Division (mathematics)

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Division is one of four basic arithmetic operations, the other is the sum, subtraction, and multiplication. The division of two natural numbers is the process of counting the number of times one number contained in another. For example, in the picture to the right, 20 apples are divided into four groups of five apples, which means that twenty divided by five gives four , or four is the result of the twenty-fifth division . This is denoted as 20/5 = 4 , 20 ÃÆ' Â · 5 = 4 , or 20 / 5 = 4 .

The division can be viewed either as a quotition or as a partition. In the quotation, 20 ÃÆ' Â · 5 means the amount of 5 to be added to get 20. In partition, 20 ÃÆ' Â · 5 means the size of each of the 5 sections in which a set of size 20 is divided.

The division is the opposite of multiplication; if a ÃÆ'â € " b = c , then a = c ÃÆ' Â · b , as long as b is not zero. The division with zero is undefined for real numbers and most other contexts, because if b = 0 , then a can not be inferred from b and c , because then c will always equal zero regardless of a . In some contexts, division by zero can be defined though to some extent, and boundaries involving the division of real numbers when close to zero are defined.

In the division, dividend is shared by the divisor to get quotient . In the example above, 20 are dividends, five are divisors, and four are quotient. In some cases, the divisor may not be entirely contained by dividends; for example, 10 ÃÆ' Â · 3 leaves left of one, because 10 is not a multiple of three. Sometimes the remainder is added to the quotient as a fractional part, so 10 ÃÆ' Â · 3 is equal to 3 1 / 3 or 3.33Ã,.Ã,. , but in the context of integer division, where numbers do not have fractional parts, the remainder are stored separately or discarded.

In addition to dividing apples, divisions can be applied to other physical and abstract objects. Divisions have been defined in several contexts, such as for real and complex numbers and for more abstract contexts such as for vector spaces and fields.

Division is the most difficult mentally of the four basic operations of arithmetic, but its discipline and mastery provides educational bridges from arithmetic to number theory and algebra. The teaching of objective concepts divides integers introducing students into fractional arithmetic. Unlike addition, subtraction, and multiplication, the set of all integers does not close below the division. Dividing the two integers can produce the remainder. To complete the remaining division, the number system is extended to include a fraction or rational number because it is more commonly called. As students progress to algebra, the abstract theory of divisions that are integrated from arithmetic naturally extends to the division of algebraic, polynomial, and matrix variables.

Video Division (mathematics)

Notation

Pembagian sering ditampilkan dalam aljabar dan sains dengan menempatkan dividen di atas pembagi dengan garis horizontal, juga disebut bar fraksi, di antara mereka. Sebagai contoh, a dibagi dengan b ditulis

${\ displaystyle {\ frac {a} {b}}}  ÂÂ$

This can be read aloud as " a divided by b ", " a by b " or " b ". The way to disclose all sharing on one line is to write dividend (or a numerator), then a slash, then divider (or denominator), like this:

${\ displaystyle a/b \,}$

This is a common way of determining division in most computer programming languages ​​as it can be easily typed as a simple sequence of ASCII characters. Some math software, such as MATLAB and GNU Octave, allow the operand to be written in reverse order by using the backslash as the divisional operator:

${\ displaystyle b \ backslash a}$

Variasi tipografi di antara dua bentuk ini menggunakan solidus (pecahan garis potong) tetapi menaikkan dividen, dan menurunkan pembagi:

${\ displaystyle {} ^ {a}/{} _ {b}}  ÂÂ$

Each of these shapes can be used to display fractions. Fraction is the division expression in which both dividends and divisors are integers (usually called numerators and denominators ), and there is no implication that the division should be evaluated further. The second way to show the division is to use an obelus (or division mark), common in arithmetic, in this way:

${\ displaystyle a \ div b}$

This form is rare except in basic arithmetic. ISO 80000-2-9.6 states that it should not be used. Obelus is also used alone to represent the division operations themselves, such as labels on the calculator keys. Obelus was introduced by Swiss mathematician Johann Rahn in 1659 in Teutsche Algebra.

In some non-English speaking countries, "a divided by b" is written a Ã,: Ã, b . This notation was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum. Leibniz dislikes having separate symbols for ratios and divisions. However, in English colon usage is restricted to express the concept of related ratios (then " a is b ").

