## subtraction

**-**

## Subtraction

# Subtraction

The definitions of subtraction

Subtraction, in arithmetic, is a mathematical operation for finding the difference of two numbers. Thus, 7— 5 = 2, which means seven subtract five equals two. It is also commonly referred to as the operation that is the inverse of addition. Subtracting a number has the effect of nullifying the addition of that same number. For example, if one adds 11 to 41 then subtracts 11, the result is 41 again.

Symbolically, for any numbers a and b, (a + b) — b= a

Or, one can subtract first, then add: (a — b) + b = a

Thus, one can say that subtraction and addition are inverse operations.

## The definitions of subtraction

The second of the two rules above can be taken as a definition of subtraction. That is, the difference of two numbers, a− b, is the number that must be added to the second number, b, to equal the first, a. Before the widespread use of electronic cash registers, grocery clerks and others, making change, would use this definition directly. If, for example, a customer bought groceries worth $3.70 and gave the clerk $5.00 to pay for them, the clerk would not subtract in order to compute the change. Instead, he (or she) would count it out. Starting with “three seventy,” he would give the customer a nickel, and say, “three seventy-five.” He would give the customer a quarter and say, “four.” Then he would give the customer a dollar and say, “five.” This method is still in use when no calculator is available.

A second definition of subtraction is often given in algebra books after the introduction of negative numbers. There, a − b is defined as a + (—b); i.e., to subtract a number, add its opposite. This definition is mathematically efficient, but it hides many of the ways in which subtraction is used, as in the change-making example above.

## Terminology

in a subtraction problem it is useful to have names for the various parts. In a− b, the entire expression is called a difference, and the answer to a− b is the difference, or occasionally the remainder. These two terms arise from two ways in which subtraction is used in practical problems. The first number, a, is called the minuend. This is the part that is diminished or made smaller when something is subtracted from it (that is, when that something is positive). The second number, b, is the subtrahend or the part subtracted.

## Properties

it matters very much which of the two parts of a difference are named first. Taking $500 from an account with $300 in it is very different from taking $300 from $500. For this reason, subtraction is *not* commutative: a− b does not equal b− a.

Subtraction is not associative either: (a− b)− c does not equal a− (b− c).

An example will demonstrate this: (20− 10)− 3 is 7, but 20− (10− 3) is 13.

Often one encounters expressions such as 5x^{2}− 2− 3x^{2}, with no indication of which subtraction is to be done first. Since subtraction is non-associative, it matters. To avoid this ambiguity one can agree that subtractions, unless otherwise indicated, are to be done left-to-right. This is a rather limiting agreement, therefore, it may be more convenient to use some other order. Another agreement, which is the common agreement of algebra, is to treat the minus sign as a plus-the-opposite-of sign. Thus, one would interpret the example above as 5x^{2} + (—2) + (—3x^{2}). In this interpretation it becomes a sum, whose terms can be combined in any order one pleases.

In certain sets, subtraction is not a closed operation. The set of natural numbers, for instance, is not closed with respect to subtraction. If a merchant will not extend credit, one cannot buy an article whose price is greater than the amount of money one has.

## Uses of subtraction

The most familiar meaning for subtraction is called “take away.” To find the answer if you take 5 eggs from a dozen, you subtract: 12 — 5 = 7.

Subtraction is also used to compare two numbers. When one wants to know how much colder − 7.3° is than 13.8°, one computes the difference −7.3 − 13.8 to get −21.1°.

A third use of subtraction is to figure out a missing addend. If one has a certain sum of money, say $45.20 and wants to buy an article costing $85.50, subtraction is used to compute that needed amount: 85.50 − 45.20.

