Step by step
The question
Question: What is the 100th digit after the decimal point of 1/13?
1 Find the block
10 divided by 13 is 0, and 10 is left.
We divide 1 by 13 the long way.
Each time we bring down a 0.
100 divided by 13 is 7, and 9 is left.
90 divided by 13 is 6, and 12 is left.
120 divided by 13 is 9, and 3 is left.
30 divided by 13 is 2, and 4 is left.
40 divided by 13 is 3, and 1 is left.
1 left: that is where we started.
So the digits repeat from here.
So 1/13 is 0 point 0 7 6 9 2 3, and the same 6 digits again and again.
The block is 6 long.
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
6 times 16 is 96.
So 16 full blocks take us to the 96th digit.
100 minus 96 is 4 more.
3 Count into the block
Careful!
The 16 only counts full blocks.
The 4 left over gives the place.
4 left means place 4 in the block: 9.
4 Check
Let's check by counting on.
A block ends at the 96th digit.
Then we count on: 0, 7, 6, 9.
The 100th is 9.
So it is right!
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.