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What are the first 10 digits after the decimal point (technically the hexadecimal point...) when the fraction 1/7 is written in base 16?

 Mar 9, 2019
 #1
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"What are the first 10 digits after the decimal point (technically the hexadecimal point...) when the fraction 1/7 is written in base 16?"

 

\(x = \frac{1}{7}\\x_d=0.\overline{142857}\\x_h=0.\overline{249}\)

 

I'll leave you to work out the other 7 digits!!

 

 Mar 10, 2019
edited by Alan  Mar 10, 2019
edited by Alan  Mar 10, 2019
 #2
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+1

Here is another way of doing it:

1/7 =0.142857...........etc.

 

1- Multiply the above fraction by 16 and subtract the integer part:

16 x 0.142857......... = 2.2857142857142857142857142857143 - 2 =0.28571428..........etc. Keep this 2 and:

 

2 - Multiply the remaining fraction by 16:

0.2857142857...... x 16=4.571428571428571428571428571 - 4    =0.57142857.............etc. Keep this 4 and:

 

3- Multiply the remaining fraction by 16:

0.57142857......      x 16 =9.14285714285714285714285714 - 9     =0.142857142...........etc. Keep this 9 and:

 

4 - List them from top to bottom with a decimal in front:

0.249249249249........As you can see the decimal reverts to 1/7 =0.14285712..........etc. in (3) above.

 Mar 10, 2019

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