Step by step
The question
Question: What is the 40th digit after the decimal point of 2/7?
1 Find the block
20 divided by 7 is 2, and 6 is left.
We divide 2 by 7 the long way.
Each time we bring down a 0.
60 divided by 7 is 8, and 4 is left.
40 divided by 7 is 5, and 5 is left.
50 divided by 7 is 7, and 1 is left.
10 divided by 7 is 1, and 3 is left.
30 divided by 7 is 4, and 2 is left.
2 left: that is where we started.
So the digits repeat from here.
So 2/7 is 0 point 2 8 5 7 1 4, 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 6 is 36.
So 6 full blocks take us to the 36th digit.
40 minus 36 is 4 more.
3 Count into the block
Careful!
The 6 only counts full blocks.
The 4 left over gives the place.
4 left means place 4 in the block: 7.
4 Check
Let's check by counting on.
A block ends at the 36th digit.
Then we count on: 2, 8, 5, 7.
The 40th is 7.
So it is right!
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.