Subtraction is one of the four basic 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. In common usage, it refers to the simpler properties when operations In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values. There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values,; it is the inverse of 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. Therefore, 3 + 2 = 5. Besides counts of fruit,, 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 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. Plus and minus are Latin terms meaning "more" and "less", respectively in 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 ( e.g. + 2 2 ) or postfix notation ( e.g. 2 2 + ), but many programming languages use it due to its familiarity.

The traditional names for the parts of the formula In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language

c − b = a

are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are uncommon in modern usage^[1]. Instead we say that c and −b are terms, and treat subtraction as addition of the opposite. The answer is still called the difference.

Subtraction is used to model four related processes:

1. From a given collection, take away (subtract) a given number of objects. For example, 5 apples minus 2 apples leaves 3 apples.
2. 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.
3. 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.
4. 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 is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, it is often useful to view or even define subtraction as a kind of 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. Therefore, 3 + 2 = 5. Besides counts of fruit,, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms The word term is from the Latin terminus which literally means "boundary line, limit", from the Proto-Indo-European root *ter- "peg, post, boundary": 7 and -3. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such or commutative In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the—in fact, it is anticommutative In mathematics, anticommutativity refers to the property of an operation being anticommutative, i.e. being non-commutative in a precise way. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence in physics: they are often called antisymmetric operations—but addition of signed numbers is both.

Basic subtraction: integers

Imagine 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. 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 is either an edge if they are adjacent vertices, or of length 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 thickness. Another example is FET transistors, in which the channel width may be larger than channel 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 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. Therefore, 3 + 2 = 5. Besides counts of fruit,:

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 1 is a number, numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement. For example, a line segment of "unit length" is a line segment of length 1, 2 2 (pronounced /ˈtuː/ ( listen)) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3, and 3 3 is a number, numeral, and glyph. It is the natural number following 2 and preceding 4. 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 In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century, 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 The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6 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.

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 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 is preferred, whereby the subtraction is replaced by an addition in a modular arithmetic.

The method by which elementary school An elementary 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. Primary school is the preferred term in the United Kingdom, India, Pakistan, Australia, Latin America, South 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 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. Traditional mathematics education has been challenged by several reform, 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^{[citation needed]}. 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^{[citation needed]}. 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^{[citation needed]}.

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 ... s₁

from minuend

m_k m_k−1 ... m₁,

where each s_i and m_i is a digit, proceeds by writing down m₁ − s₁, m₂ − s₂, 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 m₃ = 7, m₂ = 0 and m₁ = 4. The subtrahend digits are s₃ = 5, s₂ = 1 and s₁ = 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 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.

See also

• Elementary 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
• Decrement An increment is an increase of some amount, either fixed or variable. For example one's salary may have a fixed annual increment or one based on a percentage of its current value. A decrease is called a decrement
• Negative and non-negative numbers Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, −1.414, −1. A positive number is one that is greater than zero, such as , 1.414, 1. Zero itself is neither positive nor negative. The

Algorithms

• Method of complements
• Subtraction without borrowing

Notes and references

1. ^ *Linderholm, Carl (1971). Mathematics made difficult. Wolfe. p. 42. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-7234-0415-1.

• 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

External links

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

• Subtraction Game at cut-the-knot Cut-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, the Encyclopedia Britannica's Internet Guide Award,
• Subtraction on a Japanese abacus selected from Abacus: Mystery of the Bead

Categories: Elementary arithmetic | Binary operations

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