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Perfect Numbers
If a number’s proper divisors (the divisors other than the number itself) add up to the number itself, we call the number perfect (partly because this is SO rare!). Examples: 6 is perfect, since its proper divisors are 1, 2, & 3 and 1 + 2 + 3 = 6. 28 is perfect because the proper divisors are 1,2,4,7, and 14, and these add to 28.
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 50 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. :-)).
*One of the most amazing coincidences in all of mathematics is the perfect numbers are directly linked to Mersenne Primes. But that is a story for another day.
Limerick Translation
Is Zero (0) Even or Odd? (Or Neither or Both?)
A whole number is even if it is divisible by 2. Equivalently (since sometimes there is confusion with 0 and the word ‘divisible’), a whole number is even if it can be written as two times some whole number. E.g., 22 is even because it can be written as 2 time somes whole number (namely 11), and 7 is NOT even because it can’t be written as 2 times some whole number.
That’s all there is to it. SO, to ask “Is o even?” is to ask “Can it be written as 2 times some other whole number?” AND, since that answer is yes (2 x 0 = 0), we say that 0 is even.
If we’re careful (and have already defined even), it’s OK to say a whole number is odd if it isn’t even. We could – maybe should – get more technical, but I won’t. A whole number is either even or odd, never both.
SO, since 2 IS even, it is NOT odd. And therefore it is NOT both (and not neither, for that matter. 🙂 ).
As long as we’re here, let’s quickly review positive and negative. An integer is positive if it’s greater than 0, and it’s negative if it’s less than 0. Since 0 is not greater than or less than itself, it is NEITHER positive nor negative.
NOTE – and PREVIEWS 🙂 We have used terms like ‘integer’ and ‘whole number’. Do you remember all the distinctions for the various sets of numbers we use? That will show up as a Math Tidbit very soon. So might the discussion of the why we don’t divide by 0.
The Pythagorean Theorem – Amazing & Beautiful!
Most of us ‘memorize’ (what we think is) the Pythagorean Theorem. And it seems to be one of the few things we claim to remember from math classes. My guess is, that if asked right now what the theorem is, most of us would answer (somewhat triumphantly)
a2 + b2 = c2 !!
And, we’d be partially right. To explore further, we first need some precision. This is actually more than just being picky. 🙂
Precisely, . . .
The Pythagorean Theorem is actually an IF-THEN statement/theorem, namely, IF you have a right triangle with sides a & b, and hypotenuse c, THEN the relationship a2 + b2 = c2 will always be true. If you think about it, that’s actually amazing! It always works!
Mathematically, that means that if you square (multiply by itself) one of the shorter sides of a right triangle, and add it to the square of the other shorter side, the sum will always equal the square of the hypotenuse.
The standard example is the right triangle with sides 3 & 4, and hypotenuse 5. Then (3×3) + (4×4) = (5×5), or 9 + 16 = 25. (Or try it with sides 5, 12, 13).
Amazing – even beautiful, right, but often BORING (and unappreciated) to students.
The Geometric Interpretation!
It was WELL into my career before I was shown this! (Why?!?) Look at the picture below:
Notice the RED right triangle, and the squares built on each of the 3 sides. What is the area of the yellow square, which has side a? Right – it’s a x a or ‘a squared’. Similarly, figure the areas of the blue and green squares, as ‘b squared’ and ‘c squared’, respectively.
Another way of ‘seeing’ the Theorem is that the AREAS of the yellow and blue squares together EQUAL the area of the green square!! Now, that’s beautiful, amazing, and interesting!!
The Generalized Theorem!
Once one sees the geometric interpretation above, a very fascinating fact can be appreciated (without having to ‘do the math’). The Pythagorean Theorem is not just true for squares!! Look at the figure below:
With the same orginal right triangle (labeled slightly differently here, though that is no problem), the theorem is still true for these figures (regular pentagons). The area of A, added to the area of B, is EQUAL to the area of C !! Doesn’t that send shivers of joy up and down your mathematical spine?!? 🙂
One final sidelight:
Returning to the IF-THEN statement at the top of the page . . . it turns out that the IF-THEN statement is also true the ‘other way’ (called the converse, as in “. . and conversely . . “). IF you have a triangle with sides a,b, and c, and the a2 + b2 = c2 relationship is true THEN you must have a right triangle! (It won’t work any other way.)
What is PI really? Enlightening background and history.
This needs to be said carefully, but pi is technically a ratio. More below, but first: a ratio is a comparison of two quantities, as in ‘the ratio of boys to girls in that class is 5 to 4.’ Such a comparison (using the example) is often written 5:4, or sometimes 5/4. And that can be viewed as the fraction number 5/4 or 1.25.
Ratios and numbers are pretty much interchangeable (all numbers can be written as ratios, but not quite all ratios can be written as numbers*), so this is usually not a problem. But, in the case of pi (and other places), it tends to obscure things.
SO: Take anything that can be viewed as a perfect circle: a wedding ring, a manhole cover, a circular pizza pan – you name it. It will always be the case, no matter how large or small, that the distance around that circle (circumference) is a little over 3 times the distance across the circle (diameter). That is, their ratio is a little more than 3 to 1. This has been known since before the time of Christ.
So this particular ratio [of circumference to diameter] is approximately 3/1 then, but not quite. (Though the ancient Babylonians used that approximation – see Pi Triva Quiz ). The ratio is almost 22/7, but not quite. And, it is nearly 3.14 -or even 3.1416*, but not quite.
For centuries, math types struggled to find the exact value of this ratio. (Archimedes, who lived over 200 years before Christ*, proved it was between 22/7 [3.1428 . .] and 223/71 [3.1408 . .] – a remarkable feat!). In the 1700s, it gradually became popular and standard to use (as a shortcut) the Greek letter π (pi) to denote/name the ratio, and thus make things easier (the value of notation!). So, when one hears ‘pi is almost 22/7, but not exactly’, they mean (whether or not they realize it) the equivalent ‘ratio’ sentence in the paragraph above.
Finally -also in the 18th century- Johann Lambert proved that the ratio we call pi will NEVER be able to be exactly calculated or represented. (In math terms, pi is irrational.) Thus you find more and more precise values being calculated (see Pi Trivia Quiz) with new records all the time, though we know it will never be exact.
By the way, using the vocabulary and symbol above, you can now see where the ‘mysterious’ formula for the circumference (C) of a circle comes from! The equation – or formula – so many have memorized, namely C = π * d means precisely the same as ‘the circumference of any circle is a little more than 3 times the diameter.’ If you know one, you can find the other.
Isn’t mathematics wonderful? 🙂