## “10”: The Base of Daily Decimal

> 10 is the base of our decimal numeric system — it means 10 ones is ten, 10 tens is hundred, 10 hundreds is thousand, etc. We use decimal system on a daily basis.

> Digit “0” was invented in Oriental world but it took a long time to be accepted in the Western hemisphere. For number 10, people initially use a (blank) bar or a circle symbol as there’s nothing in units-place; then it has become a standard to use ‘0’ once ‘0’ emerged. Roman numerals, which did not embrace the idea of “place value”, were used until recent (where we use X, C etc for 10, 100, .. — there never has been there a zero in Roman numeral).

> 10 is a triangle number as well as a centric triangular number.

10 = 1 + 2 + 3 + 4 = 1 + 3 + 6

As shown, 10 is the sum of the first four counting numbers, as well as the sum of the first three triangle numbers [1,3,6 are triangle numbers].

> If a number has exactly two prime divisors, then it is called a semi-prime. Number 10 is a semi-prime as

10 = 2 × 5

The first semi-prime is 6. Any semi-prime will have 4 divisors (including number 1 and itself).

> 10 can be represented as the sum-of-squares as below:

10 = 1

^{2}+ 3^{2}= (1^{2}+ 1^{2}) (1^{2}+ 2^{2})

> There is something interesting in the entry just above. If two numbers, n and m can be both written as the sum of two squares, then their product can also be written as the sum of two squares. 10 is the first of such examples, apart from the trivial 4 = (1^{2} + 1^{2}) (1^{2} + 1^{2}) = 2^{2} + 0.

The general formula is illustrated below:

if n = (a

^{2}+ b^{2}), m = c^{2}+ d^{2}, then

nm = (ac-bd)^{2}+ (ad+bc)^{2}

Note that the representation of (nm )as the sum of two squares is usually NOT unique.

> Ancient Greeks has suggested to use the square root of 10 as an approximate value of *π* (where of a circle, *π* is the ratio of circumference to radius).

> In the metric system, the conversion among different units of measurement are based on the power of 10; for example: 1 metre = 100 centimetre; 1 kilometre (km) = 1000 metre. In monetary system, 1$ = 100 c and 1 dime = 10c (though “cent” has been out of use).

In early history, as a fast calculation mechanism, people resort to logarithms; of which the most commonly used was 10 (therefore base-10 logarithm obtained its name as the common logarithm).

> Considered that a number is raised to the 10^{th} power.

The last digit of a 10

^{th}power of an integer is the same as the last digit of the square.

For example: 2^{10} = 1024 and 2^{2} = 4;

For another one, 7^{10} = 282475249, which has the last digit 9, whereas it’s known that 7^{2} = 49 too.