# Dictionary Definition

subtraction

### Noun

1 an arithmetic operation in which the difference
between two numbers is calculated; "the subtraction of three from
four leaves one"; "four minus three equals one" [syn: minus]

2 the act of subtracting (removing a part from
the whole); "he complained about the subtraction of money from
their paychecks" [syn: deduction] [ant: addition]

# User Contributed Dictionary

## English

### Pronunciation

- /səbˈtɹækʃən/, /s@b"tr

# Extensive Definition

Subtraction is one of the four
basic arithmetic
operations; it is the inverse of addition, meaning that if we
start with any number and add any number and then subtract the same
number we added, we return to the number we started with.
Subtraction is denoted by a minus
sign in infix
notation.

The traditional names for the
parts of the formula

- c − b = a

Subtraction is used to model
four related processes:

- From a given collection, take away (subtract) a given number of objects. For example, 5 apples minus 2 apples leaves 3 apples.
- From a given measurement, take away a quantity measured in the same units. If I weigh 200 pounds, and lose 10 pounds, then I weigh 200 − 10 = 190 pounds.
- Compare two like quantities to find the difference between them. For example, the difference between $800 and $600 is $800 − $600 = $200. Also known as comparative subtraction.
- To find the distance between two locations at a fixed distance from starting point. For example if, on a given highway, you see a milage marker that says 150 miles and later see a milage marker that says 160 miles, you have traveled 160 − 150 = 10 miles.

In mathematics, it is often
useful to view or even define subtraction as a kind of addition, the addition of the
opposite. We can view 7 − 3 = 4 as the
sum of two terms:
seven and negative three. This perspective allows us to apply to
subtraction all of the familiar rules and nomenclature of addition.
Subtraction is not associative or commutative— in
fact, it is anticommutative—
but addition of signed numbers is both.

## Basic subtraction: integers

Imagine a line segment
of length b with the left
end labeled a and the right end labeled c. Starting from a, it
takes b steps to the right to reach c. This movement to the right
is modeled mathematically by addition:

- a + b = c.

From c, it takes b steps to
the left to get back to a. This movement to the left is modeled by
subtraction:

- c − b = a.

Now, imagine a line segment
labeled with the numbers 1, 2, and
3.
From position 3, it takes no steps to the left to stay at 3, so
3 − 0 = 3. It takes 2 steps to the left
to get to position 1, so 3 − 2 = 1.
This picture is inadequate to describe what would happen after
going 3 steps to the left of position 3. To represent such an
operation, the line must be extended.

To subtract arbitrary natural
numbers, one begins with a line containing every natural number
(0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to
get to 0, so 3 − 3 = 0. But
3 − 4 is still invalid since it again
leaves the line. The natural numbers are not a useful context for
subtraction.

The solution is to consider
the integer number line
(…, −3, −2, −1, 0, 1, 2, 3, …). From
3, it takes 4 steps to the left to get to −1, so

- 3 − 4 = −1.

## Algorithms for subtraction

There are various algorithms
for subtraction, and they differ in their suitability for various
applications. A number of methods are adapted to hand
calculation; for example, when making change, no actual
subtraction is performed, but rather the change-maker counts
forward.

For machine calculation, the
method
of complements is preferred, whereby the subtraction is
replaced by an addition in a modular arithmetic.

The method by which Elementary
school children are taught to subtract varies from country to
country, and within a country, different methods are in fashion at
different times. In traditional
mathematics, these are taught to children in elementary school
for use with multi-digit numbers, starting in the 2nd or last 1st
year, and the fourth or fifth grade for decimals. Such standard
methods have sometimes been omitted from some American standards-based
mathematics curricula in the belief that manual computation
fosters failure and is irrelevant in the age of calculator; in
texts such as TERC, students are
encouraged to invent their own methods of computation.

American schools currently
teach a method of subtraction using borrowing and a system of
markings called crutches. Although a method of borrowing had been
known and published in textbooks prior, apparently the crutches are
the invention of William A. Browell who used them in a study in
November of 1937. This system caught on rapidly, displacing the
other methods of subtraction in use in America at that
time.

European children are taught,
and some older Americans employ, a method of subtraction called the
Austrian method, also known as the additions method. There is no
borrowing in this method. There are also crutches (markings to aid
the memory) which vary according to country.

