# Dictionary Definition

subtrahend n : the number to be subtracted from
the minuend

# User Contributed Dictionary

## English

### Noun

subtrahend (plural subtrahends)- A number or quantity to be subtracted from another.
- In the subtraction 10 − 4, 4 is the subtrahend.

# 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 Subtractionsubtrahend in Arabic: طرح

subtrahend in Aymara: Jakhuqawi

subtrahend in Belarusian: Адніманне

subtrahend in Breton: Lamadur

subtrahend in Bulgarian: Изваждане

subtrahend in Catalan: Resta

subtrahend in Czech: Odčítání

subtrahend in Danish: Subtraktion

subtrahend in German: Subtraktion

subtrahend in Modern Greek (1453-):
Αφαίρεση

subtrahend in Spanish: Resta

subtrahend in Esperanto: Operacioj per
nombroj

subtrahend in Basque: Kenketa

subtrahend in Persian: تفریق

subtrahend in French: Soustraction

subtrahend in Scottish Gaelic: Toirt air
falbh

subtrahend in Korean: 뺄셈

subtrahend in Indonesian: Pengurangan

subtrahend in Icelandic: Frádráttur

subtrahend in Italian: Sottrazione

subtrahend in Latin: Subtractio

subtrahend in Lithuanian: Atimtis

subtrahend in Dutch: Aftrekken

subtrahend in Japanese: 減法

subtrahend in Norwegian: Subtraksjon

subtrahend in Novial: Subtraktione

subtrahend in Polish: Odejmowanie

subtrahend in Portuguese: Subtração

subtrahend in Quechua: Qichuy

subtrahend in Russian: Вычитание

subtrahend in Simple English: Subtraction

subtrahend in Slovenian: Odštevanje

subtrahend in Finnish: Vähennyslasku

subtrahend in Swedish: Subtraktion

subtrahend in Tagalog: Pagbabawas

subtrahend in Tamil: கழித்தல் (கணிதம்)

subtrahend in Thai: การลบ

subtrahend in Turkish: Çıkarma

subtrahend in Urdu: تفریق (ریاضی)

subtrahend in Yiddish: אראפנעם

subtrahend in Chinese: 減法