by Larry Campbell | Jan 22, 2016 | Explanations, More Math Tidbits |
Here are just a few of the mathematically special dates in 2016. I hope you’ll have fun celebrating them – I know I will!
1. Pi Day 2016 !! We might as well start with what we called last year the ONCE IN A CENTURY date! As you may remember, the common decimal approximation for pi is 3.14. So March 14 (3/14) (which also happens to be Einstein’s birthday!) is called Pi Day and is actually celebrated every year. But, if you carry pi’s decimal expansion (3.1415926535 . . . going on forever without repeating!) a little further for your approximation, you get 3.1415 – which was GREAT last year!! BUT, if you round the approximation above to 4 places, you would get 3.1416. So, on Pi Day this year (3/14/16), there you could almost have just as big a celebration as last year. 🙂 Mark your calendar now! 🙂
2. Days involving Powers and Exponents. Since the last two digits this year are 16, and since 16 is equal to both 2^4 (2 x 2 x 2 x 2) and 4^2 (4 x 4), this lends itself to lots of nice patterns with exponents. (See also 2.5) We’ve shared one of them already, namely 1/4/16, which is 2^0/2^2/2^4 (consecutive even exponents. Another will be 4/8/16 (2^2/2^3/2^4, consecutive exponents). In fact, any date this year with only 1’s, 2’s, 4’s and/or 8’s for the month/day will be some kind of ‘powers of 2 day’.
2.5 Base/Exponent Day: Feb 4 (2/4/16) and April 2 (4/2/16). In each case, the month raised to the power of the day equals the year. These are pretty rare. The phenomenon won’t happen again until 2025. (What day?)
3. Perfect Square Day This is somewhat similar to #2 above. On April 4 or 4/4/16. Since 4 x 4 = 16, 16 is the perfect square of 4.
4. Year-Product Days: Jan 16, Feb 8, Apr 4, and Aug 2. Also similar. Look carefully at these four dates (1/16/16, 2/8/16, 4/4/16, and 8/2/16), and you’ll note that the product of the first two numbers (mo/day) is the third one (year). These are called “Year – Product Days”, and this is the first year we’ve had (at least) FOUR of them since 2012 (when there were 6!) .
5. Consecutive Multiples of 4 Day: Aug 12. That day – 8/12/16 – is composed of three consecutive multiples of 4. We haven’t had one of these since April 8, 2012 and we only have one more this century! (What is the day? Why is that the last one?)
6. June 28 – The BEST mathematical day each year! We did this last year, but this is MY favorite numerical day each year. There are TWO reasons to celebrate 6/28, each one significant!
A. 6/28 is 2-PI Day (Take 3.14 or 3/14, and double each number). Note that there is no 3-PI Day or any other similar multiple. (Why?)
B. 6/28 is the only day each year where the month (6) and day (28) are different perfect numbers!! (For a quick review of perfect numbers, visit the July14 Math Tidbit and note the material in red font. Visiting this link will also remind you why July 28 is special each year as well.)
7. Over 300 others! I mentioned that I happen to believe every day is special. If you’re so inclined, send in your birthday – or any other favorite – date, and I’ll let you know why it’s special! 🙂 [I’ll respond directly and -with your permission – possibly list it in a future mailing.]
by Larry Campbell | Nov 18, 2015 | More Math Tidbits |
Pick any positive integer you want, and look at its divisors (or factors). For example, the divisors of 15 are 1, 3, 5, and 15; the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24; and the divisors of 29 are 1 & 29. Now disregard the number itself , and consider the other factors (these are called proper divisors, FYI.) So the proper divisors of 15 are 1, 3, and 5.
Now, if you were to ADD those proper divisors, it should make sense that the SUM of these divisors could only be less than, equal to, or greater than the original number. (In the example above, 1 + 3 + 5 = 9, which is less than the starting number 15.) Let’s look at these three cases, in order of how rare they are:
1. If a number’s proper divisors add up to the number itself, we call the number perfect (partly because this is SO rare!). Example: 6 is perfect, since its proper divisors are 1, 2, & 3 and 1 + 2 + 3 = 6.
Interesting Facts about perfect numbers:
*The first 4 perfect numbers are 6, 28, 496, and 8128. These 4 were known to the anient Greeks.
*There are currently only 48 known perfect numbers. (Many of them are HUGE.) All of them are even.
*It can be proven that all (even) perfect numbers end in 6 or 8.
*No one knows if there are any odd perfect numbers. (Find one, and make us both famous. :-)).
2. If a number’s proper divisors add up to less than the number, we call the number deficient. Example: The proper divisors of 9 are 1 & 3. Since 1 + 3 adds to less than 9, we call 9 deficient.
Interesting Facts about deficient numbers:
*All prime numbers are deficient. (If you think about this, you can explain why.)
*In some sense, deficient numbers are ‘rarer’ than abundant numbers (see below). They’re both ‘infinite’, but there are almost always fewer deficient numbers than abundant in a given stretch of 100 numbers.
3. If a number’s proper divisors add up to more than the number, we call the number abundant. Example: The proper divisors of 12 are 1,2,3,4, & 6. Since they add up to 16, which is more than 12, we call 12 abundant.
Interesting Facts about abundant numbers:
*There are odd abundant numbers, but they are relatively rare. The smallest odd abundant number is 945. (But it can be proven that there are an infinite number of odd perfect numbers!)
*If you have an abundant number, then 4 times that number will also be abundant.
