# Subtraction

Subtraction is essentially the opposite of addition in that we are taking one number away from another rather than adding them together. Put another way, we are finding the *difference* between two numbers rather than their *sum*. In simple cases (i.e. with very small numbers), the fingers can be used to count back from a number providing the number to be subtracted is not greater than ten. Take the subtraction of seven from fifteen as an example. Count back from fifteen, using the fingers of both hands to keep track of how many you have subtracted. Decrementing (taking one away from) fifteen seven times gives *fourteen*, *thirteen*, *twelve*, *eleven*, *ten*, *nine*, *eight* - by which point you have used seven fingers and can stop counting back and the answer is eight. Like the *counting on* method used for simple addition, this process is somewhat laborious and only works for small numbers.

The simplest case of subtraction involves two numbers, one of which (sometimes called the *subtrahend*) is taken away from the other (sometimes referred to as the *minuend*). The notation used for representing the subtraction of one number from another shows the minuend and subtrahend separated by a *minus sign* ("-"), with the minuend on the left-hand side of the expression. For example, the addition we looked at above (fifteen minus seven) would be written as: "15 − 7". When more than two numbers are involved, the operation could be written as a series of subtraction operations. For example, the expression fifteen minus seven minus five would be written as: "15 − 7 − 5". Subtraction is not so straightforward as addition however, because subtraction is not *commutative*. This means that the order in which the numbers appear is critical to the outcome, unlike with addition. To illustrate this, the following expressions involving the same numbers yield very different results:

15 − 7 = 8

7 − 15 = −8

Note that in this notation (as with addition), the result follows an *equals sign*. Also in contrast to addition, subtraction is not associative in the sense that while the subtraction of two numbers (or subtrahends) from another number (the minuend) can be handled as two separate subtractions, we must subtract the first (left-most) subtrahend from the minuend first. We can then subtract the second subtrahend from the result. We could not get the same result by first subtracting the second subtrahend from the first, and then subtracting the result from the minuend. Both the order of the numbers and the order in which we carry out the individual subtraction operations must stay the same. To illustrate this, the following expressions involving the same numbers yield different results:

(15 − 7) − 5 = 3

15 − (7 − 5) = 13

As with addition, there are many techniques that have been used to carry out subtraction as mental arithmetic, and we will not go into them here. As is the case with addition, however, certain "subtraction facts" may be learned or derived from other known facts. A child may recall for example that ten minus five equals five, and reason that ten minus six must result in an answer that is one less and correctly conclude that the answer is four. Often with the passage of time, these derived facts are also committed to memory and can be recalled very quickly. We can extend the idea of presenting "addition facts" in tabular form to create a table of "subtraction facts", enabling us to fluently recall the result of subtracting one number from another where both numbers can take any value from one two ten. Such a facility can give us a basis on which to derive the answer to more complex subtraction problems. The "subtraction fact" table is presented below. The intersection of each row and column gives the *difference* between the number at the start of the row and the number at the top of the column. Note that for subtractions involving zero, the subtraction of zero from any number simply yields that number. The subtraction of any number from zero yields a number with the same absolute value but the opposite sign. In all cases where one positive (i.e. greater than zero) integer is subtracted from another, the *sign* of the result (i.e. whether it will be positive or negative) will depend on whether the subtrahend is smaller than the minuend (in which case the result is positive) or larger (in which case the result is negative). If both minuend and subtrahend have the same value, the answer will always be zero.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 | 5 |

6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 |

7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 | 3 |

8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 | 2 |

9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 1 |

10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |

The availability of a "lookup-table" does not significantly extend the range of subtraction problems that can quickly be solved beyond the simple technique of *counting back* using fingers, although it may be slightly quicker. Memorisation of the subtraction facts, together with the use of various techniques for deriving hitherto unknown facts, allows us to solve more complex subtraction problems. Some people can subtract one or more relatively large (three or four-digit) numbers from another relatively large number using only mental arithmetic. As with addition, as the numbers become larger it is necessary to use more sophisticated methods to derive the correct result. While the availability of pocket calculators may appear to render these techniques somewhat redundant, a working knowledge of them is necessary in order to engender a thorough understanding of more advanced mathematical concepts. While the methods used have varied as ideas about education have changed, they represent a basic set of problem solving skills that facilitate the acquisition of the more sophisticated skills required to solve more complex problems.

