by Larry Campbell | Nov 20, 2014 |
COUNTER-INTUITIVE?
There are several places in mathematics where ‘the truth’ isn’t quite what it might feel like it should be. This is obviously good and bad.
On the one hand, I’m VERY aware -from years of teaching- that this phenomenon can be frustrating, and contribute to the all-too-familiar feeling that math is just a set of rules that make no sense. 🙁
On the other hand, ‘well informed is well armed’ as they say, and it can help not to be misled, intentionally or otherwise. (See Example 3, e.g.) Intuition can be a valuable ally or a tricky foe.
I’ve picked 3 (of many), interesting* examples of areas where ‘the truth’ is counter-intuitive. The first 2 are related to probability (likelihood), and are prompted by our ‘how likely?’ discussions of earlier this month. The 3rd is DEFINITELY an area where a consumer can be tricked, if not careful.
*I know ‘interesting’ is in the eye of the beholder, but I hope you’ll think so, too. 🙂
To see the three examples, see Counter-Intuitive Examples.
by Larry Campbell | Oct 23, 2014 |
Primes, Twin Primes, & Infinity
Most of us can semi-remember (can’t we?) the definition of a prime number from somewhere in our school days. A prime number is a (positive, whole) number that is only divisible by itself and 1. (Technically, the more precise definition is ‘has exactly two divisors, but there’s little difference as long as we remember that 1 is NOT considered a prime.) There are lots of things that could be explored further here, but those are for another day. 🙂
Interestingly enough, we KNOW that the primes ‘go on forever’ ! Another way to say that is that they are infinite. Another interesting way to say it is to note that there will then be NO largest prime!! (There will always be another one bigger!). This fact was proved by Euclid (of geometry fame) roughly 200 years before Christ. It is often considered one of the most beautiful and elegant proofs in mathematics (it can be understood by high schoolers) and that it should be over 2000 years old only adds to its intrigue. 🙂
Closely related to primes – and also of interest to number geeks – are twin primes. Twin primes are pairs of primes that are only 2 apart, like 5 & 7, 11 & 13, 101 & 103, etc. [FYI, 2 & 3 are both primes, but are not considered twin primes, since 2 is a unique and special case of a prime. (Why?)]
And here is the fascinating part: While primes themselves have been known to be infinite since well before Christ, the question of whether or not twin primes ‘go on forever’ is still unknown and unsettled!! No one has succeeded in proving (or disproving!) this fact to this day!! This is one of the beauties of math that the one question has been settled for so long, and yet the other (similar) question still goes unsettled!
BONUS Challenge:
If anyone out there can prove or disprove the above question for twin primes, it will get you a 6 FREE copies of (any/all of) my book(s)!! – AND make you (and me? :-)) very famous!! Good luck!!
by Larry Campbell | Sep 18, 2014 |
A Standing Ovation for a Math Talk in which Nothing was Said!!
(Yes, I know some of you would gladly give a standing ovation for a math talk that said nothing!)
In 1903, Frank Cole delivered a talk at the annual American Mathematical Society (AMS) meeting. He never spoke a word. Here’s the short version of what happened:
At that time, it was thought that (2^67 – 1) [2 multiplied by itself 67 times, and then subtract 1] was prime. Mr (Dr?) Cole spent his entire ‘talk’ doing two things:
1. First he multiplied 2 by itself 67 times and subtracted one, correctly getting 147,573,052,589,676,412,927.
2. He then moved to another blackboard (if you’re too young to know what those are, ask someone over 50 or so), and multiplied (761838257287) x (193707721). His (correct) answer was the same number he got in #1 above.
The audience stared for a second, then burst into spontaneous applause, realizing that Cole had shown that the number in question had (at least) 2 other divisors (besides itself and 1), and therefore could not be prime!!
EPILOGUE: It turns out Cole had no mysterious secret. He had merely spent his Sunday afternoons for 20 years checking primes of this form and happened to get lucky! Even I roll my eyes at that! I mean – I know there was no Sunday afternoon football then, but double-checking certain types of primes?! 🙂
***
For you math-types (or anyone else!) who would like to see a few more details, you may download and visit this 5-slide Powerpoint: MersenneStandingOvation
by Larry Campbell | Aug 21, 2014 |
Cool Primes: 37 & 73 🙂
Our son Adam just turned 37 (yes, we’re old!). In honor of this event, lets look at a couple of cool properties of 37 and its ‘mirror image’ 73. They may seem like boring numbers (though how can any number – let alone a prime – be boring?! :-)), but they’re not . . . .
1. First for 37: It has this interesting divisibility property: It is the only two-digit* number the ‘goes into’ all 9 of 111, 222, 333, 444, 555, 666, 777, 888, and 999.
* There is ONE one-digit number (besides 1!) and ONE 3-digit number that also divide all nine of these, and with some thought, they should ‘easy’ to find.
2. As for 73: Some of you know that Sheldon on The Big Bang considers 73 to be the ‘best number’. Whether you’re a Big Bang fan or not, you might find his reasoning both interesting and hilarious. See the clip here: Sheldon & 73
A little extra lagniappe . . . Look at the two numbers 3773 and 7337. Neither of them is prime, but it turns out that they are both divisible by 11** (as well as several others.)
**I find this fascinating (am I the only one? 🙂 ): It turns out (and can be proven) that any four-digit palindromic number (some forwards as backwards) will be divisible by 11!
by Larry Campbell | Jul 24, 2014 |
Today’s date (7/28/14) is (numerically) special – for several reasons:
1. Each year, July 28 (7/28) is the last day of the year year where the Day # (28, in this case) is four times the Month # (7, in this case). (There are only 7 of those each year! *)
2. Today (7/28/14) is the last day this year where all three numerals (month, day, and year) are each multiples of 7. There are/were only FOUR of those this year* (and there haven’t been any others since ’07!), and only TWO of those where the month/day/year are all different, as well !!*
* Can you name all the dates in question?
Bonus Tidbit:
28 happens to be what is called a perfect number. The factors of 28, not counting itself (1,2,4,7,14) all add up to 28. This is a rare phenomenon (hence the name ‘perfect’ :-)), and there were only four such numbers* known to the ancient Greeks. Even now, there are less than 50 of these numbers that are known – most of them VERY big!
* (Added on 12/29/14) These four were/are 6, 28, 496, and 8128.
(EXPLORE: There is one perfect number LESS THAN 28 ! Can you find it?!?)
If you have answers to any of the above questions, and are so inclined, feel free to share them (use the Contact Form on the website, or reply to your Weekly Sharing e-mail). We’ll share answers and respondents in a future Sharing. 🙂
