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Contents of this page |
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How to decide which decimal
is larger There are two expert strategies. |
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Can two different decimals
be equal? Only if there is an infinite string of 9's. |
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Rational and real numbers Differences between the set of fractions and the set of decimals. Terminating and repeating decimals are rational numbers. |
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Density and completeness Between any two decimal numbers, there is another one. |
How to decide which decimal is larger
Deciding which of two decimals is larger is easy for those who understand place value fully. There are two ways: by equalizing the length with zeros and then comparing fractions with same denominator or by left-to-right digit-by-digit comparison. There is one exception when dealing with decimals of infinite length (see below).
Strategy 1: Equalizing length with zeros
Equalizing length with zeros is probably the most common strategy taught in Australian schools, although it is used infrequently in other countries, such as Japan where it is regarded as not taking advantage of the decimal system. To find out which of two decimals is the larger, add zeros to the shorter until they have the same lengths and then compare as whole numbers. For example, to compare 0.4 and 0.457, add zeros to 0.4 to get three decimal places (0.400) and then compare 400 with 457. This strategy works because 400 thousandths is being compared to 457 thousandths.
Another example: To compare 4.032 with 4.10006 add zeros to equalize the lengths, getting 4.03200 and 4.10006. The first number is 4 ones + 3200 hundred-thousandths, the second is larger being 4 ones + 10006 hundred-thousandths.
| 4.03200 | 4 ones + 03200 hundred-thousandths (smaller) |
| 4.10006 | 4 ones + 10006 hundred-thousandths (larger) |
Note that this strategy cannot be used for infinite decimals.
Strategy 2: Left to right comparison
This strategy is to compare columns from left to right, until a digit in one decimal is larger than the corresponding digit in the other (and the former will then be the larger number).
A simple example: to compare 23.87 with 23.863
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Tens |
Ones |
Tenths |
Hundredths |
Thousandths |
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2 |
3 |
8 |
7 |
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2 |
3 |
8 |
6 |
3 |
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same |
same |
same |
top is larger so stop |
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The left-to-right digit-by-digit comparison strategy depends on the fact that no matter how large the values in the later columns of a decimal number, they can never add up to change an earlier value. In the example above, no matter what digits came after the hundredths in the second decimal, they could never make it larger than the 7 hundredths in the top decimal.
| 23.87 | 23.8 + 7 hundredths |
| 23.863 | 23.8 + 6 hundredths + less than one hundredth |
Beware of zeros and blanks!: Children who lack place value
understanding have trouble with this strategy as shown in the
following 2 examples:
- the strategy does not help children to find that 23.870 is equal to 23.87, unless they understand that 23.870 is 23.87 + 0 thousandths
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same | same | same | 0 vs blank! Unsure? |
- the strategy does not help children to find that 23.873 is greater than 23.87 unless they understand that 23.87 is 23.87 + 0 thousandths
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blank |
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same | same | same | 3 Vs blank! Unsure? |
Can two different decimals be equal?
Usually it is not possible for two decimal numbers to be equal
unless they have exactly the same digits in the same columns.
For example, we know at a glance that 2.126 and 2.5025 are not
equal. This is quite different to fractions, which might look
different but be equal (e.g. 6/12 and 25/50). It is the reason
why the left-to-right comparison strategy works.
There is one exception with strings of nines in infinite decimals.
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The number 0.9999999999999..... (with the nines repeating infinitely) is exactly equal to 1 |
Many people find this hard to accept - they generally think it
is a little less than 1. This is because the repeating decimal
is actually the sum of an infinite series. Here is a proof which
may convince you. For convenience, we let k denote the number
0.9repeating. Then we multiply it by 10. This is easy because
all the digits just move one column to the left. Note that because
there are an infinite number of 9's, there is no gap at the end
(actually there is no end!). Then we subtract k from 10 x k to
get 9x k and divide the answer by 9 (this is very easy!) to get
k itself. This shows that k = 1.
