Inside the 128

Numbers which won't add up

THERE'S ONE TOPIC which appears in my postbag with the same kind of regularity as those infernal competition offers from Reader's Digest. Every mortal has his cross to bear, and the rounding error seems to be mine.

On this particular occasion it is John Jennings of Staines who is suffering from rounding errors. He clearly did not read my article in the May 1982 issue of Sinclair User when I first wrote about the problem! He includes a copy of the program which is giving him trouble and I have listed it in table one. The screen display when the program is executed is shown in table two.

Table 1. John Jennings' test program
demonstrating the effect of rounding errors

10 LET X=3
20 FOR I=1 TO 10
30 LET X=X-0.5
40 PRINT X
50 IF X=2.5 THEN PRINT "Two point five"
60 IF X=-1 THEN PRINT "Minus one"
70 NEXT I

The output from the program listed
in Table 1

2.5
Two point five
2
1.5
1
0.5
2.3283064E-10
-0.5
-1
-1.5
-2

John writes: I typed this test program into the Spectrum Plus demonstration model at a local branch of W H Smith and got the same result as with my own machine.

I really want to know whether there is a fault in my machine or whether this is a quirk of the Spectrum.

It is not a fault with the machine and it is only a quirk of the Spectrum in the sense that the computer tries in a rather half-hearted fashion to hide a problem with the nature of arithmetic from ordinary users. The problem is related to the fact that it is impossible to express some fractions exactly without rounding errors.

For example, we all know the following statements are incorrect:

1/3 = 0.3
1/3 = 0.33
1/3 = 0.333

No matter how many trailing threes we place at the end of the number the result is still incorrect, even if only slightly. We are so familiar with this example that we no longer consider it worthy of comment or investigation.

In this regard we are no different from the ancient Romans who no doubt knew that their system of numerals was cumbersome but were happy to persist with it.

Unfortunately, because the decimal representation serves us well in everyday life we somehow assume that the inaccuracy in the number one-third is a property of the number rather than the method by which we choose to represent it. In fact it is easy to show that it is the method of representation which is at fault, not the number itself.

If we had been created with six fingers on each hand rather than five then we would presumably count in groups of 12. In maths jargon we would count to base 12. When counting to base 12 the following statement is true:

1/3 = 0.4

That is easy to understand if you remember that the column following the decimal point when counting in base 12 is the twelfths column, and four-twelfths equals one-third.

However, counting to base 12 is not immune to problems. The fraction one-fifth, for example, which is equal to 0.2 in decimal can only be exactly represented in base 12 by 0.24972497 ... followed by endless repetition of the four digits 2497.

No matter what base you choose there will always be some fractions which cannot be represented exactly. There are even some numbers - such as pi, the ratio of the circumference of a circle to its diameter - which cannot be exactly represented in any base.

Of course those inexactitudes do not normally matter because we are happy to accept a number which is approximately correct as exact The trick is to build in acceptance of tiny but irrelevant differences. In particular you should not, as John Jennings has done in line 50 and 60 of his program, demand equality in any comparison.

The program illustrates how the Spectrum disguises the problem of rounding errors in is printing routine except for numbers which are very close to zero. Thus we do not know if there are rounding errors in the calculation of 2, 1.5, 1, 0.5, -0.5, -1, -1.5 and -2 because the print routine rounds off the numbers to the first half dozen or so significant figures, discarding any rounding error as it does so.

But the number zero has no leading significant figures by definition, so the Spectrum has no way of knowing where truth ends and rounding error begins.

Hence it prints what it has stored away, which is a number which is not quite zero but is smaller than one divided by a thousand million. The poor old computer, because it isn't human, doesn't realise that John Jennings is going to be very perplexed by this result.