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Whole numbers are not enough!
The whole numbers, also called the counting numbers, provide the means to specify "discrete" quantities, e.g. to say that 25 children are in the classroom. Children are separate entities; it does not make sense to talk about anything other than a whole number of children. For counting children and other discrete quantities, the whole numbers are all that is required.
Not all uses of number are like this. Other quantities, such as children's height or an amount of cake are "continuous". The need to describe continuous quantities often occurs in commerce and everyday activities involving measurement where amounts of a basic measurement unit intermediate between two whole numbers need to be specified and communicated.
The first way in which people solved this problem was to invent new smaller units. For example, the basic unit of time, the day, was divided into hours and then into minutes and then into seconds. Gradually people agreed that there would be 24 hours in a day and that the hours would be of equal length. The hour itself was divided into 60 minutes, which were again divided into 60 seconds. Today even smaller units of time are often needed, for example to describe the race times of runners and swimmers or to keep communications through computers synchronized.
Measuring more and more precisely by creating smaller and smaller units has the advantage that only whole number arithmetic is needed. For example, problems about hours and minutes require only knowledge of whole numbers and the conversion 60 minutes = 1 hour. However, smaller and smaller units need to be invented and the conversions remembered. The metric system makes this easier, with all conversion factors based on ten.
The second solution to the problem of describing measuring more precisely is by extending the number system itself to include both common fractions and decimal fractions (referred to in this resource as decimals).
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Using fractions or decimals, continuous quantities can be described to any desired accuracy: |
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the area of the floor is thirty-six square metres and eight tenths
of another square metre, that is 36 |
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the cost of carpet for a square metre is $40.35 (forty dollars
and thirty-five cents), or forty dollars and thirty-five hundredths
of another dollar, that is 40 |
square metres, or 36.8 square metres, or more commonly, 36.8 m2
dollars,
or 40.35 dollars.Note that there are two units (the dollar and the cent) being used in the money example, but not in the area example. The area example shows that there is no need for the smaller unit (the tenth of a square metre) to have a name: the number system itself provides all the information required, because the whole number and fractional parts are not separate, but are together in the one number.
There are mathematical advantages to extending the number system rather than creating smaller units. This is not shown with addition and subtraction but with multiplication and division. In the table below, the addition problem can be solved equally easily by thinking about dollars and cents as two (linked) different units or by thinking of the quantity as one decimal number. However, the multiplication is very hard to do in the first way, but easy in the second.
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Problem |
Calculation with subunits |
Calculation with single numbers |
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Per square metre, carpet costs $40.35 and underlay costs $17.20. How much to get both? |
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Per square metre, carpet costs $40.35. How much for 36.8 square metres? |
How to multiply 40 dollars and 35 cents by 36 sq. metres and 8 tenths of a square metre ? |
40.35 x 36.8 |
Fractions or Decimals?
Fractions and decimals both serve the same purpose of describing parts of a whole. The idea of a fraction is more basic. Indeed the fraction concept of a tenth is required to understand decimals. For most uses, decimals have many advantages:
- their relative size is easy to determine (for example, it is much easier to see that 0.75 is bigger than 0.65, than to see that 3/4 is bigger than 13/20),
- the rules of calculation are very similar to the rules for whole numbers (because decimals are an extension of the same place value, base ten system) whereas the rules for calculation with fractions are very different,
- decimals convert to percentages easily (both being based on tens and hundreds),
- decimals are totally compatible with the metric system of measurement,
- decimals fit on small calculator screens and are typed easily but fractions are awkward.
In other sections of this resource, it is shown that use of the decimal system is not easy to learn and there are many points which beginners find hard. However, good understanding is essential for dealing with measurements and with number.
When is a point a decimal point?
A full stop is often used just as a general separator between numbers, and not as a decimal point separating ones from tenths and hundredths etc. This can be a source of potential confusion for children as they learn about decimals. By discussing examples with students as they arise in other contexts, teachers can alert them to notations which may cause confusion as they look similar to decimals. Confusing decimal points with general separators is also a source of error when children try to use calculators with time and other non-decimal contexts.
Remember: The word "decimal" comes from the Latin word meaning ten.
Examples:
Time
A full stop is often used as a separator between units of
time (hours, minutes, seconds, etc).
- 1.30, as often shown in TV guides and on some digital watches, is used to show that the specified time is 30 minutes past the hour of 1 o'clock,
- time for a long distance runner in a marathon might be reported as 2.15.32, meaning the time taken was 2 hours 15 minutes and 32 seconds,
- the world record for running 100 metres is 9.84, meaning the time taken was 9 seconds and 84 hundredths of a second.
Only in the third example above is the dot actually being used as decimal
point. The other examples are not decimal: for example the time after
1.59 is 2.00, whereas the two place decimal number after 1.59 is 1.60.
Children sometimes are misled by the similarity with decimals to use
calculators for time calculations. They may, for example, try to use subtraction
on a simple calculator to find the time difference between 2.10 pm and
3.45pm. Because time is not a base 10 system, this will give the wrong
answer (i.e. 135 mins instead of 95 mins). Children will also frequently
make errors with half an hour, saying for example that half past 6 is
6:50, again thinking it is a decimal system (what a sensible idea!)
Cricket, baseball and football
- In cricket, 4.3 is used to indicate that a bowler has bowled 4 complete overs and 3 balls out of the 6 for another over . This means that 4.3 (not 4.5) is 4 and a half overs.
- In baseball, 7.2 is used to indicate that a pitcher has pitched 7
complete innings and
of
another complete inning (i.e. the pitcher has been able to get "out"
2 of the 3 batters needed for a complete innings). - The goals and points of Australian rules football scores are sometimes separated by what looks like a decimal point. But 12.7.79 indicates 12 goals (each worth 6 points) and 7 behinds (worth 1 point each) yielding a total score of 79 points.
Documents
- A diagram in a book is referenced as Diagram 3.9, meaning it is the ninth diagram shown in chapter 3 of the book. Diagram 3.10 will come next. With decimals, 3.9 is not followed by 3.10
- Textbooks will have Exercise 4.10 following Exercise 4.9
- The second agenda item for a meeting is listed as, 2.1, 2.2, 2.3,............. 2.10, 2.11, 2.12 meaning there are twelve sub-items for item 2
Library classification system
The Dewey decimal system used to classify books is a decimal system.
For example, 510.12 is a specified classification code for books concerned
with a particular aspect of mathematics. Books are ordered on the shelves
in the library exactly as decimal numbers are ordered, and so a book numbered
510.8 is to be found after books numbered 510.12, not before as some wrongly
believe.
The advantage is that new books can always be inserted into an appropriate place.
(Although the ordering is the same as with decimals, the distance between numbers is not the same - for example,it is not meaningful to say that books in the 509 section are "as close" to books in the 510's as are books in the 511's. Their topics may be more or less similar.