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Call this number k |
k | = |
0.99999999999....... |
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Multiply it by ten |
10 x k | = |
9.99999999999....... |
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Subtract k from 10k |
9 x k | = |
9.0000000000..... |
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Divide both sides by 9 |
k | = |
1 |
Because 0.9repeating is exactly equal to 1, other decimals ending in strings of 9's are also equal to terminating decimals. For example,
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2.9999999999999999999repeating |
= |
3 |
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2.4999999999999999999999repeating |
= |
2.5 |
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0.39999999999999999999999999repeating |
= |
0.4 |
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0.9991199999999999999999999repeating |
= |
0.99912 |
Note: These are exact mathematical results. The issue of significant
figures (glossary) is not
relevant because these decimals could not occur from measurements.
Note: The software used in present day websites does not properly
display the notation needs for repeating decimals, so we have
used the word "repeating". There are two commonly used
correct notations as shown below, using either a bar across the
top of the repeating part, or dots.
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4.999999999999.... |  |
| 0.8777777777777.... | ||
| 0.654545454545..... | ||
| 8.32132132132..... | ||
| 7.6661428571428... |
Rational and Real Numbers: Fractions and Decimals
Many people think that fractions and decimals are different types
of numbers and want to treat them separately. However, it is possible
to express any fraction as a decimal, so the difference may be only
skin deep. Furthermore, most people have only come across numbers
which can be written as both fractions and decimals, and are not
aware that other types of numbers exist. Firstly, some definitions:
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A rational number is any number which can be expressed as a fraction. |
A real number is any number which can be expressed as a decimal. |
| Examples of
rationals 1/2 0.75 (it can be written as 3/4), -16 (it can be written as -16/1) 22/3 -458/111 0.3repeating (it is equal to 1/3) |
Examples of
reals all the numbers in the left column (i.e. all rational numbers) plus others (called irrational numbers) like pi, e, the square root of 2, the square root of 17 and infinite decimals which do not repeat. |
So, all rational numbers are real numbers too! Why bother having these fancy definitions if they are just the same?
They are not the same! Real numbers come in 2 flavours: rational and irrational. It is the existence of the irrational numbers that necessitates the definitions above. Students first meet irrational numbers in secondary school. The lengths of some line segments are irrational. This means that they cannot be described as a fraction of the measurement unit. An infinite decimal is needed. For example, the length of the hypotenuse of a right-angled triangle which has sides of 1cm and 3cm is is 3.162277660...cm (a decimal which does not terminate or repeat, equal to the square root of 10).
Which decimals are rational numbers? Those that terminate or repeat.
- Decimals which terminate (i.e. have only a finite number of places) are always equal to fractions - i.e. they are equal to one whole number divided by another. Examples:
| 0.5 = 1/2 | 0.375 = 3/8 | -12.34 = -1234/100 |
- Decimals which repeat (i.e. have a certain string of numbers which are repeated in the same order to infinity) are always equal to fractions too - i.e. they are equal to one whole number divided by another. Examples:
| 0.3repeating = 1/3 | 0.142857142857142857repeating =1/7 |
| 0.1repeating =1/9 | 0.1234343434repeating = 1222/9900 |
Which decimals are irrational? Those that are infinite without repeating.
- The irrational numbers are decimals that neither repeat nor terminate. This means that an infinite decimal such as 0.101101110111101111101111110....... (if it continues in this pattern which is not a fixed group repeating) will be an irrational number.
Calculating the decimal digits of pi, which do not repeat, is often used to test super computers. (Pi is the ratio of the length of a circumference of a circle to its radius.) Recently, 51.5396 billion decimal places of pi have been calculated at the University of Tokyo, taking 29 hours on one computer and 37 on another. Current information about pi can be found at "The Ridiculously Enhanced Pi Page" (http://www.exploratorium.edu/learning_studio/pi/). There is no apparent pattern in the digits. Amongst the first 50 billion digits, 8 occurs most often and 3 the least often. ( "Pi-eyed after all these years" The Age 27 January 1998 page A14)
Hardly any real numbers are rational. Most numbers are irrational.
Almost all the numbers that we know about are rational. However there are many, many more irrational numbers than rational numbers. If a number could truly be picked at random, it is mathematically certain that it would be an irrational number. Even though there are an infinite number of fractions (rational numbers) and an infinite number of decimals (real numbers) in a certain way it is known that there are very many more decimals than fractions. This property and the associated theory of infinite numbers can be found in many popular accounts of important mathematics. For example, Courant and Robbins (1996) is a classic.
