The 100th digit
In the decimal expansion of , what is the 100th digit after the decimal?
Details and assumptions
In the decimal expansion , the 2nd digit after the decimal is 4.
The answer is 8.
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1 solution
By doing long division, we get that 1 . 0 0 0 0 0 0 ÷ 7 = 0 . 1 4 2 8 5 7 remainder 0 . 0 0 0 0 0 1 . As such, this tells us that the pattern will repeat, since 0 . 0 0 0 0 0 1 ÷ 7 = 0 . 0 0 0 0 0 0 1 4 2 8 5 7 remainder 0 . 0 0 0 0 0 0 0 0 0 0 0 1 .
Thus, 7 1 = 0 . 1 4 2 8 5 7 .
To calculate the 100th digit after the decimal place, we see that it is periodic with period 6. Since 1 0 0 = 6 × 1 6 + 4 , the 100th digit will be equal to the 4th digit, which is 8.