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In mathematics, subtraction is one of the four basic arithmetic operations. It is usually denoted by an infix minus sign.
The traditional names for the terms of the subtraction in the formula
- c − b = a
are minuend (c) − subtrahend (b) = difference (a).
Subtraction is used to model the following three physical processes. First, from a given collection, take away (subtract) a given number of objects. Second, combine a given measurement with an opposite measurement. For example, combine a movement right followed by a movement left, or combine a deposit and a withdrawal. Third, compare two objects to find their difference. For example, to find the difference between $800 and $600, subtract $800 − $600 = $200.
Mathematically, it is often useful to view subtraction as a kind of addition, addition of the opposite. Thus, we can view 7 − 3 = 4 as the sum of seven and negative three. This allows and requires us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not commutative or associative, but addition of signed numbers is. Professional mathematicians seldom used the words minuend and subtrahend but rather consider 7 − 3 = 4 to be the sum or two terms, seven and negative three.
Basic subtraction (case two above)
Imagine a straight line of length b painted on the ground with the left end labeled a and the right end labeled c.
Starting from position a, it takes you b steps to the right to reach position c. This movement to the right, called addition, can be stated as:
- a + b = c
From position c, it takes you b steps to the left to get back to a. This movement to the left, called subtraction, can be stated as:
- c − b = a
Now, imagine a line labelled with the numbers 1, 2, and 3.
From position 3, it takes no steps to the left to stay at position 3, so
- 3 − 0 = 3
From position 3, it only takes 1 step to the left to get to position 2, so
- 3 − 1 = 2
From position 3, it takes you 2 steps to the left to get to position 1, so
- 3 − 2 = 1
What would happen if you continued the process by going 3 steps to the left of position 3? For our example, you would walk off the end of the line which is not allowed. So, for this operation to be valid, the line must be extended.
For subtraction of natural numbers, the line would have every natural number (0, 1, 2, 3, 4, ...) on it.
Using the natural number line, from position 3, it takes you 3 steps to the left to get to position 0, so
- 3 − 3 = 0
But, for natural numbers, 3 − 4 is invalid since it leaves the line. So, for this operation to be valid, the line must be extended.
Using the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …), from position 3, it takes you 4 steps to the left to get to position −1, so
- 3 − 4 = −1
Algorithms for subtraction
External links
Printable Worksheets: One Digit Subtraction, Two Digit Subtraction, and Four Digit Subtraction
See also
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