Subtraction
Encyclopedia
In arithmetic
, subtraction is one of the four basic binary operation
s; 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
, in contrast to the use of the plus sign for addition.
Since subtraction is not a commutative operator, the two operands are named. The traditional names for the parts of the:c − b = a
are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are uncommon in modern usage. Instead we say that c and −b are terms, and treat subtraction as addition of the additive inverse
. The answer is still called the difference.
Subtraction is used to model four related processes:
In mathematics
, it is often useful to view or even define subtraction as a kind of addition
, the addition of the additive inverse
. We can view 7 − 3 = 4 as the sum of two terms
: 7 and -3. 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
and left-associative—but addition of signed numbers is both.
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
:
From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:
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 number
s, 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:
becomes problematic. For example, 3 − (−2) (i.e. subtract −2 from 3) is not immedious from either a natural number
view or a number line view, because it is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view subtraction as addition of signed numbers. Extra minus signs simply denote additive inversion
. Then we have 3 − (−2) = 3 + 2 = 5. This also helps to keep the ring
of integers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.
; 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
, a specific process is taught to children at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
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. Brownell who used them in a study in November 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:
from minuend
where each s_{i} and m_{i} is a digit, proceeds by writing down m_{1} − s_{1}, m_{2} − s_{2}, and so forth, as long as s_{i} does not exceed m_{i}. Otherwise, m_{i} is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit m_{i+1} by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit s_{i+1} by one.
Example: 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are m_{3} = 7, m_{2} = 0
and m_{1} = 4. The subtrahend digits are s_{3} = 5, s_{2} = 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 not 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 on the 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.
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
, subtraction is one of the four basic binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s; it is the inverse of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, 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
Plus and minus signs
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...
in infix notation
Infix notation
Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation or postfix notation Infix notation is the common arithmetic and logical formula notation,...
, in contrast to the use of the plus sign for addition.
Since subtraction is not a commutative operator, the two operands are named. The traditional names for the parts of the:c − b = a
are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are uncommon in modern usage. Instead we say that c and −b are terms, and treat subtraction as addition of the additive inverse
Additive inverse
In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
. The answer is still called the difference.
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 mileage marker that says 150 miles and later see a mileage marker that says 160 miles, you have traveled 160 − 150 = 10 miles.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, it is often useful to view or even define subtraction as a kind of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, the addition of the additive inverse
Additive inverse
In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
. We can view 7 − 3 = 4 as the sum of two terms
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...
: 7 and -3. 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
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
and left-associative—but addition of signed numbers is both.
Basic subtraction: integers
Imagine a line segmentLine segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
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
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
:
- 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 number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, 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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to get to −1:
- 3 − 4 = −1.
Subtraction as addition
There are some cases where subtraction as a separate operationOperation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
becomes problematic. For example, 3 − (−2) (i.e. subtract −2 from 3) is not immedious from either a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
view or a number line view, because it is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view subtraction as addition of signed numbers. Extra minus signs simply denote additive inversion
Additive inverse
In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
. Then we have 3 − (−2) = 3 + 2 = 5. This also helps to keep the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of integers "simple" by avoiding the introduction of "new" operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.
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 calculationElementary arithmetic
Elementary arithmetic is the simplified portion of arithmetic which is considered necessary and appropriate during primary education. It includes the operations of addition, subtraction, multiplication, and division. It is taught in elementary school....
; for example, when making change, no actual subtraction is performed, but rather the change-maker counts forward.
For machine calculation, the method of complements
Method of complements
In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. This method was commonly used in mechanical calculators and is still used in modern computers...
is preferred, whereby the subtraction is replaced by an addition in a modular arithmetic.
The method by which elementary school
Elementary school
An elementary school or primary school is an institution where children receive the first stage of compulsory education known as elementary or primary education. Elementary school is the preferred term in some countries, particularly those in North America, where the terms grade school and grammar...
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
Traditional mathematics
Traditional mathematics is a term used to describe the predominant methods of Mathematics education in the United States in the early-to-mid 20th century. The term is often used to contrast historically predominant methods with non-traditional approaches to math education...
, a specific process is taught to children at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
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. Brownell who used them in a study in November 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:
- s_{j} s_{j−1} ... s_{1}
from minuend
- m_{k} m_{k−1} ... m_{1},
where each s_{i} and m_{i} is a digit, proceeds by writing down m_{1} − s_{1}, m_{2} − s_{2}, and so forth, as long as s_{i} does not exceed m_{i}. Otherwise, m_{i} is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit m_{i+1} by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit s_{i+1} by one.
Example: 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are m_{3} = 7, m_{2} = 0
and m_{1} = 4. The subtrahend digits are s_{3} = 5, s_{2} = 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 not 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 on the 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.
See also
- Elementary arithmeticElementary arithmeticElementary arithmetic is the simplified portion of arithmetic which is considered necessary and appropriate during primary education. It includes the operations of addition, subtraction, multiplication, and division. It is taught in elementary school....
- Decrement
- Negative and non-negative numbersNegative and non-negative numbersA negative number is any real number that is less than zero. Such numbers are often used to represent the amount of a loss or absence. For example, a debt that is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase...
- Method of complementsMethod of complementsIn mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. This method was commonly used in mechanical calculators and is still used in modern computers...
External links
Printable Worksheets: One Digit Subtraction, Two Digit Subtraction, and Four Digit Subtraction- Subtraction Game at cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- Subtraction on a Japanese abacus selected from Abacus: Mystery of the Bead