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Fakta-fakta Menarik Seputar Matematika

Dimulai oleh Chanz99, Oktober 18, 2011, 02:32:31 PM

« sebelumnya - berikutnya »

0 Anggota dan 1 Pengunjung sedang melihat topik ini.

Fariz Abdullah

Kutip dari: mhyworld pada Maret 19, 2012, 06:26:26 PM
Sebenarnya masalah serupa juga dihadapi oleh angka 0, namun kita sudah terbiasa untuk mengabaikannya.
contoh :
0 + 0 = 0; bukannya 2 (0)
jika anda mengatakan 3 x 0 = 0; maka berapakah 6 x 0? apakah menghasilkan bilangan yang dua kali lebih besar, atau sama besarnya?


Logis juga sih..Saya cuman ingin mengatakan bahwa infinity bukan real number..Karena itu jika 1/0 = infinity, itu tidak merujuk bilangan tertentu..Karena itu kita bisa bilang undefined juga..

Walaupun 0 melambangkan void atau emptyness, tetapi 0 sendiri adalah real number..Bahkan termasuk bilangan asli (natural number) menurut Aksioma Peano, dan mengikuti aksioma aritmetika peano (operasi penjumlahan, pengurangan, perkalian dan pembagian).

Belakangan George Cantor memperkenalkan operasi aritmetika untuk infinity, walaupun ada beberapa kritik, menyangkut paradox yang melingkupinya. ref : [pranala luar disembunyikan, sila masuk atau daftar.]

Dilihat dari grafik y = 1/x, untuk x mendekati 0, memang y mendekati infinity..Karena itu dengan pendekatan limit, nilai 1/x = infinity. Tetapi secara filosofi matematika, nilai infinity ini tetap undefined, mengingat tidak ada real number yang dimaksud.

Beberapa pendapat masalah ini bisa dibaca di : [pranala luar disembunyikan, sila masuk atau daftar.]

CMIIW...
[move]DOUBT EVERYTHING AND FIND YOUR OWN LIGHT[/move]

mhyworld

nambah satu pendapat lagi tentang pembagian dengan 0.
[pranala luar disembunyikan, sila masuk atau daftar.]
KutipDivision by zero is the operation of taking the quotient of any number x and 0, i.e., x/0. The uniqueness of division breaks down when dividing by zero, since the product 0·y=0 is the same for any y, so y cannot be recovered by inverting the process of multiplication. 0 is the only number with this property and, as a result, division by zero is undefined for real numbers and can produce a fatal condition called a "division by zero error" in computer programs.

To the persistent but misguided reader who insists on asking "What happens if I do divide by zero," Derbyshire (2004, p. 36) provides the slightly flippant but firm and concise response, "You can't. It's against the rules." Even in fields other than the real numbers, division by zero is never allowed (Derbyshire 2004, p. 266).

There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^*!=0 in the extended complex plane C-* is defined to be a quantity known as complex infinity. This definition expresses the fact that, for z!=0, lim_(w->0)z/w=infty (i.e., complex infinity). However, even though the formal statement 1/0=infty is permitted in C-*, note that this does not mean that 1=0·infty. Zero does not have a multiplicative inverse under any circumstances.

Although division by zero is not defined for reals, limits involving division by a real quantity x which approaches zero may in fact be well-defined.
once we have eternity, everything else can wait

mhyworld

#17
Sedangkan untuk infinity,
KutipInfinity, most often denoted as , is an unbounded quantity that is greater than every real number. The symbol had been used as an alternative to M (1000) in Roman numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets.

Informally, 1/=0, a statement that can be made rigorous using the limit concept,


Similarly,


where the notation indicates that the limit is taken from the positive side of the real line.

In Mathematica, is represented using the symbol Infinity.
once we have eternity, everything else can wait