Subtraction

Subtraction (which is signified by the minus sign −) is one of the four arithmetic operations along with addition, multiplication and division.

Subtraction is an operation that represents removal of objects from a collection.

While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication.

All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond.

General binary operations that follow these patterns are studied in abstract algebra.

[3] Subtraction is usually written using the minus sign "−" between the terms; that is, in infix notation.

"Subtraction" is an English word derived from the Latin verb subtrahere, which in turn is a compound of sub "from under" and trahere "to pull".

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: From c, it takes b steps to the left to get back to a.

This movement to the left is modeled by subtraction: Now, a line segment labeled with the numbers 1, 2, and 3.

This picture is inadequate to describe what would happen after going 3 steps to the left of position 3.

Such a case uses one of two approaches: The field of real numbers can be defined specifying only two binary operations, addition and multiplication, together with unary operations yielding additive and multiplicative inverses.

In general, the expression can be defined to mean either (a − b) − c or a − (b − c), but these two possibilities lead to different answers.

In what is known in the United States as traditional mathematics, a specific process is taught to students 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.

Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches.

[11][12] Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after William A. Brownell published a study—claiming that crutches were beneficial to students using this method.

[13] This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

There are also crutches (markings to aid memory), which vary by country.

Starting with a least significant digit, a subtraction of the subtrahend: from the minuend 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.

The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one.

Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7.

There is an additional subtlety in that the student always employs a mental subtraction table in the American method.

The Austrian method often encourages the student to mentally use the addition table in reverse.

Example:[citation needed] A variant of the American method where all borrowing is done before all subtraction.

Another method that is useful for mental arithmetic is to split up the subtraction into small steps.