In Summary
Every number that you are likely to meet is a real number. (Only students of higher mathematics courses study numbers which are not real numbers. They have the appropriate name of imaginary and complex numbers.)
Every real number is either rational or irrational. To determine
what sort of real number a certain decimal is, ask these questions:
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does it stop (terminate)? Yes, then it is rational - it is equal to a fraction. |
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if it does not terminate then it is infinite. Does it repeat? Yes, then it is rational (it is equal to a fraction). |
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if it is infinite and does not repeat, then it is irrational (it is not equal to any fraction). |
Proofs of the mathematical results here and more information can
be found in most elementary number theory text books. For example,
Courant and Robbins (1996) is a classic.
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Density and Completeness
Unlike the set of counting numbers, both the sets of decimals and fractions have the property that between any two there is another one. In between the two whole numbers 2 and 3, for example, there is no whole number, but in between any two fractions there is always another fraction and in between two decimals there is always another decimal.
Completeness is a more advanced mathematical property of the real numbers, which is studied in university mathematics courses. More information can be found in most elementary number theory text books, including Courant and Robbins (1996).
The following exercise illustrates the density of the numberline.
Exercises:
1) Find fractions between these fractions:
(a) 1/2 and 1/3
(b) 1/10 and 3/31
(c) 3 1/7 and 3 10/71
(these are two approximations to pi)
2) Note how much easier it is to find decimals between the corresponding
pairs of decimals
(a) 0.5 and 0.3repeating
(b) 0.1 and 0.096774193...
(c) 3.142859142859.... and 3.140845......
Answers
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ANSWERS TO EXERCISES 1 (a) Because 1/2 = 6/12,
and 1/3 = 4/12, then 5/12 is between them. (Note that looking at
the numbers of sixths didn't help as they are 2/6 and 3/6.) But
there are more....
e.g. 1/2 = 12/24 and 1/3 = 8/24 so between the two are 9/24, 10/24
(we already have this one!), and 11/24. You can keep going if you
want! There are others like 2/5, 3/7, 4/9, 5/11....
1 (b) Because 1/10 = 62/620 and 3/31 = 60/620 then 61/620 is between
the two of these. (Note that looking at the number of 310ths didn't
help as they are 31/310 and 30/310 with no whole number between
31 and 30. So we needed to make smaller pieces; I halved each of
the 310 pieces to make 620.) But there are more....
e.g. 1/10 = 124/1240 and 3/31 = 120/1240 so between are 121/1240,
122/1240 ( we already have this one!) and 123/1240. You can continue
for as long as you want!
OR: Because 1/10 = 0.1 and 3/31 = 0.0967741.., then some decimals
between them are 0.099, (which is 99/1000) and 0.097 (which is 97/1000)
and 0.0968 (which is 968/10000). Keep going!
1 (c) Consider the fraction parts of 1/7 and 10/71. First we may
choose to look at the number of 497 ths (7x71). Then 1/7 = 71/497
and 10/71 = 70/497, with no whole number between 70 and 71. So instead,
make smaller pieces; try halving each of the 497 to make 994 pieces.
Then 1/7 = 142/994 and 10/71 = 140/994, so the obvious fraction
between is 141/994. The original question had the whole number 3
as well as the fractional part so one of the answers is 3 141/997.
Continue looking for more if you wish!
2 (a) Lots, e.g. 0.4 or 0.35 or 0.37859462 or 0.452687541 or 0.49999
2 (b) Lots, e.g. 0.099 or 0.097 or 0.096775 or 0.096774194
2 (c) Now the decimal corresponding to 3 1/7
is 3.142859142859...(NB we know that it must repeat as it is a rational
number and they will always terminate or repeat) and the decimal
corresponding to 3 10/71 is 3.14084507.....(NB
the repeating digits did not appear on my calculator but they must
be there!). Now to find a decimal in the interval between the two
is much easier as we just need to compare the digits from left to
right until a difference occurs. The smaller number has 0 thousandths
while the larger has 2, so any number which starts 3.141
will do, e.g. 3.141 or 3.1417583619. Also, we could
choose a number which starts with 3.1409 (closer to the smaller
end) or one which starts with 3.1427 (closer to the larger end).
Lots to choose from and easy to find.
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