Karena buku-buku teks AS abad ke-19 telah menggunakan $            {\displaystyle         b        )        \text{Â}        a      }        \{\backslash \; displaystyle\; b)\; ~\; a\}\;  ÂÂ$ atau ${\ displaystyle b {\ overline {) a}}}  ÂÂ$ untuk menunjukkan a dibagi dengan b , terutama ketika membahas pembagian panjang. Sejarah notasi ini tidak sepenuhnya jelas karena berevolusi seiring waktu.

Maps Division (mathematics)

Komputasi

Metode manual

Sharing is often introduced through the idea of ​​"sharing" a set of objects, such as a pile of candy, into the same number of parts. Distributing multiple objects at a time in each round of sharing to each section leads to the idea of ​​"chunking", that is, division by repetitive reduction.

More systematic and more efficient (but also more formalized and more rule-based, and more removed from a thorough overall picture of what divisions are achieved), someone who knows the multiplication table can divide two integers using pencil and paper using the short division method, if The divider is simple. Long division is used for larger integer dividers. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm through the place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

One can calculate the division by the abacus by repeatedly placing the dividend on the abacus, and then subtracting the offset divisor per digit in the result, counting the number of possible divisions on each offset.

One can use a logarithmic table to divide the two numbers, by reducing the logarithm of two numbers, then searching for the result antilogarithm.

One can calculate the division by the slide rule by aligning the divisor on a C scale with dividend on a D scale. The quotient can be found on the D scale where it is parallel to the left index on a C scale. However, the user is responsible for mentally monitoring the decimal point.

With a computer or with the help of a computer

The modern computer calculates the division by a faster method than the long division. For division by remainder, see Division Algorithm.

In modular arithmetic (modulo primes) and real numbers, nonzero numbers have multiplicant invers. In this case, the division by x can be counted as a product with multiplication inverses of x . This approach is often the most efficient.

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Properties

Pembagian adalah distributif-kanan atas penambahan dan pengurangan. Itu berarti:

${\ displaystyle {\ frac {a b} {c}} = (a b) \ div c = {\ frac {a} {c}} {\ frac {b} {c}}}  ÂÂ$

dengan cara yang sama seperti dalam perkalian ${\ displaystyle (a b) \ kali c = a \ kali c b \ times c}  ÂÂ$ . Tetapi pembagian tidak berdistribusi kiri, yaitu kita punya

${\ displaystyle {\ frac {a} {b c}} = a \ div (b c) = \ kiri ({\ frac {b} {a}} {\ frac {c} {a}} \ right) ^ {- 1} \ neq {\ frac {a} {b}} {\ frac {a} {c}}}  ÂÂ$

not like multiplication.

Jika ada beberapa divisi berturut-turut, urutan operasi berjalan dari kiri ke kanan, yang disebut asosiatif kiri:

${\ displaystyle a \ div b \ div c = (a \ div b) \ div c = a \ div (b \ times c) = a \ times b ^ {- 1} \ kali c ^ {- 1}}  ÂÂ$ .

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Pembagian Euclidean

The division of Euclidean is a mathematical formulation of the ordinary process of division. It confirms that, given two integers, a , dividends , and b , divider , so < b ? 0, there is a unique integer q , quotient , and r , the rest, up to a = bq r and 0 <= r & lt; | b |, where | b | indicates the absolute value b .

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Integer

Dividing integers in computer programs requires special care. Some programming languages, such as C, treat integer divisions as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and each computer algebra system return rational numbers as the answer, as in case 3 above. These languages ​​also provide a function to get results from other cases, either directly or from case outcomes 3.

The names and symbols used for integer divisions include div,/, \, and%. Variable definitions of the division of integers when dividends or negative dividers: rounding may go to zero (called T-division) or toward -? (F-division); a more rare style can occur - see Modulo operation for details.

Sharing rules can sometimes be used to determine quickly whether one integer divides exactly into another.

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From rational numbers

Hasil membagi dua bilangan rasional adalah bilangan rasional lainnya ketika pembagi tidak 0. Pembagian dua bilangan rasional p / q dan r / s dapat dihitung sebagai

${\ displaystyle {p/q \ over r/s} = {p \ over q} \ kali {s \ over r} = {ps \ over qr}.}  ÂÂ$

The fourth quantity is an integer, and only p is possible 0. This definition ensures that the division is a multiplicative inversion operation.

From a real number

The division of two real numbers yields another real number when the divisor is not 0. It is defined as a / b = c if and only if a = cb and b ? 0.

With zero

Any division of numbers by zero (where the divisor is zero) is undefined. This is because zero multiplied by a finite number always yields a zero product. The inclusion of such expressions to most calculators yields an error message.