While the subtraction of one relatively small number from another can often be carried out using mental arithmetic, subtraction problems involving several three- or four-digit numbers without the aid of a calculator may necessitate the use of a pencil (or pen) and paper. As with addition, subtraction involving several multiple-digit numbers can be done using a *columnar* method in which the subtrahends for one column are subtracted from the minuend for that column, and the result is written beneath the column. The process starts with the right-most column and proceeds from right to left. The main difference is that when adding, we can potentially have a result for a single column that exceeds ten. If that happens, we place just the right-most digit of the result beneath the current column and place remaining digit (the "carry") beneath the column immediately to the left of the current column to be in the addition for that column. With subtraction, the opposite situation can arise in which we end up with a negative number as a result of subtracting the subtrahends from the minuend. In this case, we must "borrow" an amount from the column immediately to the left of the current column in order to achieve a non-negative result. This amount is deducted from the minuend in the column from which it is being borrowed, which is annotated by striking through the minuend and writing its new value above it. The following example should clarify how this works:

0 0 7 0

0 7 9 1 5

− 0 4 8 7

− 0 0 3 6

= 7 3 9 2

In the above problem, we are subtracting 487 and 36 from 7915. In the first (rightmost) column, subtracting 7 and 6 from 5 will lead to a negative result, so we borrow the 1 from the column immediately to the left of the current column (or if you like, the "tens" column) to give us 15. This produces a result in the current column of 2 (15 − 7 − 6 = 2), but leaves us with 0 in the "tens" column. We now have to take 8 and 3 away from 0 which will obviously give us a negative result. Borrowing 1 from the "hundreds" column would also give a negative result, so we borrow 2 from this column to leave 7 in place of the original 9, and give us a result beneath the current column of 9 (20 − 8 − 3 = 9). The next ("hundreds") column is more straightforward (7 − 4 = 3), and the last ("thousands") column involves no further subtraction so the 7 does not change. Subtraction involving real numbers can be carried out in similar fashion, provided you remember to maintain the correct position for the decimal point (the decimal point in all of the numbers involved should be vertically aligned). The example below illustrates how this works.

0 0 7 5 4 . 1 1

0 4 8 7 6 . 2 3 0 0

− 0 1 2 9 . 7 6 5 0

− 0 0 9 7 . 5 3 6 4

= 4 6 4 8 . 9 2 8 6

Notice in the above example that the subtrahend has no digits in the third and fourth columns to the right of the decimal point. We have substituted zeros, but in the right-most columnar subtraction operation there is nothing in the column immediately to the left of it from which to borrow. We therefore need to borrow 2 from the second column after the decimal point (if you like, the "hundredths" column). This would give us 20 in the "thousandths" column, but we need to borrow one from this number to give us 10 in the last column (the "ten-thousandths" column). We can then subtract 4 from 10 to leave 6 as the result for the "ten-thousandths" column. In the "thousandths" column, 19 − 5 − 6 leaves 8 as the result. From that point onwards, the borrowing follows a more normal pattern.

There are a number of methods that can be used for manual calculations involving subtraction, and the method one person finds easiest may not suit another person. You should try different methods until you find one you like, institutional constraints notwithstanding. As with addition, subtraction may be applied to quantities other than pure numbers, such as length, area, volume and so on. Again, you must make sure that the quantities involved are expressed in the same units. You cannot subtract centimetres from metres or grams from kilograms. The values must be expressed using the same units. Note also that you will sooner or later encounter expressions in which one or more of the subtrahends are negative. The minuend itself could also be negative, and care must always be taken when evaluating such expressions. In the following example, the subtraction of a negative subtrahend from a positive minuend is equivalent to *adding* the two numbers together as if they were both positive:

15 − (− 6) = 21

This expression effectively equates to:

15 + 6 = 21

The two minus signs in this case equate to a single plus sign. Something similar happens when you *add* a negative number to a positive number, in which case the result is the same as subtracting the absolute value of the second number from the first:

15 + (− 6) = 9

This expression effectively equates to:

15 − 6 = 9