Difference between revisions of "Real Numbers and their Decimal Expansions"
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Decimal expansion | Decimal expansion for the real numbrs <math>\frac{1}{7} , \frac{10}{3} , \frac{7}{8}</math> is explained below. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| colspan="8" style="text-align: center" |<math>\frac{1}{7} </math> | |||
| colspan="7" style="text-align: center" |<math>\frac{10}{3}</math> | |||
| | | | ||
| colspan=" | | colspan="5" style="text-align: center" |<math> \frac{7}{8}</math> | ||
| | |- | ||
| | | | ||
| colspan=" | | colspan="7" style="border-bottom: solid 5px blue" |0.142857..... | ||
| rowspan="15" | | |||
| | | | ||
| colspan="5" style="border-bottom: solid 5px blue"|3.3333... | |||
| rowspan="15" | | |||
| | |||
| colspan="4" style="border-bottom: solid 5px blue"|0.875 | |||
|- | |- | ||
| rowspan="13" style="border-right: solid 5px blue; vertical-align:top | | rowspan="13" style="border-right: solid 5px blue; vertical-align:top" |7 | ||
|1 | |1 | ||
|0 | |0 | ||
| colspan="5" rowspan="2" | | | colspan="5" rowspan="2" | | ||
|3 | | rowspan="9" style="border-right: solid 5px blue; vertical-align:top" |3 | ||
|1 | |1 | ||
|0 | |0 | ||
| | | colspan="3" rowspan="2" | | ||
| | | rowspan="7" style="border-right: solid 5px blue; vertical-align:top" |8 | ||
| | |7 | ||
|0 | |||
| colspan="2" rowspan="2" | | |||
|- | |- | ||
| rowspan="12" | | | rowspan="12" | | ||
|7 | |7 | ||
| | | rowspan="8" | | ||
| | |||
|9 | |9 | ||
| | |6 | ||
| | |4 | ||
|- | |- | ||
| | | | ||
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|0 | |0 | ||
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|1 | |1 | ||
|0 | |0 | ||
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| | | rowspan="5" | | ||
|6 | |||
|0 | |||
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|- | |- | ||
|2 | |2 | ||
| | | | ||
8 | 8 | ||
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|9 | |9 | ||
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| | |6 | ||
|- | |- | ||
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|0 | |0 | ||
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|1 | |1 | ||
|0 | |0 | ||
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| rowspan="3" | | |||
|4 | |||
|0 | |||
|- | |- | ||
|1 | |1 | ||
|4 | |4 | ||
| | | rowspan="4" | | ||
| | |||
|9 | |9 | ||
| | |4 | ||
|0 | |||
|- | |- | ||
| rowspan="7" | | | rowspan="7" | | ||
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|0 | |0 | ||
| colspan="2" rowspan="2" | | | colspan="2" rowspan="2" | | ||
|1 | |||
|0 | |||
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|0 | |0 | ||
|- | |- | ||
|5 | |5 | ||
| | | | ||
6 | |||
| | | | ||
|9 | |9 | ||
| colspan="5" rowspan="7" |Reminders: 6,4,0. | |||
Divisor: 8 | |||
|- | |- | ||
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|0 | |0 | ||
| rowspan="2" | | | rowspan="2" | | ||
| | | | ||
|1 | |1 | ||
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|3 | |3 | ||
|5 | |5 | ||
| | | colspan="6" rowspan="5" |Reminders: 1,1,1,1... | ||
| | Divisor: 3 | ||
|- | |- | ||
| rowspan="3" | | | rowspan="3" | | ||
|5 | |5 | ||
|0 | |0 | ||
|- | |- | ||
|4 | |4 | ||
|9 | |9 | ||
|- | |- | ||
| | | | ||
|1 | |1 | ||
| | |- | ||
| | | colspan="8" |Reminders: 3,2,6,4,5,1,3,2,6,4,5,1... | ||
Divisor: 7 | |||
| | |||
|} | |} | ||
In the above division operation | |||
* The remainders either become <math>0</math> after some stage, or start repeating themselves. | |||
* The number of entries in the repeating string of remainders is less than the divisor (in <math>\frac{10}{3} </math> , one number repeats itself and the divisor is <math>3</math> , in <math>\frac{1}{7} </math> there are six entries <math>326451</math> in the repeating string of remainders and the divisor is <math>7</math> ) | |||
*If the remainders repeat, then we get a repeating block of digits in the quotient (for <math>\frac{10}{3} </math> , <math>3</math> repeats in the quotient and for <math>\frac{1}{7} </math> , repeating block <math>142857</math> in the quotient) | |||
The above pattern using the examples above is true for all rationals of the form <math>\frac{p}{q} </math> (<math>q \ne 0</math>). | |||
On division of <math>p </math> by <math>q </math>, two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of remainders. | |||
The decimal expansion of real numbers can be classified into three types. They are: | |||
* [[Terminating decimal expansions|Terminating Decimals]] | |||
* [[Non-terminating recurring decimal expansions|Non-terminating and Repeating Decimals]] | |||
* [[Non-terminating recurring decimal expansions|Non-terminating and Non-repeating Decimals]] |
Latest revision as of 22:17, 6 May 2024
Decimal expansion for the real numbrs is explained below.
0.142857..... | 3.3333... | 0.875 | ||||||||||||||||||
7 | 1 | 0 | 3 | 1 | 0 | 8 | 7 | 0 | ||||||||||||
7 | 9 | 6 | 4 | |||||||||||||||||
3 |
0 | 1 | 0 | 6 | 0 | |||||||||||||||
2 |
8 |
9 | 5 | 6 | ||||||||||||||||
2 |
0 | 1 | 0 | 4 | 0 | |||||||||||||||
1 | 4 | 9 | 4 | 0 | ||||||||||||||||
6 |
0 | 1 | 0 | 0 | ||||||||||||||||
5 |
6 |
9 | Reminders: 6,4,0.
Divisor: 8 | |||||||||||||||||
4 | 0 | 1 | ||||||||||||||||||
3 | 5 | Reminders: 1,1,1,1...
Divisor: 3 | ||||||||||||||||||
5 | 0 | |||||||||||||||||||
4 | 9 | |||||||||||||||||||
1 | ||||||||||||||||||||
Reminders: 3,2,6,4,5,1,3,2,6,4,5,1...
Divisor: 7 |
In the above division operation
- The remainders either become after some stage, or start repeating themselves.
- The number of entries in the repeating string of remainders is less than the divisor (in , one number repeats itself and the divisor is , in there are six entries in the repeating string of remainders and the divisor is )
- If the remainders repeat, then we get a repeating block of digits in the quotient (for , repeats in the quotient and for , repeating block in the quotient)
The above pattern using the examples above is true for all rationals of the form ().
On division of by , two main things happen – either the remainder becomes zero or never becomes zero and we get a repeating string of remainders.
The decimal expansion of real numbers can be classified into three types. They are: