aritmatika + disjoint union A1={1,2,3,4} ∧ A2={2,4,5,7} ⇒ the disjoint union of A1 + A2 means the disjoint union of A1 + A2 = {(1,1), (2,1), (3,1), … and … sets A1 and A2. (4,1), (2,2), (4,2), (5,2), teori himpunan (7,2)} kurang Kurang 9 − 4 berarti 9 dikurangi 4. 8−3=5 − aritmatika tanda negatif −3 berarti negatif dari angka 3. −(−5) = 5 Negative
    • aritmatika set-theoretic complement A − B means the set that contains all the elements of A that are not {1,2,4} − {1,3,4} = {2} minus; without in B. set theory multiplication 3 × 4 means the multiplication of 3 Kali 7 × 8 = 56 by 4. aritmatika Cartesian product the Cartesian X×Y means the set of all ordered product of … and …; pairs with the first element of {1,2} × {3,4} =× the direct product of each pair selected from X and the {(1,3),(1,4),(2,3),(2,4)} … and … second element selected from Y. teori himpunan cross product u × v means the cross product of (1,2,5) × (3,4,−1) = Cross vectors u and v (−22, 16, − 2) vector algebra division÷ bagi 2 ÷ 4 = .5 6 ÷ 3 atau 6/3 berati 6 dibagi 3. 12/4 = 3/ aritmatika square root √x berarti bilangan positif yang akar kuadrat √4 = 2 kuadratnya x. bilangan real√ complex square root if z = r exp(iφ) is represented in the complex square polar coordinates with -π < φ ≤ π, √(-1) = i root of; square root then √z = √r exp(iφ/2). bilangan complex absolute value |x| means the distance in the real |3| = 3, |-5| = |5||| absolute value of line (or the complex plane) between x and zero. |i| = 1, |3+4i| = 5 numbers factorial n! is the product 1×2×...×n. 4! = 1 × 2 × 3 × 4 = 24!
    • faktorial combinatorics probability distribution X ~ D, means the random variable X ~ N(0,1), the standard~ has distribution X has the probability distribution D. normal distribution statistika material implication A ⇒ B means if A is true then B is⇒ implies; if .. then also true; if A is false then nothing is said about B. → may mean the same as ⇒, or it x = 2 ⇒ x2 = 4 is true, but→ may have the meaning for functions given below. x2 = 4 ⇒ x = 2 is in general false (since x could be −2). propositional logic ⊃ may mean the same as ⇒, or it⊃ may have the meaning for superset given below.⇔ material equivalence if and only if; iff A ⇔ B means A is true if B is true x + 5 = y +2 ⇔ x + 3 = y and A is false if B is false. propositional logic↔ logical negation The statement ¬A is true if and¬ not only if A is false. ¬(¬A) ⇔ A A slash placed through another x ≠ y ⇔ ¬(x = y) propositional logic operator is the same as "¬" placed˜ in front. logical conjunction or meet in a lattice The statement A ∧ B is true if A n < 4 ∧ n >2 ⇔ n = 3∧ and and B are both true; else it is false. when n is a natural number. propositional logic, lattice theory logical disjunction or join in a lattice The statement A ∨ B is true if A or n≥4 ∨ n≤2 ⇔n≠3∨ or B (or both) are true; if both are false, the statement is false. when n is a natural number. propositional logic, lattice theory
    • exclusive or xor⊕ The statement A ⊕ B is true when either A or B, but not both, are (¬A) ⊕ A is always true, A ⊕ A is always false. propositional logic, true. A ⊻ B means the same.⊻ Boolean algebra universal quantification ∀ x: P(x) means P(x) is true for all∀ for all; for any; for each x. ∀ n ∈ N: n2 ≥ n. predicate logic existential quantification ∃ x: P(x) means there is at least∃ there exists one x such that P(x) is true. ∃ n ∈ N: n is even. predicate logic uniqueness quantification∃! there exists exactly one ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ N: n + 5 = 2n. predicate logic definition:= is defined as x := y or x ≡ y means x is defined to be another name for y (but note cosh x := (1/2)(exp x + that ≡ can also mean other things, exp (−x))≡ such as congruence). A XOR B :⇔ everywhere P :⇔ Q means P is defined to be (A ∨ B) ∧ ¬(A ∧ B) logically equivalent to Q.:⇔ set brackets {a,b,c} means the set consisting of{,} the set of ... a, b, and c. N = {0,1,2,...} teori himpunan set builder notation {x : P(x)} means the set of all x for {n ∈ N : n2 < 20} ={:} the set of ... such which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {0,1,2,3,4} that ...
