Decimals as a mathematical system
How to decide which decimal is largerDeciding 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 Another example: To compare 4.032 with 4.10006 append 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.
Note that this strategy cannot be used for infinite decimals. Strategy 2: Left to right comparison A simple example: to compare 23.87 with 23.863
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.
Beware of zeros and blanks!: Children who lack place value understanding have trouble with this strategy as shown in the following 2 examples:
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.
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 9 x k and divide the answer by 9 (this is very easy!) to get k itself. This shows that 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,
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.
Rational and Real Numbers: Fractions and DecimalsMany 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:
So, all rational numbers are real numbers too! Why bother having these
fancy definitions if they are just the same?
Which decimals are irrational? Those that are infinite without repeating.
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". 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.
In Summary Every real number is either rational or irrational. To determine what sort of real number a certain decimal is, ask these questions:
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. Density and CompletenessUnlike 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: Answers to Exercises1 (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.... 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.... 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. |