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At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one.:7 This number of times is not always an integer, and this led to two different concepts.
The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.
For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c ÷ b means a × b = c, as long as b is not zero—if b = 0, then this is a division by zero, which is not defined.[a]:246
Both forms of divisions appear in various algebraic structures. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate. Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units; e.g., within the ring of integers the units are 1 and –1.
In its most simple form, division can be viewed either as a quotition or a partition. In terms of quotition, 20 ÷ 5 means the number of 5s that must be added to get 20. In terms of partition, 20 ÷ 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into four groups of five apples, meaning that twenty divided by five is equal to four. This is denoted as 20 / 5 = 4, 20 ÷ 5 = 4, or 20/ = 4. Notationally, the dividend is divided by the divisor to get a quotient. In the example, 20 is the dividend, five is the divisor, and four is the quotient.
Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 ÷ 3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, so 10 ÷ 3 is equal to 31/ or 3.33..., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. When the remainder is kept as a fraction, it leads to a rational number. The set of all rational numbers is created by every possible division using integers. In modern mathematical terms, this is known as extending the system.
Unlike multiplication and addition, Division is not commutative, meaning that a ÷ b is not always equal to b ÷ a. Division is also not associative, meaning that when dividing multiple times, the order of the division changes the answer to the problem. For example, (20 ÷ 5) ÷ 2 = 2, but 20 ÷ (5 ÷ 2) = 8, where the parentheses mean that the operation inside the parentheses is performed before the operations outside.
Division is, however, distributive. This means that (a+b) ÷ c = (a ÷ c) + (b ÷ c) for every number. Specifically, division has the right-distributive property over addition and subtraction. That means:
This is the same as multiplication: . However, division is not left-distributive:
This is unlike multiplication.
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, a divided by b is written
This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), like this:
This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters. Some mathematical software, such as MATLAB and GNU Octave, allows the operands to be written in the reverse order by using the backslash as the division operator:
A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. The obelus was introduced by Swiss mathematician Johann Rahn in 1659 in Teutsche Algebra.:211
In some non-English-speaking countries colon is used to denote division. This notation was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum.:295 Leibniz disliked having separate symbols for ratio and division. However, in English usage the colon is restricted to expressing the related concept of ratios.
Since the 19th century US textbooks have used or to denote a divided by b, especially when discussing long division. The history of this notation is not entirely clear because it evolved over time.
Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e., division by repeated subtraction.
More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper using the method of short division, if the divisor is simple. Long division is used for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones 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.
A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.
A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.
By computer or with computer assistanceEdit
In modular arithmetic (modulo a prime number) and for real numbers, nonzero numbers have a multiplicative inverse. In these cases, a division by x may be computed as the product by the multiplicative inverse of x. This approach is often the most efficient one.
Division in different contextsEdit
The Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder, such that a = bq + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b.
Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:
- Say that 26 cannot be divided by 11; division becomes a partial function.
- Give an approximate answer as a decimal fraction or a mixed number, so or This is the approach usually taken in numerical computation.
- Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also . This simplification may be done by factoring out the greatest common divisor.
- Give the answer as an integer quotient and a remainder, so To make the distinction with the previous case, this division, with two integers as result, is sometimes called Euclidean division, because it is the basis of the Euclidean algorithm.
- Give the integer quotient as the answer, so This is sometimes called integer division.
Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.
Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
Of rational numbersEdit
The result of dividing two rational numbers is another rational number when the divisor is not 0. The division of two rational numbers p/q and r/s can be computed as
All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.
Of real numbersEdit
Division of two real numbers results in another real number when the divisor is not 0. It is defined such a/b = c if and only if a = cb and b ≠ 0.
Of complex numbersEdit
Dividing two complex numbers results in another complex number when the divisor is not 0, which is found using the conjugate of the denominator:
This process of multiplying and dividing by is called 'realisation' or (by analogy) rationalisation. All four quantities p, q, r, s are real numbers, and r and s may not both be 0.
Division for complex numbers expressed in polar form is simpler than the definition above:
Again all four quantities p, q, r, s are real numbers, and r may not be 0.
One can define the division operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation, polynomial long division or synthetic division.
One can define a division operation for matrices. The usual way to do this is to define A / B = AB−1, where B−1 denotes the inverse of B, but it is far more common to write out AB−1 explicitly to avoid confusion. An elementwise division can also be defined in terms of the Hadamard product.
Left and right divisionEdit
Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A−1B. For this to be well defined, B−1 need not exist, however A−1 does need to exist. To avoid confusion, division as defined by A / B = AB−1 is sometimes called right division or slash-division in this context.
Note that with left and right division defined this way, A / (BC) is in general not the same as (A / B) / C and nor is (AB) \ C the same as A \ (B \ C), but A / (BC) = (A / C) / B and (AB) \ C = B \ (A \ C).
To avoid problems when A−1 and/or B−1 do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., A / B = AB+ and A \ B = A+B, where A+ and B+ denote the pseudoinverse of A and B.
In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique. Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b. Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element).
"Division" in the sense of "cancellation" can be done in any magma by an element with the cancellation property. Examples include matrix algebras and quaternion algebras. A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses. In an integral domain, where not every element need have an inverse, division by a cancellative element a can still be performed on elements of the form ab or ca by left or right cancellation, respectively. If a ring is finite and every nonzero element is cancellative, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, and division by any nonzero element is possible. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.
Division by zeroEdit
Division of any number by zero in most mathematical systems is undefined because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message. However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as wheels. In these algebras, the meaning of division is different from traditional definitions.
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