Division (Mathematics)

Alright, let's dissect this. You want me to take this… Wikipedia article on division and inject it with some of my particular brand of… clarity. And make it longer. And preserve all those little blue links you've scattered about like lost thoughts. Fine. Just don't expect me to enjoy it.

Arithmetic Operation: Division

"Divided" redirects here. For other uses, see Divided (disambiguation).

This entire edifice of information, this meticulously constructed edifice of numbers and operations, feels… incomplete. It needs more. More depth, more grit. Like a perfectly clean room that’s missing the lived-in chaos that makes it real. This article, like so many others, begs for additional citations for verification. It’s a plea for substance, for the grounding that comes from reliable sources. Please, help improve this article by adding citations to reliable sources. Otherwise, this unsourced material, this airy speculation, may be challenged and, frankly, removed. And who wants that? Find sources: "Division" mathematics – news · newspapers · books · scholar · JSTOR (October 2014). Don't let this flicker out like a cheap bulb.

Here, we see the stark, almost brutal simplicity of division: 20 / 4 = 5. Illustrated, rather quaintly, with apples. "Twenty divided by four equals five." It’s a statement of fact, devoid of the messy human emotion that often accompanies such calculations.

Arithmetic Operations

Addition (+)

term + term summand + summand addend + addend augend + addend

sum

Subtraction (−)

term − term minuend − subtrahend

difference

Multiplication (×)

factor × factor multiplier × multiplicand

product

Division (÷)

dividend divisor numerator denominator

fraction quotient ratio

Exponentiation

base exponent

base power

power

n th root (√)

radicand degree

root

Logarithm (log)

log base ( anti-logarithm )

logarithm

Division, you see, is one of the fundamental pillars of arithmetic. The others, addition, subtraction, and multiplication, stand beside it, but division… division has a certain edge. It’s the operation where something is divided – the dividend – by another, the divisor, and the result is what we call the quotient.

At its most basic, when we’re dealing with natural numbers, division can be seen as a rather stark calculation of containment. How many times does one number fit inside another? It’s a cold, hard count. Take those 20 apples, for instance, divided amongst 4 people. Each person, with chilling efficiency, receives 5 apples. It’s a partition, yes, but one devoid of generosity, merely a distribution of quantity. And this quantity, this divisor, doesn't always need to be a clean integer.

Then there's division with remainder, or Euclidean division, for those who prefer their numbers with a bit more… fragmentation. It gives you an integer quotient – the whole number of times the divisor fits – and a remainder, the stubborn, leftover bit. Twenty-one apples, divided by four people. Each gets five, and there’s one apple left, a lonely testament to the imperfection of the division. It’s a concept that can be extended to natural numbers in general.

But for division to truly sing, to always yield a single, unblemished number, we must venture beyond the confines of natural numbers. We must embrace rational numbers or even the vast expanse of real numbers. In these more forgiving number systems, division becomes the elegant inverse of multiplication. If a = c / b, then a × b = c. Simple. Elegant. Unless, of course, b is zero. Then we face the abyss of division by zero, a void that remains undefined. [a] [4] Back to the apples: 21 apples divided by 4 people. No more lonely leftovers. Each person receives five and a quarter apples. The remainder is absorbed, dissolved into the larger whole.

These forms of division, in their varied manifestations, weave through the intricate tapestry of algebraic structures. Those that allow for Euclidean division, that embrace the remainder, are known as Euclidean domains. Think of polynomial rings in a single indeterminate, where multiplication and addition coalesce into formulas. Then there are the fields and division rings, where division by any non-zero element is not just possible, but a given. Within a ring, the elements that permit division are called units – the simple 1 and -1 in the realm of integers, for instance. And for those seeking a more abstract understanding, the quotient group offers a generalization where the result of "division" is not a number, but a group.