Both these methods break up
the subtraction as a process of one digit subtractions by place
value. Starting with a least significant digit, a subtraction of
subtrahend:

- sj sj−1 ... s1

- mk mk−1 ... m1,

Example:
704 − 512. The minuend is 704, the
subtrahend is 512. The minuend digits are m3 = 7, m2 = 0 and m1 =
4. The subtrahend digits are s3 = 5, s2 = 1 and s''1 = 2. Beginning
at the one's place, 4 is not less than 2 so the difference 2 is
written down in the result's one place. In the ten's place, 0 is
less than 1, so the 0 is increased to 10, and the difference with
1, which is 9, is written down in the ten's place. The American
method corrects for the increase of ten by reducing the digit in
the minuend's hundreds place by one. That is, the 7 is struck
through and replaced by a 6. The subtraction then proceeds in the
hundreds place, where 6 is no less than 5, so the difference is
written down in the result's hundred's place. We are now done, the
result is 192.

The Austrian method will not
reduce the 7 to 6. Rather it will increase the subtrahend hundred's
digit by one. A small mark is made near or below this digit
(depending of school). Then the subtraction proceeds by asking what
number when increased by 1, and 5 is added to it, makes 7. The
answer is 1, and is written down in the result's hundred's
place.

There is an additional
subtlety in that the child always employs a mental subtraction
table in the American method. The Austrian method often encourages
the child to mentally use the addition table in reverse. In the
example above, rather than adding 1 to 5, getting 6, and
subtracting that from 7, the child is asked to consider what
number, when increased by 1, and 5 is added to it, makes
7.

## References

- Browell, W. A. (1939). Learning as reorganization: An experimental study in third-grade arithmetic, Duke University Press.

- Subtraction in the United States: An Historical Perspective, Susan Ross, Mary Pratt-Cotter, The Mathematics Educator, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.) http://math.coe.uga.edu/TME/Issues/v10n2/5ross.pdf

## See also

## Notes and references

## External links

Printable Worksheets: One Digit Subtraction, Two Digit Subtraction, and Four Digit Subtractionsubtraction in Arabic:
طرح

subtraction in Aymara:
Jakhuqawi

subtraction in Belarusian:
Адніманне

subtraction in Breton:
Lamadur

subtraction in Bulgarian:
Изваждане

subtraction in Catalan:
Resta

subtraction in Czech:
Odčítání

subtraction in Danish:
Subtraktion

subtraction in German:
Subtraktion

subtraction in Modern Greek
(1453-): Αφαίρεση

subtraction in Spanish:
Resta

subtraction in Esperanto:
Operacioj per nombroj

subtraction in Basque:
Kenketa

subtraction in Persian:
تفریق

subtraction in French:
Soustraction

subtraction in Scottish
Gaelic: Toirt air falbh

subtraction in Korean:
뺄셈

subtraction in Indonesian:
Pengurangan

subtraction in Icelandic:
Frádráttur

subtraction in Italian:
Sottrazione

subtraction in Latin:
Subtractio

subtraction in Lithuanian:
Atimtis

subtraction in Dutch:
Aftrekken

subtraction in Japanese:
減法

subtraction in Norwegian:
Subtraksjon

subtraction in Novial:
Subtraktione

subtraction in Polish:
Odejmowanie

subtraction in Portuguese:
Subtração

subtraction in Quechua:
Qichuy

subtraction in Russian:
Вычитание

subtraction in Simple
English: Subtraction

subtraction in Slovenian:
Odštevanje

subtraction in Finnish:
Vähennyslasku

subtraction in Swedish:
Subtraktion

subtraction in Tagalog:
Pagbabawas

subtraction in Tamil:
கழித்தல் (கணிதம்)

subtraction in Thai:
การลบ

subtraction in Turkish:
Çıkarma

subtraction in Urdu: تفریق
(ریاضی)

subtraction in Yiddish:
אראפנעם

subtraction in Chinese:
減法

# Synonyms, Antonyms and Related Words

abatement, abridgment, absence, abstraction, addition, alienation, alleviation, approximation, attenuation, awayness, blank, contraction, dampening, damping, decrease, decrement, decrescence, deduction, deflation, depreciation, depression, deprivation, detachment, differentiation,
diminishment,
diminution, disarticulation,
disassociation,
disconnectedness,
disconnection,
discontinuity,
discount, disengagement, disjointing, disjunction, dislocation, disunion, division, divorce, divorcement, dying, dying off, equation, evolution, extenuation, extrapolation, fade-out,
incoherence,
integration,
interpolation,
inversion, involution, isolation, lack, languishment, lessening, letup, lowering, luxation, miniaturization,
mitigation, multiplication, neverness, nonexistence, nonoccurrence, nonpresence, notation, nowhereness, parting, partition, practice, proportion, rebate, reduction, relaxation, removal, sagging, scaling down, segmentation, separation, separatism, simplicity, subdivision, transformation, want, weakening, withdrawal, zoning