division

Division is when you take a number or item and break it into parts of all equal size.
The bigger the denominator/smaller the numerator the smaller the parts.
A fraction is a way of writing division.
Note: Division by zero is undefined.

See also: Addition, Subtraction, Multiplication

A logical fallacy in which is it assumed that because the whole has some property, the parts have that property as well. This is easily confused with the Composition fallacy.

Example: "The human brain is intelligent, so the neurons are intelligent."

To prove the fallacy, show that the property in question is that of the whole and not of the parts. You may want to show how the parts combine to give the whole its differing property, or show how it is impossible that the parts have the property.

The process by which the House of Commons votes on a matter.

This is the process:

1. The Speaker of the House asks that all in favour of the bill shout 'Aye' and all against to shout 'Nay' (all the M.P.s shout at the same time).
2. If there is no definitive difference between the two then the Speaker shouts 'Division' (the House often is very raucous at this point).
3. The M.P.s in the chamber exit out into the lobbies and make their way to different sides depending on what they think. If they think 'Aye' then they go to one side and if they think 'Nay' they go to the other side.
4. In the lobbies they make their way to the exit where stalls are with clerks who take down the name of the M.P.. A tally is then made and the result is announced in the House.

In the case of a dead heat then the Speaker votes but she/he follows very strict rules of impartiality

A binary operation in mathematics, the inverse operation of multiplication. Its operands are its dividend and its divisor; its result is its quotient.

A word of warning:

A long time ago, the author, quite an enthusiast of E2, sought a way to make a contribution. A great comfort and pleasure in his life was Mathematics; for example he spent many hours reading about the Extraction of Roots in the Encyclopedia Britannica, and the occasional mathematical problem within his means was all the sustenance he needed.

Having seen many attempts at the Rigorization of Mathematics, directed at the Youths in High Schools today, the Foundations seemed important. Certainly they are poorly understood, if the many writeups here on, for example, division by zero are any indication. So the task became the telling of what he knew of division - particularly its DEFINITION, which is not commonly known.

Various concepts of division exist - it is how we check multiplication; it is an idealization of physical partitioning into equal groups; it creates formal symbols for which multiplication by the divisor returns the dividend - in this way we can divide by numbers which are not factors of the dividend... There is a mess of algorithms and notations and definitions, but in the end it seems that division is not very important or interesting. The author is at a loss.

(If there is any interest, he will tell about formal division — how mathematicians narrowly escape dividing by zero —, division viewed as a continuous function, or division into factors as a method for understanding an algebraic structure. Let him know.)

Operation

Division is breaking a dividend into parts each equal in magnitude to the divisor and counting them. Equivalently, one can break a dividend into a number of equal parts represented by the divisor and measure each part's size. But perhaps the most precise and simple definition is that division is the inverse operation of multiplication.

(a ÷ b = c) if and only if (c × b = a).

Dividing by a number in a group (such as a nonzero real number), then, is equivalent to multiplying by its multiplicative inverse.

(a × b^-1) × b = a × (b^-1 × b) = a × 1 = a;
applying the definition of division, a/b = ab^-1 if the latter is defined.

Addition distributes on the left over division because it distributes over multiplication.

When a/c + b/c = q, (a/c + b/c)c = qc,
and since multiplication is distributive, a + b = qc;
so (a + b)/ c = q.

It is not always possible, however, to distribute (a + b)/c = q; for example, as integers (1 + 3)/2 = 2, but 1 and 3 are not divisible by two.

Notation

Division is usually written horizontally with the operator in the middle. There are at least three such ways to write a quotient:

• As a fraction, a divided by b is written with a above and b below a horizontal bar. By analogy, fractions are often written diagonally or horizontally, as ^a/_b.
• As a ratio, a divided by b is written horizontally, as a : b.
• The generic symbol for division is the division sign ÷, a colon superimposed on a horizontal bar, which reads as "divided by" instead of "over" or "to".

In long division, the long division symbol or the colon is used to separate the divisor from the dividend.

Algorithms

The division algorithm (see abiessu's writeup there) allows splitting division problems into smaller problems.

• Long division is the classic example. One uses the decimal representation of the dividend to split a complicated division problem into simpler division problems and subtraction problems. In each new division problem, the dividend has at most one more digit than the divisor. (Short division is long division with fewer steps written down.)
• Egyptian division is long division in base 2. The quotients along the way are all either 1 or 0, so they can be trivially found by comparison.
• The usual method for polynomial division is similar. Short division of polynomials is called synthetic division.

People usually divide small two digit numbers by one digit numbers in their heads using shortcuts and multiplication tables. If all else fails, you can guess and check.

Multiplication by the multiplicative inverse

To divide fractions, you may multiply the dividend by the reciprocal of the divisor. To divide integers modulo n, multiply the dividend by the multiplicative inverse of the divisor. In other cases, another method is probably simpler.

See Also

• Complex number division
• slide rule
• how to determine whether a number is divisible by n
• How to really impress people using division by 7

Di*vi"sion (?), n. [F. division, L. divisio, from dividere. See Divide.]

1.

The act or process of diving anything into parts, or the state of being so divided; separation.

I was overlooked in the division of the spoil. Gibbon.

2.

That which divides or keeps apart; a partition.

3.

The portion separated by the divining of a mass or body; a distinct segment or section.

Communities and divisions of men. Addison.

4.

Disunion; difference in opinion or feeling; discord; variance; alienation.

There was a division among the people. John vii. 43.

5.

Difference of condition; state of distinction; distinction; contrast.

Chaucer.

I will put a division between my people and thy people. Ex. viii. 23.

6.

Separation of the members of a deliberative body, esp. of the Houses of Parliament, to ascertain the vote.

The motion passed without a division. Macaulay.

7. Math.

The process of finding how many times one number or quantity is contained in another; the reverse of multiplication; also, the rule by which the operation is performed.

8. Logic

The separation of a genus into its constituent species.

9. Mil. (a)

Two or more brigades under the command of a general officer.

(b)

Two companies of infantry maneuvering as one subdivision of a battalion.

(c)

One of the larger districts into which a country is divided for administering military affairs.

10. Naut.

One of the groups into which a fleet is divided.

11. Mus.

A course of notes so running into each other as to form one series or chain, to be sung in one breath to one syllable.

12. Rhet.

The distribution of a discourse into parts; a part so distinguished.

13. Biol.

A grade or rank in classification; a portion of a tribe or of a class; or, in some recent authorities, equivalent to a subkingdom.

Cell division Biol., a method of cell increase, in which new cells are formed by the division of the parent cell. In this process, the cell nucleus undergoes peculiar differentiations and changes, as shown in the figure (see also Karyokinesis). At the same time the protoplasm of the cell becomes gradually constricted by a furrow transverse to the long axis of the nuclear spindle, followed, on the completion of the division of the nucleus, by a separation of the cell contents into two masses, called the daughter cells. -- Long division Math., the process of division when the operations are mostly written down. -- Short division Math., the process of division when the operations are mentally performed and only the results written down; -- used principally when the divisor is not greater than ten or twelve.

Syn. -- compartment; section; share; allotment; distribution; separation; partition; disjunction; disconnection; difference; variance; discord; disunion.

 

© Webster 1913.