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19Oct/1119

Seeking interesting math factoids about the number 11

I'll be doing a brief talk at a conference next month, on 11/11/11 at 11:11AM, about the number 11. If you know interesting factoids about 11 (or about 1111...) that I could include in the talk I'd greatly appreciate it.

One factoid is that 1/(1+(1/(1+1/(1+... converges to the golden ratio.

Another is that if a number is divisible by 11, reversing its digits will result in another multiple of 11.

Other ideas?

Thanks.

1. 7*11*13 = 1001. That leads to all sorts of goodies, I think. Any number of the form abcabc is abc*1001, so it has 11 as a factor (along with 7 and 13).

2. Sue, it’s more subtle than that. 11×932 as a random example = 10252. Reverse the digits of the product to get 25201. 20251 = 11×2291.

I know the answer has to do with the alternating sum test for divisibility by 11. I don’t recall that proof and I’m too lazy to look it up right now.

3. Slightly hidden use of repunits (1, 11, 111, etc)

11-3×3=2
1111-33×33=22
111111-333×333=222

I love the idea of an 11 theme day. I’ll keep thinking.

4. It’s pretty basic, but I’ve always found this to be pleasing:
11×11 = 121
111×111 = 12321
1111×1111 = 1234321

111111111×111111111 = 12345678987654321

5. Showing how powers of 11 yield the rows in Pascal’s Triangle is always fun, especially when you don’t ‘carry’.

The proof of the alternating digital sum property is very similar to the proof of the divisibility-by-9 rule. Write out the number in its expanded base-10 representation. Replace every 10 with (11-1). Each 10^n will then become (11-1)^n.

When each of those binomial powers are expanded, every term except the last will be divisible by 11. In each case, the last term is the ‘digit’ times (-1)^n. So the entire sum is divisible by 11 if and only if that alternating sum is.

6. There are eleven different types of real intervals: ∅, {a}, (a,b), [a,b), (a,b], [a,b], (a,∞), [a,∞), (−∞,b), (−∞,b], and (−∞,∞).

7. 11 is a repunit prime number. see http://en.wikipedia.org/wiki/Repunit
(well, elEVEN is not EVEN but I don’t know if it counts as a factoid)

8. 11 : composite(3)+3 = 8+3 = 11
11 : Smallest number which is square begin and end with the same digit 11^2=121

On November 11, 2011, at 48 minutes and 47 8/9 seconds before Noon the time will be:
11/11/11 11:11:11.1111111111111…

11 x 11 = 65 + 56 (palindromic equality)

11^2 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4

11^2 x 9182736455463728191 = 1111111111111111111111

“Five plus Six” or ”Two plus Nine” each has 11 English letters. Hence it is an “honest” number

11 is the largest number which is not expressible as the sum of two composite numbers

11 is the smallest strobogrammatic prime

11 is the only palindromic prime with an even number of digits

Another curiosities in http://primes.utm.edu/curios/page.php?short=11

9. Hi,
expanding Mark’s example:

multiplication by 11 : put digits of the number you’re multiplying and in between each digit, the sum of the two digits beside each other.

11 * xy = x (x+y) y
apply the carry of (x+y) to x, if there is one.

11 * xyz = x (x+y) y (y+z) z
apply any carries from right to left.

etc…

10. Thanks, everybody, for the great ideas!

11. I like the second proof here of why the alternating sum thing works to test for divisibility by 11.

http://www.artofproblemsolving.com/Wiki/index.php/Divisibility_rules/Rule_for_11_proof