Introduction

The most straightforward way to conceptualize division is through the lenses of quotition and partition. From the quotition viewpoint, 20 / 5 signifies the number of fives that must be aggregated to reach twenty. Alternatively, in terms of partition, 20 / 5 represents the magnitude of each of five distinct sets that can be formed from a collection of twenty items. Consider, for example, twenty apples, meticulously divided into five groups, each containing precisely four apples. This translates to the statement: "twenty divided by five is equal to four." This is formally expressed as 20 / 5 = 4, or ⁠20/5⁠ = 4. [2] In this scenario, the number 20 assumes the role of the dividend, 5 acts as the divisor, and 4 emerges as the quotient.

Unlike the other fundamental arithmetic operations, the division of natural numbers can sometimes result in a remainder that does not divide evenly into the dividend. For instance, dividing 10 by 3 leaves a remainder of 1, as 10 is not an exact multiple of 3. In some contexts, this remainder is incorporated into the quotient as a fractional part, rendering 10 / 3 as ⁠3 + 1/3⁠ or 3.33... . However, within the framework of integer division, where fractional components are absent, the remainder is either maintained separately or, in certain exceptional cases, disregarded or rounded. [5] When the remainder is preserved as a fraction, it gives rise to a rational number. The complete set of rational numbers is constructed by augmenting the integers with all conceivable outcomes derived from the division of integers.

It is crucial to note that, unlike multiplication and addition, division lacks the property of being commutative. This means that the expression a / b does not invariably yield the same result as b / a . [6] Furthermore, division is generally not associative. This implies that when performing a sequence of divisions, the order in which these operations are executed can significantly alter the final result. [7] For example, consider the expression (24 / 6) / 2. Performing the operations within the parentheses first yields 4 / 2, which equals 2. However, if we evaluate 24 / (6 / 2), we first compute 6 / 2 as 3, and then 24 / 3, which results in 8. The disparity between 2 and 8 underscores the non-associative nature of division.

Division is conventionally understood as being left-associative. This convention dictates that when a series of division operations are encountered, the calculation proceeds from left to right:

a / b / c = (a / b) / c = a / (b × c) ≠ a / (b / c) = (a × c) / b.

{\displaystyle a/b/c=(a/b)/c=a/(b\times c);\neq ;a/(b/c)=(a\times c)/b.}

Division exhibits right-distributive properties over addition and subtraction. This means that:

(a ± b) / c = (a ± b) / c = (a / c) ± (b / c) = a/c ± b/c.

{\displaystyle {\frac {a\pm b}{c}}=(a\pm b)/c=(a/c)\pm (b/c)={\frac {a}{c}}\pm {\frac {b}{c}}.}

This property mirrors that of multiplication, where (a + b) × c = a × c + b × c. Conversely, division is not left-distributive. This is demonstrated by the inequality:

a / (b + c) = a / (b + c) ≠ (a / b) + (a / c) = ac/bc + ab/bc.

{\displaystyle {\frac {a}{b+c}}=a/(b+c);\neq ;(a/b)+(a/c)={\frac {ac+ab}{bc}}.}

Consider, for instance, the expression 12 / (2 + 4). This simplifies to 12 / 6, which equals 2.

{\displaystyle {\frac {12}{2+4}}={\frac {12}{6}}=2,}

However, when we evaluate (12 / 2) + (12 / 4), we get 6 + 3, which sums to 9.

{\displaystyle {\frac {12}{2}}+{\frac {12}{4}}=6+3=9.}

This stark contrast highlights the absence of left-distributivity in division, unlike multiplication, which is both left- and right-distributive, and therefore exhibits the distributive law.

Notation

Further information: Division sign

The visual representation of division, much like the emotion it often evokes, can be varied and sometimes unsettling. The obelus, often recognized as the division sign (÷), can appear as a variant of the minus sign, as seen in an excerpt from an official Norwegian trading statement form.

