Difference between revisions of "142857"
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| − | + | The number '''142857''' has the curious property that, if it is multiplied by any number from 1 to 6, the result is a number with the same digits in the same order (a [[cyclic permutation]]). | |
| + | |||
| + | For example: | ||
| + | :142857 × 1 = 142857 | ||
| + | :142857 × 2 = 285714 | ||
| + | :142857 × 3 = 428571 | ||
| + | :142857 × 4 = 571428 | ||
| + | :142857 × 5 = 714285 | ||
| + | :142857 × 6 = 857142 | ||
| + | |||
| + | |||
| + | This property is closely related to the facts that | ||
| + | :1/7 = 142857/999999 | ||
| + | |||
| + | and that 1/7 expressed as a [[decimal]] is | ||
| + | |||
| + | :0.142857142857142857142857142857... | ||
| + | |||
| + | Investigating the number 142857 can lead a student into an understanding of repeating decimals and [[rational numbers]]. | ||
| + | |||
| + | |||
| + | == Similar Numbers == | ||
| + | 142857 is the only multi-digit decimal number with this property. However, if leading zeros are allowed on a number, some other examples are: | ||
| + | :0588235294117647 (16 digits) | ||
| + | :052631578947368421 (18 digits) | ||
| + | :0434782608695652173913 (22 digits) | ||
| + | :0344827586206896551724137931 (28 digits) | ||
| + | :0212765957446808510638297872340425531914893617 (46 digits) | ||
| + | |||
| + | In octal (base-8 notation), the numbers 25 and 1463 have this property. There are no examples in hexadecimal (base-16). | ||
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| + | [[Category:Mathematics]] | ||
Revision as of 01:42, October 28, 2011
The number 142857 has the curious property that, if it is multiplied by any number from 1 to 6, the result is a number with the same digits in the same order (a cyclic permutation).
For example:
- 142857 × 1 = 142857
- 142857 × 2 = 285714
- 142857 × 3 = 428571
- 142857 × 4 = 571428
- 142857 × 5 = 714285
- 142857 × 6 = 857142
This property is closely related to the facts that
- 1/7 = 142857/999999
and that 1/7 expressed as a decimal is
- 0.142857142857142857142857142857...
Investigating the number 142857 can lead a student into an understanding of repeating decimals and rational numbers.
Similar Numbers
142857 is the only multi-digit decimal number with this property. However, if leading zeros are allowed on a number, some other examples are:
- 0588235294117647 (16 digits)
- 052631578947368421 (18 digits)
- 0434782608695652173913 (22 digits)
- 0344827586206896551724137931 (28 digits)
- 0212765957446808510638297872340425531914893617 (46 digits)
In octal (base-8 notation), the numbers 25 and 1463 have this property. There are no examples in hexadecimal (base-16).