12. 11 is a deficient, lazy caterer, odd, odious, palidromic, repunit, square-free, and Ulam number. It is a prime, a palindromic prime, a strobogrammatic prime, and a twin prime. 11 is the smallest prime such that 2p-1 is not prime. 11 is part of a sexy prime quadruple; n, n+ 6, n + 12, and n + 18 are all prime numbers.
11, when written in base 10 is “11”, but when written in base 11 it is “10”.
11 is the largest number which is not expressible as the sum of two composite numbers
11 is the smallest prime for which the sum of digits equals the number of digits
11 is the only prime comprising an even number of identical digits
11 is the only palindromic prime with an even number of digits
11 divides all palindromes with an even number of digits.
11 can be partitioned 56 different ways.
11 is a strobogrammatic calculator number; it does not change value when you rotate it 180 degrees when using an old style 7-bar calculator display.
11 is a tetradic number; one of only 17 that are less than 10,000.
11 is an 11-smooth number; it has no prime factors larger than 11.
11 is the largest known multiplicative persistence.
The number 6 can be partitioned 11 different ways.
We know that the 11th Fibonacci number is a prime number.
11 is the largest known multiplicative persistence.
There have been a total of 11 Star Trek movies.

13. Check out this web site: http://homepage2.nifty.com/m_kamada/math/10001.htm

It has factorizations of 11, 101, 1001, 10001, …

My personal favorite is: 10^67+1 = 1000000000000000000000000000000000000000­0000000000000000000000000001 (68 digits) = 11 * 9090909090909090909090909090909090909090­90909090909090909090909091 (66 digit)

14. Check out this website: http://sites.google.com/site/numeropedia/number11

I like the fact that November was originally intended to be the 9th month of the year.

15. Here’s a proof that any number n not divisable by 2 or 5 works:

Eventually there are two numbers a=111… and b=111… Which are congruent modulu n. Then a-b=111….000…0=1111…000…0=111…1*10^x=c*10^x is divisible by n. Thus c is divisible by n is such number.

Thanks for the riddle!

16. You must know about http://numbergossip.com/11 right?
I bet Tanya would be interested in this 11 gossip too.

17. 11 is:

is the third honest number, because 11 = “two plus nine”.
is the only palindromic prime with an even number of digits.
is the smallest prime with multiplicative and additive persistence of 1.

= 6 + 5 = 62 – 52
= root(5! + 1)
= LV/V = MC/C

Any power of 11 ends in a 1.

1111 contains two embedded elevens.

Any large number is divisible by 11 if the difference between the sum of its odd digits (units, hundreds, etc.) and the sum of its even digits (tens, thousands, etc.) is 0 or a number divisible by 11.

Displayed on a calculator, it reads the same whether the calculator is turned upside down or reflected on a mirror, or both.

‘Eleven plus two’ is the anagram of ‘twelve plus one’.

The Maoris, the initial inhabitants of New Zealand, used for reckoning or accounting purposes the undecimal (base-11) positional notation system.

11 x 11 = 65 + 56 (palindromic equality)
112 x 9182736455463728191 = 1111111111111111111111
– posted by Yosuke Ikeda, Yasuji Kondo, Yasuhiko Sakaitani, Ken Hirotomi, Hiroyuki Ozaki, Atushi Tanaka, Ryohei Miyadera (Kwansei Gakuin High School)
112 = 65 + 56, and 652 – 562 = (3 x 11)2,
112 = 30 + 31 + 32 + 33 + 34 (sum of consecutive powers of 3)
113 = 32 + 192 + 312

1 / 11 = 0.09 09 …
2 / 11 = 0.18 18 …
3 / 11 = 0.27 27 …
4 / 11 = 0.36 36 …
5 / 11 = 0.45 45 …
6 / 11 = 0.54 54 …
etc.

11 + 1.1 = 11 x 1.1

Magic triangle with a constant of 11;
2 + 3 + 6 = 2 + 5 + 4 = 6 + 1 + 4 = 11
2
3 5
6 1 4

18. Oh, I almost forgot …

But then, what is the continued fraction of the number Fi??

19. For a polygon of 11 sides, there are 11*5 number of diagonals.
For a polygon of 111 sides, there are 111*55 number of diagonals.
For a polygon of 1111 sides, there are 1111*555 number of diagonals.
😀
Love the number 1!