J. Paul Moulton

## subtraction

**subtraction**
•**ashen**, fashion, passion, ration
•**abstraction**, action, attraction, benefaction, compaction, contraction, counteraction, diffraction, enaction, exaction, extraction, faction, fraction, interaction, liquefaction, malefaction, petrifaction, proaction, protraction, putrefaction, redaction, retroaction, satisfaction, stupefaction, subtraction, traction, transaction, tumefaction, vitrifaction
•**expansion**, mansion, scansion, stanchion
•sanction
•**caption**, contraption
•**harshen**, Martian
•**cession**, discretion, freshen, session
•**abjection**, affection, circumspection, collection, complexion, confection, connection, convection, correction, defection, deflection, dejection, detection, direction, ejection, election, erection, genuflection, imperfection, infection, inflection, injection, inspection, insurrection, interconnection, interjection, intersection, introspection, lection, misdirection, objection, perfection, predilection, projection, protection, refection, reflection, rejection, resurrection, retrospection, section, selection, subjection, transection, vivisection
•**exemption**, pre-emption, redemption
•**abstention**, apprehension, ascension, attention, circumvention, comprehension, condescension, contention, contravention, convention, declension, detention, dimension, dissension, extension, gentian, hypertension, hypotension, intention, intervention, invention, mention, misapprehension, obtention, pension, prehension, prevention, recension, retention, subvention, supervention, suspension, tension
•**conception**, contraception, deception, exception, inception, interception, misconception, perception, reception
•Übermenschen • subsection
•**ablation**, aeration, agnation, Alsatian, Amerasian, Asian, aviation, cetacean, citation, conation, creation, Croatian, crustacean, curation, Dalmatian, delation, dilation, donation, duration, elation, fixation, Galatian, gyration, Haitian, halation, Horatian, ideation, illation, lavation, legation, libation, location, lunation, mutation, natation, nation, negation, notation, nutation, oblation, oration, ovation, potation, relation, rogation, rotation, Sarmatian, sedation, Serbo-Croatian, station, taxation, Thracian, vacation, vexation, vocation, zonation
•**accretion**, Capetian, completion, concretion, deletion, depletion, Diocletian, excretion, Grecian, Helvetian, repletion, Rhodesian, secretion, suppletion, Tahitian, venetian
•**academician**, addition, aesthetician (*US* esthetician), ambition, audition, beautician, clinician, coition, cosmetician, diagnostician, dialectician, dietitian, Domitian, edition, electrician, emission, fission, fruition, Hermitian, ignition, linguistician, logician, magician, mathematician, Mauritian, mechanician, metaphysician, mission, monition, mortician, munition, musician, obstetrician, omission, optician, paediatrician (*US* pediatrician), patrician, petition, Phoenician, physician, politician, position, rhetorician, sedition, statistician, suspicion, tactician, technician, theoretician, Titian, tuition, volition
•**addiction**, affliction, benediction, constriction, conviction, crucifixion, depiction, dereliction, diction, eviction, fiction, friction, infliction, interdiction, jurisdiction, malediction, restriction, transfixion, valediction
•**distinction**, extinction, intinction
•**ascription**, circumscription, conscription, decryption, description, Egyptian, encryption, inscription, misdescription, prescription, subscription, superscription, transcription
•proscription
•**concoction**, decoction
•**adoption**, option
•**abortion**, apportion, caution, contortion, distortion, extortion, portion, proportion, retortion, torsion
•auction
•**absorption**, sorption
•**commotion**, devotion, emotion, groschen, Laotian, locomotion, lotion, motion, notion, Nova Scotian, ocean, potion, promotion
•**ablution**, absolution, allocution, attribution, circumlocution, circumvolution, Confucian, constitution, contribution, convolution, counter-revolution, destitution, dilution, diminution, distribution, electrocution, elocution, evolution, execution, institution, interlocution, irresolution, Lilliputian, locution, perlocution, persecution, pollution, prosecution, prostitution, restitution, retribution, Rosicrucian, solution, substitution, volution
•cushion • resumption • München
•pincushion
•**Belorussian**, Prussian, Russian
•**abduction**, conduction, construction, deduction, destruction, eduction, effluxion, induction, instruction, introduction, misconstruction, obstruction, production, reduction, ruction, seduction, suction, underproduction
•**avulsion**, compulsion, convulsion, emulsion, expulsion, impulsion, propulsion, repulsion, revulsion
•**assumption**, consumption, gumption, presumption
•**luncheon**, scuncheon, truncheon
•**compunction**, conjunction, dysfunction, expunction, function, junction, malfunction, multifunction, unction
•**abruption**, corruption, disruption, eruption, interruption
•T-junction • liposuction
•**animadversion**, aspersion, assertion, aversion, Cistercian, coercion, conversion, desertion, disconcertion, dispersion, diversion, emersion, excursion, exertion, extroversion, immersion, incursion, insertion, interspersion, introversion, Persian, perversion, submersion, subversion, tertian, version
•excerption

## Subtraction

# Subtraction

Subtraction is the operation that is the inverse of **addition** . Subtracting a number has the effect of nullifying the addition of that same number. For example, if one adds 11 to 41 then subtracts 11, the result is 41 again.

Symbolically, for any numbers a and b, (a + b) - b = a

Or one can subtract first, then add: (a - b) + b = a

Thus, one can say that subtraction and addition are "inverse operations."

