perplexus.info :: Numbers : Decimal Specification

Decimal Specification (Posted on 2022-06-13)

Given that:

              1
N = ----------------------------
    1.00........00100........001
        50 zeros     50 zeros

is a decimal (base ten) rational number.

Describe the pattern after the decimal point in the decimal expansion of N.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (1 votes)

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computer solution

| Comment 1 of 3

digits 700

n=vpa(1)

n=n/vpa(10)^51+1

1/n

>> decimalSpecification
n =
1.0
n =
1.000000000000000000000000000000000000000000000000001
n =
1.000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000001
ans =
0.
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
>>

The third value shown for n is the denominator of the intended fraction.

Taking the reciprocal to get N, shows, after the decimal point:

51 9's

102 zeros

...

repeating, presumably forever.

Switching to 1400-digit precision gives:

0.
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999999999999999999999999999999999999999999999999999
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

with that 1 at the end just the indication that the next digit is higher than 5 (i.e., 9).

Edited on June 13, 2022, 9:04 am

Posted by Charlie on 2022-06-13 09:02:41

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