Numerical properties of ten

Ten is one of only five numbers that are both triangular and tetrahedral. The largest of those numbers is 7140 (see that entry for more).

Ten is also a semiprime, since it's the product of two primes (2*5).

It is the sum of the primes 2 through 5 (2+3+5) and also the product of 2 and 5, a property further discussed at the entry for 39.

It is not known if there are any numbers with the sane abundancy (ratio of sum of divisors to original number) as ten - this is an unusual lack of knowledge for such a small number. Number that have the same abundancy as another number are called friendly numbers and numbers that don't have the same abundancy as another number are solitary numbers and therefore ten is the smallest number that is not known whether or not it is friendly or solitary. 

Ten as a base

Ten is important to humanity the base of our number system and therefore somewhat has another sense of completeness (see also 4). Ten is our numerical base because ten is directly a part of the human body, as the number of fingers (and toes) on a typical human being (see photo on right). Some ancient numerical systems used different bases, but those were most often five or a multiple of ten, still showing the significance of ten. If we had seven fingers per hand, seven would be what five is now and five would only have an advantage because of its small size.

Ten is also the last number whose name doesn't really seem to mean anything - this is true for all numbers one through ten. The names five, nine, and ten are perhaps connected to the words "hand", "new", and "two hands" according to Wiktionary, but these connections are loose and very shaky.

Ten as a large number

Ten may seem small, but it’s big enough to be considered a large amount in many situations. For example, look at this:

o o o o o o o o o o

Looks like quite an amount of o's, doesn’t it? Anyway, here’s some examples to show that ten really is a (borderline) large number. Eating 3 or 4 pancakes, maybe 5, would be normal, but if you try to eat 10 pancakes you’d be very full. Having 1, 2, or 3 kids is normal, but 10 kids is outrageous. You might invite a friend over for something, maybe two or three or possibly four, but inviting 10 friends over is a lot of people, and some serious shit probably must be going on if you need to invite ten people over. A series of two or three movies is common, four or five is fairly long, but a series of 10 movies is unusually long. Just imagine a movie like The Amazing Spider-Man 10! For a last example, imagine a man ordering 10 burgers at McDonald’s. It’s laughable thinking someone would need this many McDonald’s burgers at once, unless they’re inviting their extended family over, and even then, wouldn’t you get some bigger, juicier burgers to feed their hungry bellies?

Despite all these examples, ten is so common in daily life that we rarely stop to think about ten when we hear ten, and usually forget that it’s even a large number!

Surely there was a time when man only worked with twos, thees and fours, and a number like the count of all his fingers seemed impressive in comparison. But as humans became more advanced, a need for larger numbers grew, and ten lost the sense of impressiveness it once had. Now in our modern world, everyday numbers go up to somewhere around a quintillion (10^18) - far more than people in prehistoric or even renaissance times could begin to imagine!

There are still some modern situations where 10 is small though. For example, a book with 10 pages is a very short book, and a class with ten people is a small class. Therefore, it’s best to describe ten as “borderline large”.

Ten in culture and googology

Ten has much cultural significance that stems from being the base of our numeral system. For example, a scale of 1 to 10 is often used to rate things, and the killing of one in ten soldiers was a common punishment in ancient Rome. There is also a 1977 documentary film called Powers of Ten that zooms out and then in through our universe to show the amazing scales of our world.

Ten is by far the most common base number for googolisms, followed by either 100 or 3. This stems partly from a googol, the "original googolism", being a power of ten, but mainly from ten being (surprise surprise) our base.

The prefix for 10 is deca-, both Latin and Greek. This prefix is used in words like decimal, decimate, decade, decagon, etc. Deca- (sometimes spelled deka-) is even an SI prefix meaning 10, and its reciprocal is deci-, meaning one tenth.