Placard outside a shop inBordeaux advertising subtraction of 20% from the price of the second perfume purchased.
Subtraction (which is signified by theminus sign, −) is one of the fourarithmetic operations along withaddition,multiplication anddivision. Subtraction is an operation that represents removal of objects from a collection.[1] For example, in the adjacent picture, there are5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, thedifference of 5 and 2 is 3; that is,5 − 2 = 3. While primarily associated with natural numbers inarithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects includingnegative numbers,fractions,irrational numbers,vectors, decimals, functions, and matrices.[2]
In a sense, subtraction is the inverse of addition. That is,c =a −bif and only ifc +b =a. In words: the difference of two numbers is the number that gives the first one when added to the second one.
Subtraction follows several important patterns. It isanticommutative, meaning that changing the order changes the sign of the answer. It is also notassociative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because0 is theadditive identity, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such asaddition andmultiplication. All of these rules can beproven, starting with the subtraction ofintegers and generalizing up through thereal numbers and beyond. Generalbinary operations that follow these patterns are studied inabstract algebra.
Subtraction is usually written using theminus sign "−" between the terms; that is, ininfix notation. The result is expressed with anequals sign. For example, (pronounced as "two minus one equals one") and (pronounced as "four minus six equals negative two"). Nonetheless, some situations where subtraction is "understood", even though no symbol appears; inaccounting, a column of two numbers, with the lower number in red, usually indicates that the lower number in the column is to be subtracted, with the difference written below, under a line.[4]
The number being subtracted is thesubtrahend, while the number it is subtracted from is theminuend. The result is thedifference, that is:[5]
All of this terminology derives fromLatin. "Subtraction" is anEnglish word derived from the Latinverbsubtrahere, which in turn is acompound ofsub "from under" andtrahere "to pull". Thus, to subtract is todraw from below, or totake away.[6] Using thegerundivesuffix-nd results in "subtrahend", "thing to be subtracted".[a] Likewise, fromminuere "to reduce or diminish", one gets "minuend", which means "thing to be diminished".
Imagine aline segment oflengthb with the left end labeleda and the right end labeledc.Starting froma, it takesb steps to the right to reachc. This movement to the right is modeled mathematically byaddition:
a +b =c.
Fromc, it takesb steps to theleft to get back toa. This movement to the left is modeled by subtraction:
c −b =a.
Now, a line segment labeled with the numbers1,2, and3. From position 3, it takes no steps to the left to stay at 3, so3 − 0 = 3. It takes 2 steps to the left to get to position 1, so3 − 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 arbitrarynatural numbers, 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, so3 − 3 = 0. But3 − 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 theintegernumber line (..., −3, −2, −1, 0, 1, 2, 3, ...). This way, it takes 4 steps to the left from 3 to get to −1:
Subtraction ofnatural numbers is notclosed: the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches:
Conclude that 26 cannot be subtracted from 11; subtraction becomes apartial function.
Give the answer as aninteger representing anegative number, so the result of subtracting 26 from 11 is −15.
Thefield of real numbers can be defined specifying only two binary operations, addition and multiplication, together withunary operations yieldingadditive andmultiplicative inverses. The subtraction of a real number (the subtrahend) from another (the minuend) can then be defined as the addition of the minuend and the additive inverse of the subtrahend. For example,3 −π = 3 + (−π). Alternatively, instead of requiring these unary operations, the binary operations of subtraction anddivision can be taken as basic.
Subtraction isanti-commutative, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, ifa andb are any two numbers, then
Subtraction isnon-associative, which comes up when one tries to define repeated subtraction. In general, the expression
"a −b −c"
can be defined to mean either (a −b) −c ora − (b −c), but these two possibilities lead to different answers. To resolve this issue, one must establish anorder of operations, with different orders yielding different results.
In the context of integers, subtraction ofone also plays a special role: for any integera, the integer(a − 1) is the largest integer less thana, also known as the predecessor ofa.
When subtracting two numbers with units of measurement such askilograms orpounds, they must have the same unit. In most cases, the difference will have the same unit as the original numbers.
As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is20% − 30%/30% = −1/3 =−33+1/3%, while the percentage point change is −10 percentage points.
Themethod of complements is a technique used to subtract one number from another using only the addition of positive numbers. This method was commonly used inmechanical calculators, and is still used in moderncomputers.
Binary digit
Ones' complement
0
1
1
0
To subtract a binary numbery (the subtrahend) from another numberx (the minuend), the ones' complement ofy is added tox and one is added to the sum. The leading digit "1" of the result is then discarded.
The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing "0" to "1" and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example:
Methods used to teach subtraction toelementary school vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States astraditional 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 includedecimal 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.[10][11] Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread afterWilliam A. Brownell published a study—claiming that crutches were beneficial to students using this method.[12] This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
Some European schools 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 memory), which vary by country.[13][14]
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 the subtrahend:
sjsj−1 ...s1
from the minuend
mkmk−1 ...m1,
where eachsi andmi is a digit, proceeds by writing downm1 −s1,m2 −s2, and so forth, as long assi does not exceedmi. 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 digitmi+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 digitsi+1 by one.
Example: 704 − 512.
The minuend is 704, the subtrahend is 512. The minuend digits arem3 = 7,m2 = 0 andm1 = 4. The subtrahend digits ares3 = 5,s2 = 1 ands1 = 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's place. In the ten's place, 0 is less than 1, so the 0 is increased by 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 does not reduce the 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the 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 hundreds place.
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. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.
In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example:[citation needed]
3 − 1 = ...
We write the difference under the line.
5 − 9 = ... The minuend (5) is too small!
So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1.
15 − 9 = ... Now the subtraction works, and we write the difference under the line.
The partial differences method is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total difference.[16]
Example:
The smaller number is subtracted from the greater: 700 − 400 = 300 Because the minuend is greater than the subtrahend, this difference has a plus sign.
The smaller number is subtracted from the greater: 90 − 50 = 40 Because the minuend is smaller than the subtrahend, this difference has a minus sign.
The smaller number is subtracted from the greater: 3 − 1 = 2 Because the minuend is greater than the subtrahend, this difference has a plus sign.
The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend does not change the answer. One simply adds the amount needed to get zeros in the subtrahend.[19]
^Paul E. Peterson, Michael Henderson, Martin R. West (2014)Teachers Versus the Public: What Americans Think about Schools and How to Fix Them Brookings Institution Press, p. 163
^Janet Kolodzy (2006)Convergence Journalism: Writing and Reporting across the News Media Rowman & Littlefield Publishers, p. 180
^David Gillborn (2008)Racism and Education: Coincidence Or Conspiracy? Routledge p. 46
^Susan Ross and Mary Pratt-Cotter. 2000. "Subtraction in the United States: An Historical Perspective,"The Mathematics Educator 8(1):4–11. p. 8: "This new version of the decomposition algorithm [i.e., using Brownell's crutch] has so completely dominated the field that it is rare to see any other algorithm used to teach subtraction today [in America]."
^Ross, Susan C.; Pratt-Cotter, Mary (1999). "Subtraction From a Historical Perspective".School Science and Mathematics.99 (7):389–93.doi:10.1111/j.1949-8594.1999.tb17499.x.
