Division (mathematics)

For the digital implementation of mathematical division, see Division (digital).

In mathematics, especially in elementary arithmetic, division (÷) is an arithmetic operation. Specifically, if c times b equals a, written:

$c \times b = a\,$

where b is not zero, then a divided by b equals c, written:

$\frac ab = c$

For instance,

$\frac 63 = 2$

since

$2 \times 3 = 6\,$ .

In the above expression, a is called the dividend, b the divisor and c the quotient.

Conceptually, division describes two distinct but related settings. Partitioning involves taking a set of size a and forming b groups that are equal in size. The size of each group formed, c, is the quotient of a and b. Quotative division involves taking a set of size a and forming groups of size b. The number of groups of this size that can be formed, c, is the quotient of a and b.^[1]

Teaching division usually leads to the concept of fractions being introduced to students. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.

1 Notation
2 Computing division
3 Division algorithm
4 Division of integers
5 Division of rational numbers
6 Division of real numbers
7 Division by zero
8 Division of complex numbers
9 Division of polynomials
10 Division of matrices
- 10.1 Left and right division
- 10.2 Matrix division and pseudoinverse
11 Division in abstract algebra
12 Division and calculus
13 See also
14 References
15 External links

Notation

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them. For example, a divided by b is written

$\frac ab$

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:

$a/b\,$

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.

A typographical variation, which is halfway between these two forms, uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

^a⁄_b

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

$a \div b$

This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

In elementary mathematics the notation $b)~a$ or $b)\overline{~a~}$ is used to denote a divided by b. This notation was first introduced by Michael Stifel in Arithmetica integra, published in 1544.^[2]

Computing division

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, 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 and the method of short division, if the divisor is simple, or long division for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), we 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.

Modern computers compute division by methods that are faster than long division: see Division (digital).

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.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.

Division algorithm

Main article: Division algorithm

The division algorithm is a mathematical theorem that precisely expresses the outcome of the usual process of division of integers. In particular, the theorem asserts that integers called the quotient q and remainder r always exist and that they are uniquely determined by the dividend a and divisor d, with d ≠ 0. Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Division of integers

Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches.

Say that 26 cannot be divided by 10; division becomes a partial function.
Give the answer as a decimal fraction or a mixed number, so $\tfrac{26}{10} = 2.6$ or $\tfrac{26}{10} = 2 \tfrac 35.$ This is the approach usually taken in mathematics.
Give the answer as an integer quotient and a remainder, so $\tfrac{26}{10} = 2 \mbox{ remainder } 6.$
Give the integer quotient as the answer, so $\tfrac{26}{10} = 2.$ This is sometimes called integer division.

One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.

Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the quotient is negative: rounding may be toward zero or toward −∞.

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

Division of rational numbers

The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p/q and r/s by

${p/q \over r/s} = {p \over q} \times {s \over r} = {ps \over qr}.$

All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.

Division of real numbers

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.

Division by zero

Main article: Division by zero

Division of any number by zero (where the divisor is zero) is not defined. This is because zero multiplied by any finite number will always result in a product of zero. Entry of such an expression into most calculators will result in an error message being issued.

Division of complex numbers

Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:

${p %2B iq \over r %2B is} = {p r %2B q s \over r^2 %2B s^2} %2B i{q r - p s \over r^2 %2B s^2}.$

All four quantities are real numbers. r and s may not both be 0.

Division for complex numbers expressed in polar form is simpler than the definition above:

${p e^{iq} \over r e^{is}} = {p \over r}e^{i(q - s)}.$

Again all four quantities are real numbers. r may not be 0.

Division of polynomials

One can define the division operation for polynomials. Then, as in the case of integers, one has a remainder. See polynomial long division or synthetic division.

Division of matrices

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.

Left and right division

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).

Matrix division and pseudoinverse

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.

Division in abstract algebra

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as ${a \over b}$ are typically defined as $a \cdot {1 \over b}$ or $a \cdot b^{-1}$ where $b$ is presumed to be an invertible element (i.e. there exists a multiplicative inverse $b^{-1}$ such that $bb^{-1} = b^{-1}b = 1$ where $1$ is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form $ab = ac$ or $ba = ca$ by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division by any nonzero element is possible in such a ring. 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 and calculus

The derivative of the quotient of two functions is given by the quotient rule:

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

There is no general method to integrate the quotient of two functions.

References

^ Fosnot and Dolk 2001. Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH: Heinemann.
^ Earliest Uses of Symbols of Operation, Jeff MIller