The Mathematics 11
Competency Test
Exact and Approximate Numbers
All numbers used in technical applications can be categorized as being one of the following two
types:
(i) Approximate Numbers – values which are the result of measurement relative to a continuous
scale. Examples of this type of number are lengths, masses, times, etc. The number of digits
used to express these values results from how precisely we are able to make the required
measurement.
So, for example, when we say that a basketball player is 196 cm tall, we probably mean that
rounding to the closest whole centimetre, his height is 196 cm. Of course, with a much more
precise instrument or apparatus, we might be able to determine his height to be 196.23 cm.
Instruments exist which enable measurements of lengths to within less than a millionth of a
centimetre.
The point of this is that how many digits we use to state this player’s height in centimetres
depends on how precise our technique of measurement is, and no matter how careful we are, we
can never say that we have measured the player’s height exactly to all decimal places. (In fact,
it’s not even clear that measuring the height of a basketball player in centimetres to, say, 100
decimal places even makes sense in principle, because even if an atom in one of the hairs on his
head vibrated slightly, his height would change at that scale of measurement!)
Another example comes from the measurement of the length of time intervals. Using an ordinary
wrist watch, we might be able to determine that a runner completes a 10K race in 27 minutes and
49 seconds. A more sophisticated timing system used at athletic competitions might have
measure that same race to have taken 27 minutes 49.37 seconds. There exist readily available
clocks that can reliably measure time intervals to millionths of a second, which could potential
enable an even more precise time to be obtained for this race. The point is, whatever technique
we use to measure the elapsed time of this race, there will always be some uncertainty, however
small, in the value we state.
This uncertainty in values measured against a continuous scale gives rise to the concept of
approximate numbers. Generally when such an “approximate number” is stated, it is assumed
that the last (right-most) digit was obtained by rounding, and thus the true value of the quantity in
question could be different from the stated value by as much as half of a unit in the position of the
last stated digit.
Thus, when we stated that the basketball player’s height is 196 cm, we imply that the “6” is the
result of rounding to the nearest centimetre. So, his true height could be as much as 0.5 cm
greater or less (0.5 cm being half of one centimetre, the precision to which the height of 196 cm is
stated). That is, stating that the player’s height is 196 cm means we’ve narrowed his true height
down to the interval 195.5 cm to 196.5 cm, more or less.
Similarly when we say that the runner completed the 10K race in 27 minutes 49.3 seconds, we
imply that the ‘3’ is the result of rounding. So, the true time of the race could be different from this
value by as much as 0.05 seconds, greater or less. The 0.05 seconds is half of 0.1 seconds, the
precision to which the time, 49.3 seconds, was stated. So, the actual time has been narrowed
down to the interval 27 minutes 29.25 seconds to 27 minutes 49.35 seconds, more or less.
(ii) Exact Numbers – Other numbers that arise in technical applications are better described as
“exact numbers” because there is not the sort of uncertainty in their values that was described
above. Exact numbers arise in basically two situations:
David W. Sabo (2003)
Exact and Approximate Numbers
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(i.) They are the result of counting discrete items. Thus, one dozen eggs is exactly 12 eggs,
never 12.1 eggs or 11.9 eggs. Similarly, if we count the number of cows in a herd, or the
number of computers on a network, or the number of bytes of memory in a computer,
these are all precise whole numbers, and so are exact by definition. You can’t have a
decimal part of a cow in a herd, or a decimal part of a computer on a network, or, with the
usual hardware, or a decimal part of a byte of memory. You could never seriously say,
“rounded to the nearest whole cow, I have three cows in my herd.”
(ii.) Important fundamental numerical values may be exact by convention, international
agreement, or basic definition. Examples of this sort of exact number are most common
when we look at units of measurement.
For example, by international agreement, there are exactly 2.54 centimetres in 1 inch.
The number ‘2.54’ is not the rounding off of a lengthier decimal number. Rather, this ratio
of centimetres to inches is 2.540000…. to an infinite number of decimal places by
international agreement.
The relationships between SI units of a particular family also involve exact numbers.
There is exactly 100 centimetres in 1 metre, not because someone did an extremely
precise measurement of a metre and found that it was “exactly” 100 centimetres long, but
because the word “centimetre” is defined to mean 1/100th of a metre.
In geometry, a square has exactly four sides because that is the characteristic shape to
which the word “square” refers.
This distinction between approximate and exact values may seem to be nitpicking. However,
as we briefly describe in the next several documents in this series, being able to distinguish
these two types of values that arise in practice enables us to develop some rough but
reasonable rules for rounding off results of simple arithmetic calculations appropriately.
David W. Sabo (2003)
Exact and Approximate Numbers
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