Numbers Explained - Mersenne Primes, Perfect Numbers

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Mersenne Primes
French monk and math enthusiast Marin Mersenne (1588-1648) was playing around with powers of two and found an interesting possible source of prime numbers from them. The primes that appear from his method are today called Mersenne Primes. In this video we play with them a bit and see when they can occur. Perfect Numbers
The numbers 6, 28, 496 and 8128, for example, are "perfect" in the sense that if you sum of all their factors (except for the whole number itself) you get the original number. For example, 28 has factors 1, 2, 4, 7, 14 and 28 and, ignoring the last one, 1+2+4+7+14 does equal 28!
The Greeks noticed that each of these numbers is triangular: 6 is the 3rd triangle number, 28 is the 7th triangle number, 496 is the 31st triangle number, 8128 is the 127th triangle number, and so on, and the numbers 3, 7, 31, 127 are each primes one less than a power of two ("Mersenne Primes").
It wasn't until some 2000 years later that Euler was able to prove that every (even) perfect number must have this form. We prove it too in this video. Finger Multiplication Trick
Here's a cute trick for computing values up to your ten times tables. If you want to get up to twenty times tables as well ... use both fingers and toes! Square Roots - and Addition
Why won't square roots "mix" with addition? Surely sqrt(a+b) should equal sqrt(a) + sqrt(b)? In this video we explore this issue and prove, with a simple diagram, that sqrt(a)+sqrt(b) is always sure to be larger than sqrt(a+b).

The Truth About "Rationalizing the Denominator"
There is nothing mathematically wrong with writing square roots in the denominator of a fraction. So why do most people insist that all square roots be moved to the top?