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Note on the graphicalrepresentation of directednumbersWm. Malcolm aa Training College for Teachers of TechnicalSubjects , LondonPublished online: 30 Jul 2007.

To cite this article: Wm. Malcolm (1952) Note on the graphical representationof directed numbers, The Vocational Aspect of Education, 4:8, 62-65, DOI:10.1080/03057875280000051

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NOTE ON THE GRAPHICAL REPRESENTATION OF DIRECTED NUMBERS

By WM. MALCOLM Student, Training College for Teachers of Technical Subjects, London

A VERY POPULAR method of teaching the rules for the addition and subtraction of positive and negative numbers is by reference to operations with directed lengths along a straight line. In practice, however, this is usually more or less badly done, and the usefulness of the demonstration is unnecessarily limited.

We may begin, for example, with the following diagram:

~ ...... i . I I I I I l I , I

-5 - 4 -3 -2 -1 0 +t +2 +3 +4- +5

Fi 9. 1 where movement to the left from the point of origin is regarded as negative (--) , and movement to the right as positive (+ ) . From this it is easy to show that:

+ A + + B = + C and that - -A + --B = --C.

But we soon see that we cannot subtract + B from +A, if + B is greater than +A . The solution is simple, apparently. Our diagram is altered:

- 1 - 1

+1 +1

Fi 9. 2

so that movement anywhere along the line to the left is negative, and to the right, positive. Now, taking our point of origin wherever convenient a]ong the line, we can demonstrate the addition of any pair of numbers without difficulty; as, for instance:

Fig. 3 showing that:

+4 4- --6 = --2. At this point, it becomes clear that cardinal enumeration of the units along

the line (as in Figure I) is anuisance, and should be abandoned. We must be

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W M . M A L C O L M 6 3

free to read off our lengths as we choose. Text-books, however, seem very reluctant to give up the practice, and we are usually offered Figure i, with the qualifying information that if the units are anywhere read off in the opposite direction their sign value is reversed.

This is not merely pointless, but dangerous. I t leads to mistakes, as in Figure 4,

-5 - 4 -3 -2 -1 0 +1 +2 +3 +4- +5

Fig. 4-

where the unwary teacher thinks he has demonstrated that

+ 3 - - + 2 = + i

and is puzzled because he cannot, so easily, show that

+ 3 - - --2 = +5 .

Reference to Figure 3 will show that he has not, in fact, subtracted + 2 from +3" He has really added --2 to +3 , and is deceived by the fact that the result is the same in both cases.

In short, as far as these graphical operations are concerned, he does not know the difference between adding and subtracting; and so he lands in difficulties. How shall we answer the teacher who asks, in relation to Figure 4: ' I f this is not subtracting, what is?'

The text-books generally set out to show how such operations may be repre- sented graphically; and do so--for addition. But when they come to subtrac- tion they abandon the attempt. The student is asked to realise tha t - - 'Sub t rac - tion can be made to depend upon Addit ion' ; and then follows a purely verbal exposition, from which the rules for subtraction are obtained.

We must first realise that in such expressions as + A + +B, and + A -- --B, the symbols ' + ' and ' - - ' have more than one meaning in each case. On the one hand they mean the 'direction' of the numbers concerned; and, on the other hand, they mean 'added to' or 'subtracted from'.

In order that our graphical operations should adequately represent such expressions they must have a true logical correspondence with them, i.e. in the expression . . . + + B each ' + ' must be differently represented since its meaning is different.

As far as the ' + ' s are concerned, this is quite simple; and we seem to do it automatically. Reference, again, to Figure 3 shows the 'directional ' ' + ' as represented by movement to the right; while the 'addit ion' ' + ' is represented by tacking the second directed length on to the end of the first, and reading the answer from the point of origin to the end of the second length.

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6 4 Graphical Representation of Directed Numbers

Logically speaking, this is how we define these graphical operations. Expressed symbolically:

ok + o~ = o~ (where ---> shows that the length is directed).

I t is then possible to give a simple graphical definition of the 'subtraction'

In Figure 5 we compare + 2 with +3 . The 'difference' between them is + t

0

Fi 9. 5

or --z, depending upon how we read it. Clearly, if we read from th e end of + 2 to the end of + 3 we get the result of the subtraction sum

+ 3 -- + 2 = --~I. While if we read from the end of + 3 to the end of +2 , we get the result of the sum

+ 2 -- + 3 = -- x. Equally simply, we can read from the diagram the results of

- - 3 - - - - 2 ~- - - - - I .

and of --2 -- --3 = + z .

From this we obtain our graphical definition of the subtraction ' - - ' , as to subtract one directed length from another we lay them out from a common point of origin and read the answer from the limit of the subtrahend to the limit of the minuend.

Symbolically:

o k - o ~ = BA'. There are two cases where the result of applying this definition may look

somewhat odd. Nevertheless, it will be seen that they give a logically sound and convincing representation of the mathematical operations involved. In Figure 6

I ~ ~ ) ) ; ~.. ) ) I

I , I I ~ . < < < < i > > "~ I | I I I , I ~ I I 0

we demonstrate:

Fig. 6

+ 3 -- --5 = + 8

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W M , M A L C O L M 6 5

and in Figure 7

I < < < . . . ~ < < < ., ,

[ " ' 1 |

i ! t I , ! ! ,, ! , I I I

o

Fig. 7 we demonstrate

- -5 - - + 4 = --9. O f course, the sole point of representing subtraction of directed numbers in

this way is that it should serve as a readily comprehensible interpretation of the mathemat ical operations concerned. The student should be able to see that there is meaning in these operations in this way, and should be able to grasp the general rules for such operations from this part icular illustration of them.

I f the student has to puzzle as hard over the illustration as he would over a purely symbolic explanation, the whole thing is a waste of time.

But I believe this method will be found helpful, especially if the more difficult cases illustrated in Figures 6 and 7 are introduced, as here, by the simpler cases of Figure 5, the definition of subtraction being derived from these.

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