One of the most common and arguably elegant methods of expressing division, particularly in algebra and scientific contexts, involves positioning the dividend directly above the divisor, separated by a horizontal line—the fraction bar. This notation, like ⁠a/b⁠, is universally understood as "a divided by b" or, more colloquially, "a over b." For situations demanding a linear representation, the dividend is placed before a slash, followed by the divisor: ⁠a/b⁠. This format, familiar from computer programming languages and the fundamental characters of ASCII, offers a practical means of typing. It’s also the standard for quotient objects in abstract algebra. Some mathematical software, such as MATLAB and GNU Octave, even permits a reversed order of operands by employing the backslash as the division operator: ⁠b\a⁠.

A stylistic variation, bridging the gap between the stacked fraction and the linear slash, uses a solidus (fraction slash) but elevates the dividend and lowers the divisor: ⁠^a⁄_b⁠. Any of these notations can serve to represent a fraction—a division expression where both numerator and denominator are integers, often without the expectation of further calculation.

A more traditional, though less common in higher mathematics, method of indicating division employs the division sign (÷), also known as the obelus. This symbol, prevalent in elementary arithmetic, is represented as ⁠a ÷ b⁠. However, its usage is discouraged by ISO 80000-2-10.6, and in some non-English-speaking countries, it can be confused with the subtraction operation. [11] The obelus itself was introduced by the Swiss mathematician Johann Rahn in 1659. [10]

In certain countries outside of the Anglosphere, a colon serves to denote division: ⁠a : b⁠. This notation, introduced by Gottfried Wilhelm Leibniz in 1684, was favored by him for its ability to express both ratio and division without distinct symbols. [10] Nevertheless, within English usage, the colon is primarily reserved for the related concept of ratios.

Since the 19th century, textbooks in the United States have adopted a specific notation for division, particularly in the context of long division: ⁠b)a⁠ or ⁠b¯)a⁠. The precise origins of this notation are somewhat obscured, having evolved gradually over time. [13]

Computing

Main articles: Long division and Division algorithm

Manual Methods

Division is often initially presented as a process of "sharing out" a collection of objects, perhaps a pile of sweets, into a predetermined number of equal portions. This act of distribution, where objects are allocated in increments to each portion, introduces the concept of 'chunking'. This method involves repeatedly subtracting multiples of the divisor from the dividend.

By allowing for the subtraction of multiples larger than what the partial remainder might initially suggest, more adaptable techniques, such as the bidirectional variant of chunking, can be developed.

For a more systematic and efficient approach, two integers can be divided using pencil and paper. The method of short division is suitable for smaller divisors, while long division is employed for larger ones. If the dividend contains a fractional component, expressed as a decimal fraction, the procedure can be extended beyond the ones place to any desired degree of precision. When the divisor itself has a fractional part, the problem can be simplified by shifting the decimal point to the right in both numbers until the divisor becomes an integer. For example, the division 10 / 2.5 is equivalent to 100 / 25, which simplifies to 4.

Division can also be performed using an abacus. [14]

Logarithm tables offer another avenue for division. This method involves subtracting the logarithms of the two numbers and then finding the antilogarithm of the resulting value.

Slide rules provide a mechanical means for calculating division. This is achieved by aligning the divisor on the C scale with the dividend on the D scale. The quotient can then be read from the D scale at the position aligned with the left index of the C scale. It is crucial, however, for the user to mentally track the decimal point.

By Computer

Modern calculators and computers execute division through methods akin to long division or via more sophisticated, faster algorithms. See Division algorithm for further details.

In the realm of modular arithmetic, particularly when operating modulo a prime number, and within the context of real numbers, every non-zero number possesses a multiplicative inverse. In these systems, division by a number 'x' can be elegantly computed as a multiplication by the multiplicative inverse of 'x'. This technique is frequently associated with the more efficient computational methods employed by computers.

