Heterodiagnóstico

Let us begin with an informal discussion about subtraction. The subtraction of 15 from 37, 37−15, is the number of steps it takes to go from 15 to 37. In 71 http://dx.doi.org/10.1090/mbk/079/05

terms of the number line, each step between whole numbers has unit length. Therefore, this means (37− 15) is the length of the line segment [15, 37]:

0 15 37

(37− 15)

  

This picture exhibits the fact that the concatenation of [0, 15] and a segment of length (37−15) has length 37. Recalling the meaning of addition in terms of the concatenation of segments, we arrive at the fact that 37− 15 is the number so that 37 = 15+(37−15), or, since addition is commutative, 37−15 is the number so that 37 = (37− 15) + 15. The geometric interpretation of the latter is the following:

0 (37−15) 37

15

  

To summarize these two pictures, we see that (37− 15) is the length of the segment left behind when a segment of length 15 is removed from either end of [0, 37].

This discussion gives the motivation for the following formal definition of subtraction. First, we agree to use m ≥ n to denote “either m > n or m = n”. (Similarly, m ≤ n means “either m < n or m = n”; see Exercise 5 on page 54.) Now, if m, n are whole numbers and m ≥ n, we define the subtraction of n from m, in symbols m−n, as the whole number so that when it is added to n we get back m. In symbols, for any whole numbers m, n, with m ≥ n, if k denotes m − n, then by definition,

k + n = m.

Since addition is commutative, the formal definition can be equivalently rephrased as

(5.1) m − n is the whole number k that satisfies m = k + n.

It is common also to refer to m − n as the difference between m and n.

The formal definition (5.1) exhibits subtraction as an alternate but equiv- alent way of expressing addition, in the sense that, if m ≥ n, then instead of writing m = k + n we can write m − n = k, and vice versa.

Activity. Use the formal definition of subtraction to compute and ex- plain: 1200 − 500 = ?; 580,000,000 − 500,000,000 = ?; 580,000,000 − 20,000,000 = ?; 15× 106_{− 7 × 10}6 _{= ?}

5.2. The Subtraction Algorithm 73

It follows from the definition that m = (m − n) + n. So in terms of the number line, m − n has the following geometric meaning:

0 m − n m

n

  

Since it is equally valid that m = n + (m − n), we also have the following picture:

0 n m

m − n

  

Therefore, an equivalent definition of m − n for whole numbers m, n satis- fying m ≥ n is that m − n is the length of the remaining segment when a segment of length n is removed from either end of [0, m].

In Chapter 2, we discussed the distributive law. It should be mentioned that there is also a distributive law for subtraction: For all whole num- bers k, m, n where m ≥ n,

k(m − n) = km − kn.

If k = 3, m = 6, and n = 2, then this law says 3 × (6 − 2) = (3 × 6) − (3 × 2). The truth of this assertion for the area model can be immediately seen from following picture.

At least for the area model, the reasoning for the general case is the same; see also Exercise 15 at the end of this section.

We hasten to add that this is not a new law of operations. When we have negative numbers at our disposal in Part 3, we will show (page 411) that the distributive law for subtraction is included in the ordinary distributive law.

5.2. The Subtraction Algorithm

According to the definition of subtraction in the preceding section, the dif- ference of two numbers, such as 1658− 257, is obtained by counting the number of steps it takes to go from 257 to 1658. As is the case with the addition algorithm, the purpose of the standard subtraction algorithm is

to relieve the tedium of counting by offering a shortcut. The mathematics underlying this algorithm (to be introduced presently) is similar to that of the addition algorithm.

First, assume that we know how to do single-digit subtraction. (See the leitmotif on page 59) Now, let m, n be arbitrary whole numbers so that m > n. We want to compute m − n. Now put the number m above n, and line them up so that digits in the same column have the same place value. For example, 1658− 257 would be configured as:

1 6 5 8

− 2 5 7

? ? ? ?

As in the case of the addition algorithm, we begin with the simplest case, where each digit of m is at least as big as the digit of n in the same column. The preceding example of 1658− 257, which as we know is the same as 1658−0257, illustrates the simple case that we have in mind, because 1 ≥ 0, 6≥ 2, 5 ≥ 5, and 8 ≥ 7. In this simple case, the subtraction algorithm states that the digits of m − n are obtained by performing the single-digit subtraction in each column.1 Thus, from the four single-digit subtractions 8−7 = 1, 5−5 = 0, 6−2 = 4, and 1−0 = 1, we conclude 1658−257 = 1401. Schematically: (5.2) 1 6 5 8 − 2 5 7 1 4 0 1

Activity. Use mental math to compute 493,625 − 273,514 and 57,328,694 − 4,017,382.

In general, there will be columns in which the digit of m is less than the corresponding digit of n. For example, to compute 756 − 389, we have 6 < 9 in the ones column and 5 < 8 in the tens column. Then the subtractions within columns, 6− 9 and 5 − 8, cannot be performed using whole numbers. The subtraction algorithm now calls for trading, in the following sense (for notational simplicity, we state it explicitly for the subtraction 756− 389):

Start from the right with the ones column: change the subtraction from 6− 9 to 16 − 9 and at the same time decrease the next digit (to the left) from 5 to 4 (= 5− 1). The subtraction 4 − 8 in the tens column2 is handled the same way: change the subtraction from 4− 8 to 14 − 8 and at the same time decrease the next digit 1Recall in this context the leitmotif of Chapter 3 (page 59).

2This subtraction was 5 − 8, but is now 4 − 8 as a result of decreasing the tens digit from 5 to 4.