From complex numbers

Membagi dua bilangan kompleks menghasilkan bilangan kompleks lain ketika pembagi tidak 0, yang ditemukan menggunakan konjugasi penyebut:

${\ displaystyle {p iq \ over r is} = {(p iq) (r-is) \ over (r is) (r-is)} = {pr qs i (qr-ps) \ over r ^ {2} s ^ {2}} = {pr qs \ over r ^ {2} s ^ {2}} i {qr- ps \ over r ^ {2} s ^ {2}}.}  ÂÂ$

Proses ini mengalikan dan membagi dengan ${\ displaystyle r-is}  ÂÂ$ disebut 'realisasi' atau rasionalisasi (dengan analogi). Semua empat kuantitas p , q , r , s adalah bilangan real, dan r dan s mungkin tidak keduanya 0.

Pembagian untuk bilangan kompleks yang diekspresikan dalam bentuk polar lebih sederhana daripada definisi di atas:

${\ displaystyle {pe ^ {iq} \ over re {}} = {pe ^ {iq} e ^ {- is} \ over re ^ {is} e ^ {- is}} = {p \ over r} e ^ {i (qs)}.}  ÂÂ$

Once again the four quantities p , q , r , s < I may not be 0.

From polynomial

One can specify the division operation for polynomials in one variable above the field. Then, as in the case of integers, one has a residual. See Euclidean polynomial division, and, for handwriting calculations, long division of polynomial or synthetic division.

Matrix

One can define division operations for matrices. The usual way to do this is to define AB ^-1 at where B ^-1 shows the opposite of B , but it is much more common to write AB ^-1 explicitly to avoid confusion. The elementwise division can also be defined in terms of Hadamard products.

Left and right divisions

Since the multiplication of the matrix is ​​not commutative, we can also define the left division or the so-called slash division as A \ B = < A ^-1 B . For this to be well defined, B ^-1 does not need to exist, but A -1 is necessary. To avoid confusion, the division as defined by A / B = AB ^-1 sometimes called right dialing or slash-sharing in this context.

Note that with left and right divisions defined in this way, A /( BC ) is generally not equal to (< A / B )/ C and not ( AB ) \/i> is equal to A \ ( B \ C ) i> A /( BC ) = ( A C )/ B C i>) .

Pseudoinverse

To avoid problems when A ^-1 and/or B ^-1 does not exist, division can also be defined as multiplication with pseudoinverse, i.e. AB ^{and A \ B = A B , where A sup> indicates the pseudoinverse of A and B .}

Abstract algebra

In abstract algebra, given the magma by binary * (which can be called multiplication), the left division b by a (written a \ b ) is usually defined as the solution x in the equation a * x = b , if this is present and unique. Similarly, the right division b by a (written b / a ) is the solution is y in the equation y * a = b . The division in this sense does not require * to have a particular property (such as commutative, associativity, or identity elements).

"Distribution" in the sense of "cancellation" can be done in any magma by an element with a cancellation property. Examples include algebraic matrices and quaternion algebra. Quasigroup is a structure in which division is always possible, even without the identity element and therefore the inverse. In an integral domain, where not every element needs to have the opposite, the division by the cancellation element a can still be performed on the form elements ab ca with the left or right cancellation, respectively. If a ring is finite and any non-zero element is cancellative, then by applying the pigeonhole principle, any nonzero elements of the ring can be reversed, and division by nonzero elements is possible. To learn about when algebras (in the technical sense) have a division operation, see the page in the division of algebras. Specifically, the periodicity of Bott can be used to indicate that any real normed division algebra should be isomorphic either the real number R , the complex numbers C , quaternions H , or okton O .

Calculus

Turunan dari hasil bagi dua fungsi diberikan oleh aturan bagi-bagi:

${\ displaystyle {\ left ({\ frac {f} {g}} \ right)} '= {\ frac {f'g-fg'} {g ^ { 2}}}.}  ÂÂ$

Lihat juga

• Algoritma pembagian 400AD Sunzi
• Dibagi menjadi dua
• Bidang
• Pecahan (matematika)
• Pembagian galai
• Grup
• Elemen inversi
• Urutan operasi
• Quasigroup
• Mengulangi desimal

Catatan

Referensi

Tautan eksternal

• "Divisi". PlanetMath .
• Pembagian pada sempoa Jepang dipilih dari Abacus: Mystery of the Bead
• Teknik Bagian Pendek Tiongkok di Suan Pan
• Aturan pembagian

Source of the article : Wikipedia