Division in Different Contexts

Euclidean Division

Main article: Euclidean division

Euclidean division represents the formal mathematical description of the outcome of the standard process of dividing integers. It posits that for any two integers, 'a' (the dividend) and 'b' (the divisor), where 'b' is not equal to zero, there exist unique integers 'q' (the quotient) and 'r' (the remainder) such that a = bq + r, with the condition that 0 ≤ r < |b|, where |b| signifies the absolute value of 'b'.

Of Integers

The set of integers is not closed under the operation of division. Beyond the fundamental rule that division by zero is undefined, the quotient of two integers is not itself an integer unless the dividend is an exact integer multiple of the divisor. For instance, 26 cannot be divided by 11 to produce an integer result. In such scenarios, one of five distinct approaches may be adopted:

Assertion of Impossibility: One might simply state that 26 cannot be divided by 11, effectively rendering division a partial function.
Approximation: An approximate answer can be provided as a floating-point number. This is the prevalent method in numerical computation.
Fractional Representation: The result can be expressed as a fraction, signifying a rational number. Thus, the division of 26 by 11 yields ⁠26/11⁠ (or as a mixed number, ⁠2 4/11⁠). [5] Typically, the resulting fraction is simplified; for example, the division of 52 by 22 simplifies to ⁠26/11⁠. This simplification is often achieved by factoring out the greatest common divisor.
Integer Quotient and Remainder: The answer can be presented as an integer quotient accompanied by a remainder. For example, ⁠26/11 = 2 remainder 4⁠. To distinguish this from the previous case, this type of division, yielding two integer results, is sometimes termed Euclidean division, as it forms the foundation of the Euclidean algorithm.
Integer Quotient Only: The integer quotient alone is provided as the answer, yielding 2 in the case of 26/11. This is equivalent to applying the floor function to case 2 or 3. It is occasionally referred to as integer division and is denoted by "//".

In computer programs, the division of integers requires careful consideration. Some programming languages interpret integer division as described in case 5, returning an integer result. Conversely, other languages, such as MATLAB and every computer algebra system, produce a rational number as the outcome, aligning with case 3. These languages typically also offer functions to obtain results corresponding to the other cases, either directly or by processing the result from case 3.

Common names and symbols employed for integer division include div, /, , and %. [citation needed] Definitions can vary concerning integer division when either the dividend or the divisor is negative. Rounding may occur towards zero (termed T-division) or towards −∞ (F-division); rarer conventions may also exist – consult the modulo operation for more detailed information.

Divisibility rules can sometimes be employed to rapidly ascertain whether one integer divides exactly into another.

Of Rational Numbers

The outcome of dividing two rational numbers is another rational number, provided the divisor is not zero. The division of two rational numbers, p/q and r/s, can be calculated as:

(p/q) / (r/s) = (p/q) × (s/r) = ps/qr.

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

All four quantities involved (p, q, r, s) are integers, and only 'p' is permitted to be zero. This definition ensures that division functions as the inverse operation of multiplication.

Of Real Numbers

The division of two real numbers results in another real number, assuming the divisor is non-zero. This operation is defined such that a / b = c if and only if a = cb and b ≠ 0.

Of Complex Numbers

When dividing two complex numbers (and the divisor is non-zero), the result is another complex number. This is achieved by utilizing the conjugate of the denominator:

(p + iq) / (r + is) = ((p + iq)(r - is)) / ((r + is)(r - is)) = (pr + qs + i(qr - ps)) / (r² + s²) = (pr + qs) / (r² + s²) + i(qr - ps) / (r² + s²).

{\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}}.}

This process, involving multiplication and division by (r - is), is referred to as 'realisation' or, by analogy, rationalisation. All four quantities p, q, r, and s are real numbers, and it is imperative that r and s are not both zero.

Division of complex numbers expressed in polar form offers a more streamlined approach compared to the aforementioned definition:

(p e^(iq)) / (r e^(is)) = (p e^(iq) e^(-is)) / (r e^(is) e^(-is)) = (p/r) e^(i(q-s)).

