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Encyclopedia > Subtraction
"5 - 2 = 3" (verbally, "five minus two equals three")
"5 - 2 = 3" (verbally, "five minus two equals three")
An example problem
An example problem

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. Subtraction is denoted by a minus sign in infix notation. Image File history File links Subtraction01. ... Image File history File links Subtraction01. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) 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. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... The plus and minus signs (+ and −) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ... Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on (e. ...


The traditional names for the parts of the formula In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...

cb = a

are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are uncommon in modern usage. However, "difference" is very common.


Subtraction is used to model several closely related processes:

  1. From a given collection, take away (subtract) a given number of objects.
  2. Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal.
  3. Compare two objects to find their difference. For example, the difference between $800 and $600 is $800 − $600 = $200.

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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ...

Contents

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: Image File history File links This is a lossless scalable vector image. ... The geometric definition of a line 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. ... Look up length, width, breadth in Wiktionary, the free dictionary. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...

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:

cb = a.

Bold text

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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. Subtraction on a line segment. ... ‹ The template below is being considered for deletion. ... This article does not cite any references or sources. ... Look up three in Wiktionary, the free dictionary. ...


To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, ...). 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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...


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 The integers are commonly denoted by the above symbol. ...

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. Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ...


For machine calculation, the method of complements is preferred, whereby the subtraction is replaced by a addition in a modular arithmetic. In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. ...


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. Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives. ... Principles and Standards for School Mathematics is a document produced in 1989 by the National Council of Teachers of Mathematics [5] (NCTM) to set forth a national vision for precollege mathematics education in the US and Canada. ... Investigations in Number, Data, and Space is a complete K-5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ...


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:

 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 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.

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

See also

Algorithms

Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... An increment is an increase, either of some fixed amount, for example added regularly, or of a variable amount. ... A negative number is a number that is less than zero, such as −3. ... In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. ... Background. ...

Notes and references

External links

Printable Worksheets: One Digit Subtraction, Two Digit Subtraction, and Four Digit Subtraction


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Subtraction is considered to be a binary operation because it is performed on two numbers.
In subtraction, you start on the right (the ones column), and subtract each column in turn until you are done (subtracting the tens column, then the hundreds column, etc.).
In a column, if the digit you are subtracting (in the subtrahend) is bigger than the digit you are subtracting from (in the minuend), you have to borrow a one from the next column to the left (the borrowed digit is a ten in the column to the right).
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