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"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, 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. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Addition is the mathematical process of putting things together Subtraction is denoted by a minus sign in infix notation. The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations 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

c − b = a

are minuend (c) − subtrahend (b) = difference (a). In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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 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 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Addition is the mathematical process of putting things together We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" 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. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result In mathematics anticommutativity refers to the property of an operation being anticommutative, i

Contents

Basic subtraction: integers

Imagine a line segment of length b with the left end labeled a and the right end labeled c. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end 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. Addition is the mathematical process of putting things together

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. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 ---- In mathematics Three is the first odd Prime number, and the second smallest prime 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, . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an . . ). 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, …). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. 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 an 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. Traditional mathematics (sometimes classical math education) is a term used to describe the predominant methods of Mathematics education in the United States 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. Principles and Standards for School Mathematics is a book produced by the National Council of Teachers of Mathematics (NCTM in 2000 to set forth a national vision for precollege

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

from minuend

mk mk−1 . . . m1,

where each si and mi is a digit, proceeds by writing down m1 − s1, m2 − s2, and so forth, as long as si does not exceed mi. Otherwise, mi 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 mi+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 si+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 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

See also

Algorithms

Notes and references

External links

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

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 −2 In Mathematics and Computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers A commonly known Subtraction method is the borrowing method. To perform a - b using this method b, the Subtrahend, is Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.

Dictionary

subtraction

-noun

  1. (arithmetic) (uncountable) The process of subtracting a number from another.
  2. (arithmetic) (countable) A calculation involving subtracting.
  3. The removal of something.
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