What is the 2022nd digit after the decimal point when 3/7 is expressed as a decimal?

Question

Due in a few hours! Thank you!

Answer 1

Hi Zoe,

Do you remember when you first learned to divide?

And do you remember that when the division didn't result in a whole number, your answer included a remainder?

This is called integer division and it's probably the easiest way to solve this problem.

First, I'm going to assume that you're permitted to use a calculator to convert 3/7 to decimal digits.

Notice this results in a repeating pattern of digits; .428571428571428571.

Also notice that there are 6 digits in the repeated pattern.

Now we divide 2022 by 6, BUT we will express our answer with a quotient AND a remainder.

2022/6 = 337.R 0

Hey, wait a minute, the remainder is ZERO!

What's this mean? It means that in order to get to the 2022nd digit we went through the pattern of 6 digits EXACTLY 337 times. Therefore, the 2022nd digit is the last digit in the pattern, or 1.

I feel cheated because these numbers don't give us a chance to see how the trick usually works.

So, let's do another.

Suppose we want the 500th digit.

We divide 500 by 6 as integer division.

500/6 = 83 R 2.

What's this mean?

It means that to get to the 500th digit, we went through the entire pattern 83 times, and the remainder of 2 means we've taken 2 more steps into the next repetition.

The 500th digit is the 2nd digit in the pattern, or 2.

I hope this helps.

Answer 2

the 2022rd digit is 1

3/7= .4285714285714...

the 428571 sequence repeats, endlessly

that's 6 digits that repeat

2022 divided by 6 = 337 exactly with no remainder

1st digit is 4, 2nd 2, 3rd 8, 4th 5, 5th 7, 6th 1

7th 4

13th 4

19th 4

25th 4

any number = 6x+1 has a 4 digit, where x is an integer

2023rd digit is 4

2022nd digit is the digit just before 4, in the 428521428521... sequence = 1