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

In this representation, p, q, r, and s are all real numbers, and 'r' must not be zero.

Of Polynomials

The operation of division can be defined for polynomials in a single variable over a field. Similar to the division of integers, this operation also yields a remainder. Refer to Euclidean division of polynomials for a comprehensive understanding, and for manual computation, explore polynomial long division or synthetic division.

Of Matrices

A division operation can be defined for matrices. The conventional method involves defining A / B as AB⁻¹, where B⁻¹ denotes the inverse of B. However, it is far more common and less ambiguous to explicitly write out AB⁻¹ to prevent misinterpretation. An elementwise division can also be conceptualized through the Hadamard product.

Left and Right Division

Given that matrix multiplication does not adhere to the commutative property, one can introduce the concepts of left division (often termed backslash-division) as A \ B = A⁻¹B, and right division (slash-division) as A / B = AB⁻¹. For left division to be well-defined, the existence of B⁻¹ is not a prerequisite; however, A⁻¹ must exist. To maintain clarity and avoid confusion, the notation A / B = AB⁻¹ is sometimes referred to as right division or slash-division in this context.

With these definitions of left and right division, it is important to note that A / (BC) is generally not equivalent to (A / B) / C, nor is (AB) \ C identical to A \ (B \ C). Nevertheless, the following relationships hold: A / (BC) = (A / C) / B and (AB) \ C = B \ (A \ C).

Pseudoinverse

To circumvent potential issues arising from the non-existence of A⁻¹ and/or B⁻¹, division can also be defined through multiplication by the pseudoinverse. Specifically, A / B is defined as AB⁺ and A \ B as A⁺B, where A⁺ and B⁺ represent the pseudoinverses of matrices A and B, respectively.

Abstract Algebra

In the field of abstract algebra, within a magma characterized by a binary operation ∗ (which can be conceptually understood as multiplication), left division of b by a (denoted as a \ b) is typically defined as the unique solution 'x' to the equation a ∗ x = b, provided such a solution exists and is unique. Similarly, right division of b by a (denoted as b / a) refers to the unique solution 'y' to the equation y ∗ a = b. This interpretation of division does not necessitate that ∗ possesses specific properties like commutativity, associativity, or the presence of an identity element. A magma where both a \ b and b / a exist and are unique for all 'a' and 'b' (exhibiting the Latin square property) is classified as a quasigroup. Within a quasigroup, division in this sense is consistently achievable, even in the absence of an identity element and consequently, inverses.

"Division" in the sense of "cancellation" can be executed within any magma by an element possessing the cancellation property. Examples include matrix algebras, quaternion algebras, and quasigroups. In an integral domain, where not every element necessarily has an inverse, division by a cancellative element 'a' can still be performed on elements of the form 'ab' or 'ca' through left or right cancellation, respectively. If a ring is finite and every non-zero element is cancellative, then by an application of the pigeonhole principle, every non-zero element of the ring becomes invertible, and division by any non-zero element is thus possible. For a deeper understanding of when algebras (in the technical sense) possess a division operation, consult the page on division algebras. Notably, Bott periodicity can be employed to demonstrate 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.

Calculus

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

(f/g)' = (f'g - fg') / g².

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

Division by Zero

Main article: Division by zero

In most mathematical systems, the division of any number by zero is deemed undefined. This is because zero, when multiplied by any finite number, invariably results in a product of zero. [15] Attempting such an expression in most calculators will typically trigger an error message. However, within certain specialized mathematical structures, division by zero is indeed permissible. Examples include the zero ring and algebraic structures such as wheels. [16] Within these contexts, the interpretation and implications of division by zero diverge significantly from traditional definitions.

Notes

^ Division by zero may be defined in some circumstances, either by extending the real numbers to the extended real number line or to the projectively extended real line or when occurring as a limit of divisions by numbers tending to 0. For example: lim x →0 sin x / x = 1. [2] [3]