Your birthday is hiding in this number
Tonight's idea
Write the counting numbers in order, with no spaces, after a decimal point:
\[0.123456789101112131415\ldots\]This is Champernowne's number. It never stops. Every digit has a place: the 1 is first, the 2 is second, and the 9 is ninth. Do not count the zero before the decimal point.
Choose a string of digits, such as 2020. Why must it appear? Sooner or later, the counting numbers reach 2020. Its four digits are written together.
What about 0042? Put a 1 in front. When you write 10042, you write 0042 inside it. The extra 1 keeps those starting zeros.
Every finite string of digits appears. Put a 1 in front to make a counting number that contains it.
That is a proof, without a search. It gives a place to find your string, but that might not be the first place.
Watch one: find the first 11
The numbers 1 to 9 use nine digits. The number 10 uses positions 10 and 11. The number 11 uses positions 12 and 13.
The first 11 starts at position 12. Check the earlier neighbours: 12, 23, 34, 45, 56, 67, 78, 89, 91, 10 and 01. None is 11.
A string can cross a join between counting numbers, too. The first two digits already make 12, long before you write the counting number 12.
Your turn
- What is the 15th digit after the decimal point? Which counting number does it belong to?
- (a) How many digits do the numbers 1 to 9 use? (b) How many do 10 to 99 use? (c) At what position does the counting number 100 begin? Hint: there are 90 two-digit numbers.
- Sam says: "2020 must appear, because eventually we write the counting number 2020." Does that prove it appears? Does it prove that this is the first place it appears? Explain both answers.
- Write a birthday as four digits: two for the month, then two for the day. For example, March 4 is 0304. (a) Name a counting number containing 0304. (b) Name one containing your birthday. (c) Explain why the same trick works for any finite digit string, even 0000.
Stretch: jump to digit 1,000
What is the 1,000th digit after the decimal point? Do not write a thousand digits. Use your answer to exercise 2, then count three-digit numbers in blocks.
Hint: after the first 189 digits, another 810 digits make 270 complete three-digit numbers. Which number comes next?
Answers
- 2. The counting number 12 uses positions 14 and 15. Its second digit is 2.
- (a) 9 digits. (b) Each of the 90 numbers uses two digits: \(90\times2=180\). (c) Together they use \(9+180=189\) digits, so 100 begins at position 190.
- Yes: when 2020 is written, all four digits appear together. No: that only gives a guaranteed occurrence. The string might appear earlier, including across a join. For example, 12 appears at positions 1 and 2 before the counting number 12 is written.
- (a) 10304 contains 0304. (b) Put a 1 in front of your four digits. (c) A 1 followed by any finite digit string is a positive whole number, with no starting zero. It is eventually written in the list, including every digit of your string. For 0000, use 10000. The trick guarantees an occurrence; it does not promise the earliest one.
Stretch. 3, the first digit of 370. The numbers 1 to 99 use 189 digits. The next 270 numbers are 100 to 369, using \(270\times3=810\) digits. That reaches position \(189+810=999\). The next number is 370, so its first digit, 3, is at position 1,000.
All these strings come from one exact rule. Finding every string does not mean the digits were chosen at random.