Bonus for Teachers and Explorers
So, every positive integer, then, is either deficient, abundant, or perfect. It is fun to explore, say, the first 50 (or more) positive integers and keep tally or make a graph of how many of each. This is particularly fun (and instructive – think of the work with divisors!) for students of almost any level above 4th or 5th grade. Try it out!!
by Larry Campbell | Sep 23, 2015 | Explanations, More Math Tidbits |
Two common mistakes when it comes to primes: 1 is NOT considered a prime, and 2 is prime (the only even prime)!! The fact that 1 is not a prime is often confusing, right? After all, it is technically a number that is ‘divisible only by 1 and itself’ – which is how most people remember and word the definition of prime. So, why isn’t it considered a prime?
Actually, this is not one of those things that ‘came down off the mountain with Moses’. No high authority decreed that 1 shouldn’t be prime – indeed, there was a time (centuries back) when 1 was considered a prime! So, it’s one of those things that has developed into a convention over time, for a variety of reasons, mathematically.
One hears lots of ‘explanations’ for this, but for me, there are two main reasons. The first is the definition. The most precise definition for a prime is “a number which has exactly two divisors”. This is almost the same as the definition many of us use (no other divisors besides itself and 1), but it excludes 1 (which only has one divisor). That reason works for me . . . but I bet that still feels like it’s ‘hand-waving’, doesn’t it? The second reason – below – actually explains the big picture for me and involves a tiny bit of history.
Primes are essentially the ‘building blocks’ of all the other integers. It is true that ANY/EVERY positive integer is either already prime OR can be written as a product of primes. Examples: 30 = 2 x 3 x 5; 72 = 2 x 2 x 2 x 3 x 3; 35 = 5 x 7, etc.
Actually, even more is true. It turns out that ANY/EVERY non-prime positive integer (> 1) can be written as a product of primes in only one way! So, whereas 30 can be ‘factored’ in lots of ways ( 6 x 5 or 10 x 3 or 2 x 15 or 30 x 1), it can only be ‘factored’ into primes in one way, namely 2 x 3 x 5. (We don’t count 3 x 5 x 2 as ‘different’, by the way.)
The highlighted phrase above (‘in only one way‘) turns out to be important. Math types love ‘uniqueness’ when they can get it! This particular uniqueness for prime factorization is considered important. (It’s now actually called the ‘fundamental theorem of arithmetic’.)
So, here’s the big deal, though it will seem like ‘splitting hairs’ to some. 🙂 IF 1 were considered prime, then the ‘uniqueness’ fact above would not be true!! 🙁 You could then write 30 as 2 x 3 x 5 x 1, or even 2 x 3 x 5 x 1 x 1, and so on.
So, long story short and over-simplified: over the years and centuries, it became more important to mathematicians to keep the uniqueness and make 1 a ‘special case’, than it was to keep one a prime and lose the uniqueness. It was just a choice – and a historical development that became a convention. So 1 is NOT considered prime.
by Larry Campbell | Aug 20, 2015 | More Math Tidbits |
Here are three of several classic unsolved problems in mathematics. The problem statements are listed first, with definitions and background, as appropriate, below.
Mystery #1: Are the pairs of twin primes infinite? (Do they go on forever or end at some point?)
Mystery #2: Are the perfect numbers infinite? (Do they go on forever . . .?)
Mystery #3: Are there any odd perfect numbers?
BACKGROUND:
Mystery #1: Primes are positive integers (other than 1) that are divisible only by themselves and 1. Twin Primes are primes that are only two apart: 3 & 5, 17 & 19, 101 & 103, etc. Interestingly, it has been known for a long time that the primes themselves are infinite. Euclid proved this roughly 200 years before Christ. So, this is one of those cases where it seems like it should be easy to settle, but no one has been able to prove that they go on forever or prove that they eventually quit.
Mystery #2: Perfect numbers are those whose factors (other than the number itself) add up to the original number. Examples: The factors of 6 (other than 6 itself) are 1,2, & 3. 1 + 2 + 3 = 6, so 6 is perfect. 28 is also perfect, since 1 + 2 + 4 + 7 + 14 = 28. The next two are 496 and 8128, and these four were known to the ancient Greeks. There are currently only 48 known perfect numbers – and several of them are HUGE, but no one has succeeded in proving they go on forever or that they eventually stop.
Side Note: Perfect numbers have a surprising and fascinating connection to Mersenne Primes (mentioned in earlier Math Tidbits), but this will have to wait for another Math Tidbit sometime.
Mystery #3: All 48 known perfect numbers (see above) are even. In fact they all end in 6 or 8. It can be proven that if a number is perfect and even, it will end in 6 or 8, but no one has yet found an odd perfect number. Nor has anyone (yet) proven that there can’t be any.
by Larry Campbell | Jul 21, 2015 | Explanations, More Math Tidbits |
As it turns out, the answer to all these questions is “yes!” Even for a million! (You can find a million consecutive composite integers. That is, you can find a string of one million numbers without hitting a prime.) And – astonishingly – it turns out to be true for ANY number. Pick any number you like, no matter how big, and you can find that many consecutive numbers without a prime in them!! **
One of the ways this is phrased is that you “can find arbitrarily large gaps in the primes”.
This is particularly ironic – almost contradictory! – since the primes themselves are infinite! They NEVER run out – there is no largest one, they go on forever. And yet, we can find gaps in them as large as we’d like!!
** In each of these cases, one can even determine a number that starts the string. The reasoning is not too difficult (I used to show it to future elementary teachers all the time), but it involves learning a new symbol, and a few steps to think about. I’d be happy to elaborate for anyone who wants to know more. Feel free to contact me.
by Larry Campbell | Jun 26, 2015 | More Math Tidbits, Other Fun Stuff, Photos2 |
MATH CLOCK 🙂
A couple of notes:
1. There is an exponent of 2 buried under the minute hand. (x squared).
2. For you real math aficionados (French for ‘geek’?), there is a minor technical mistake on the clock. Can you spot it?
