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= 2r||^ The fraction will not cancel
All fractions, when converted into decimals, are either terminating or recurring: Terminating decimals have a finite number of decimal places. Refer to the earlier examples of | and ^ in this entry.
Recurring decimals have an infinite number of decimal places, but repeat a certain pattern of numbers over and over again. Recurring decimals should not be confused
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with irrational numbers, which also have an infinite number of decimal places, but do not have a repeating pattern of numbers. Irrational numbers cannot be converted into fractions. For example, the irrational number it may be expressed as a decimal to an ever-increasing number of decimal places, but cannot be expressed as a fraction, because there is no repeating pattern to the numbers. All recurring decimals can be converted into fractions. Recurring decimals are often called repeating decimals.
Example 4. Convert these fractions into recurring decimals: (a) |, (b) j Solution. For (a), write
- = 5 — 9 = 0.55555555 = 0.5
For (b), write
f = 4 = 7
= 0.571428571... Using the calculator for division; there is an infinite number of decimal places
If we look closely at the pattern, we can see that it repeats itself: 0.571428 571428 and so on. Therefore we write
| = 0.571428 With a dot above the first and last numbers that repeat
| = 0.571428 An alternative notation for a recurring group of numbers
Every fraction can be converted into a decimal or into a recurring decimal. For some fractions the decimals do no recur immediately. For example 119/990 = 0.1202. Decimals may be added, subtracted, multiplied, and divided, using a calculator.
References: Canceling, Decimal System, Denominator, Finite Decimals, Fractions, Infinite, Multiple, Numerator, Percentage.
Using calculator to give repeating 5’s
The dot above the 5 is shorthand for “0.5 recurring”
References: Decimal, Decimal System.
The decimal system is a number system based on 10 and powers of 10, and is sometimes called the denary system. It originated in India and became established in Europe by the Arabs. The decimal system is used for counting, for money, and for weights and measures. The decimal number system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to write numbers. The value of each of these digits 0-9 depends on its place in that decimal number. For example, in the decimal number 7325, the 7 represents a value of 7 x 103 = 7000, the 3 represents a value of 3 x 102 = 300, the 2 represents a value of 2x 101 = 20, and the 5 represents a value of 5 x 10° = 5. In the following table column headings are the powers of 10 and tell us the value of each position that the digits 0-9 can occupy. Here the columns are only shown as far as 104:
104 = 10,000 103 = 1000 102 = 100 101 = 10 10° = 1
When negative powers are used the column headings are for numbers between 0 and 1. These numbers are called decimals. The whole number and the decimal are separated by a decimal point.
The column headings for decimals as far as 10-4 are shown in the following table:
10-1 =0.1 10-2 = 0.01 10-3 =0.001 10-4 = 0.0001
The need for counting numbers larger than 4 or 5 probably arose for a variety of reasons. Shepherds needed to count large numbers of their sheep and cows, once they had herds, to see if any were missing. Farmers needed to know about the lengths of seasons and when the rains were expected so they could sow seeds, and when to harvest crops. They needed to count so that some sort of calendar could be set up. About 4000 years before the birth of Christ the Egyptians knew there were 365 days in a year.
It is thought that early peoples may have learned to count on their fingers very much as young children first learn to count today. Once they reached a count of 10 fingers they would need another pair of hands to which to transfer their “10” while they counted up to 10 again. That is when an assistant would have come in handy. Once they had counted up to 10, 10 times, and each 10 transferred to the assistant, they would have needed a second assistant with another pair of hands, and so on.
References: Decimal, Powers.
The word decomposition means to break down a number, or quantity, into smaller parts. The number 456 represents 400 + 50 + 6, which can be rearranged into 400 + 40 + 16. This latter form of the number is of the same value, but has undergone
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decomposition. When subtracting numbers it is sometimes helpful to use decomposition of one of the numbers. This process, which is often called “borrowing,” is explained in the following example.
Example. Work out 456 - 38.
4 $ $
4 1 8
Solution. The number 38 is arranged under the 456 using the columns 100’s, 10’s, and units (see table). The first stage of the process is to subtract the 8 from the 6 in the units column, which cannot be done using positive numbers. Now we use the decomposition of 56 which is replaces 50 + 6 by 40 + 16, and then subtract the 8 from the 16. The next stage is to subtract 3 from 4 in the tens column, and finally subtract 0 from 4 in the hundreds column.