## The definitions of subtraction

The second of the two rules above can be taken as a definition of subtraction. That is "the difference of two numbers, a - b, is the number which must be added to the second number, b, to equal the first, a." Before the widespread use of electronic cash registers, grocery clerks and others, making change, would use this definition directly. If, for example, a customer bought groceries worth $3.70 and gave the clerk $5.00 to pay for them, the clerk would not subtract in order to compute the change. Instead he (or she) would count it out. Starting with "three seventy," he would give the customer a nickel, and say, "three seventy-five." He would give the customer a quarter and say, "four." Then he would give the customer a dollar and say, "five." This method is still in use when no **calculator** is available.

A second definition of subtraction is often given in **algebra** books after the introduction of negative numbers. There a - b is defined as a + (-b), i.e. "To subtract a number, add its opposite." This definition is mathematically efficient, but it hides a lot of the ways in which subtraction is used, as in the change-making example above.

## Terminology

In a subtraction problem it is useful to have names for the various parts. In a - b the entire expression is called a "difference," and the answer to a - b is the difference, or occasionally the "remainder." These two terms arise from two ways in which subtraction is used in practical problems. The first number, a, is called the "minuend." This is the part that is diminished or made smaller when something is subtracted from it (provided that that something is positive). The second number, b, is the "subtrahend" or the part subtracted.

## Properties

It matters very much which of the two parts of a difference are named first. Taking $500 from an account with $300 in it is very different from taking $300 from $500. For this reason, subtraction is *not* commutative: a - b does not equal b - a.

Subtraction is not associative either: (a - b) - c does not equal a - (b - c).

An example will demonstrate this: (20 - 10) - 3 is 7, but 20 - (10 - 3) is 13.

Often one encounters expressions such as 5x2 - 2 - 3x2, with no indication of which subtraction is to be done first. Since subtraction is non-associative, it matters. To avoid this ambiguity one can agree that subtractions, unless otherwise indicated, are to be done left-to-right. This is a rather limiting agreement, therefore, it may be more convenient to use some other order. Another agreement, which is the common agreement of algebra, is to treat the minus sign as a plus-the-opposite-of sign. Thus one would interpret the example above as 5x2 + (-2) + (-3x2). In this interpretation it becomes a sum, whose terms can be combined in any order one pleases.

In certain sets subtraction is not a closed operation. The set of **natural numbers** , for instance, is not closed with respect to subtraction. If a merchant will not extend credit, one cannot buy an article whose price is greater than the amount of money one has.

## Uses of subtraction

The most familiar meaning for subtraction is called "take away." To find the answer if you take 5 eggs from a dozen, you subtract: 12 - 5 = 7.

Subtraction is also used to compare two numbers. When one wants to know how much colder -7.3° is than 13.8°, one computes the difference -7.3 - 13.8 to get -21.1°.

A third use of subtraction is to figure out a missing addend. If one has a certain sum of money, say $45.20 and wants to buy an article costing $85.50, subtraction is used to compute that needed amount: 85.50 - 45.20.

J. Paul Moulton

## subtract

**subtract**
•**abreact**, abstract, act, attract, bract, compact, contract, counteract, diffract, enact, exact, extract, fact, humpbacked, hunchbacked, impact, interact, matter-of-fact, pact, protract, redact, refract, retroact, subcontract, subtract, tact, tract, transact, unbacked, underact, untracked
•play-act • autodidact
•artefact (*US* artifact) • cataract
•contact
•**marked**, unremarked
•Wehrmacht
•**affect**, bisect, bull-necked, collect, confect, connect, correct, defect, deflect, deject, detect, direct, effect, eject, elect, erect, expect, infect, inflect, inject, inspect, interconnect, interject, intersect, misdirect, neglect, object, perfect, project, prospect, protect, reflect, reject, respect, resurrect, sect, select, subject, suspect, transect, unchecked, Utrecht
•prefect • abject • retroject • intellect
•genuflect • idiolect • dialect • aspect
•circumspect • retrospect • Dordrecht
•vivisect • architect • unbaked
•sun-baked

## subtract

sub·tract
/ səbˈtrakt/
•
v. [tr.]
take away (a number or amount) from another to calculate the difference:
*subtract 43 from 60.*
∎
take away (something) from something else so as to decrease the size, number, or amount:
*programs were added and subtracted as called for.*
DERIVATIVES:
sub·tract·er
n.
sub·trac·tive
/ -tiv/ adj.

## subtract

**subtract** †withdraw; deduct. XVI. f. *subtract-*, pp. stem of L. *subtrahere*, f. SUB- + *trahere* draw.

So **subtraction** †withdrawal XIV; taking of one quantity *from* another XV. — late L. *subtractiō*, *-ōn-*.

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**subtraction**