    • teori himpunan{|} himpunan kosong∅ himpunan kosong ∅ berarti himpunan yang tidak memiliki elemen. {} juga berarti {n ∈ N : 1 < n2 < 4} = ∅ teori himpunan hal yang sama.{}∈ set membership is an element of; is a ∈ S means a is an element of the (1/2)−1 ∈ N not an element of set S; a ∉ S means a is not an element of S. 2−1 ∉ N∉ everywhere, teori himpunan⊆ subset is a subset of A ⊆ B means every element of A is also element of B. A ∩B ⊆ A; Q ⊂ R⊂ teori himpunan A ⊂ B means A ⊆ B but A ≠ B.⊇ superset is a superset of A ⊇ B means every element of B is also element of A. A ∪ B ⊇ B; R ⊃ Q⊃ teori himpunan A ⊃ B means A ⊇ B but A ≠ B. set-theoretic union A ∪ B means the set that contains∪ the union of ... and ...; union all the elements from A and also all those from B, but no others. A⊆B ⇔ A∪B=B teori himpunan set-theoretic intersection A ∩B means the set that contains ∩ intersected with; intersect all those elements that A and B have in common. {x ∈ R : x2 = 1} ∩N = {1} teori himpunan set-theoretic A B means the set that contains complement all those elements of A that are {1,2,3,4} {3,4,5,6} = {1,2}
    • minus; without not in B. teori himpunan function application f(x) berarti nilai fungsi f pada Jika f(x) := x2, maka f(3) = of elemen x. 32 = 9. teori himpunan precedence () grouping Perform the operations inside the (8/4)/2 = 2/2 = 1, but parentheses first. 8/(4/2) = 8/2 = 4. umum function arrow f: X → Y means the function f Let f: Z → N be defined byf:X→Y from ... to maps the set X into the set Y. f(x) = x2. teori himpunan function composition fog is the function, such that if f(x) = 2x, and g(x) = x + 3, O composed with (fog)(x) = f(g(x)). then (fog)(x) = 2(x + 3). teori himpunan natural numbers N N N means {0,1,2,3,...}, but see the article on natural numbers for a {|a| : a ∈ Z} = N ℕ different convention. numbers integers Z Z Z means {...,−3,−2,−1,0,1,2,3,...}. {a : |a| ∈ N} = Z ℤ numbers rational numbers Q Q 3.14 ∈ Q Q means {p/q : p,q ∈ Z, q ≠ 0}. π∉Q ℚ numbers real numbers π∈R R R R means {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. numbers √(−1) ∉ R
    • ℝ complex numbers C C C means {a + bi : a,b ∈ R}. i = √(−1) ∈ C ℂ numbers infinity ∞ is an element of the extended number line that is greater than all ∞ infinity real numbers; it often occurs in limx→0 1/|x| = ∞ numbers limits. pi π berarti perbandingan (rasio) A = πr² adalah luas Π pi antara keliling lingkaran dengan diameternya. lingkaran dengan jari-jari (radius) r Euclidean geometry norm ||x|| is the norm of the element x|| || norm of; length of of a normed vector space. ||x+y|| ≤ ||x|| + ||y|| linear algebra summation sum over ... from ... ∑k=14 k2 = 12 + 22 + 32 + 42 = ∑ to ... of n ∑k=1 ak means a1 + a2 + ... + an. 1 + 4 + 9 + 16 = 30 aritmatika product ∏k=14 (k + 2) = (1 + 2)(2 + product over ... from ∏k=1n ak means a1a2···an. 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × ... to ... of 6 = 360 aritmatika ∏ Cartesian product the Cartesian ∏i=0nYi means the set of all (n+1)- product of; the ∏n=13R = Rn tuples (y0,...,yn). direct product of set theory derivative f (x) is the derivative of the … prime; derivative of … function f at the point x, i.e., the slope of the tangent there. If f(x) = x2, then f (x) = 2x kalkulus
    • indefinite integral or antiderivative indefinite integral of ∫ f(x) dx means a function whose ∫x2 dx = x3/3 + C …; the antiderivative derivative is f. of …∫ kalkulus definite integral ∫ab f(x) dx means the signed area integral from ... to ... between the x-axis and the graph ∫0b x2 dx = b3/3; of ... with respect to of the function f between x = a and x = b. kalkulus gradient ∇f (x1, …, xn) is the vector of partial If f (x,y,z) = 3xy + z² then∇ del, nabla, gradient of derivatives (df / dx1, …, df / dxn). ∇f = (3y, 3x, 2z) kalkulus partial derivative With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, If f(x,y) = x2y, then ∂f/∂x = partial derivative of with all other variables kept 2xy kalkulus constant.∂ boundary ∂{x : ||x|| ≤ 2} = boundary of ∂M means the boundary of M {x : || x || = 2} topology perpendicular x ⊥ y means x is perpendicular to is perpendicular to y; or more generally x is If l⊥m and m⊥n then l || n. orthogonal to y. geometri⊥ bottom element x = ⊥ means x is the smallest the bottom element ∀x : x ∧ ⊥ = ⊥ element. lattice theory entailment A ⊧ B means the sentence A entails the sentence B, that is|= entails every model in which A is true, B is A ⊧ A ∨ ¬A model theory also true. inference|- infers or is derived x ⊢ y means y is derived from x. A → B ⊢ ¬B → ≦A from
    • propositional logic, predicate logic normal subgroup is a normal subgroup N ◅ G means that N is a normal ◅ of subgroup of group G. Z(G) ◅ G group theory quotient group {0, a, 2a, b, b+a, b+2a} / {0, G/H means the quotient of group / mod G modulo its subgroup H. b} = {{0, b}, {a, b+a}, {2a, b+2a}} group theory isomorphism Q / {1, −1} ≈ V, is isomorphic to G ≈ H means that group G is where Q is the quaternion ≈ isomorphic to group H group and V is the Klein group theory four-group.`