> 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 = 12 + 32 = (12 + 12) (12 + 22)
> 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 = (12 + 12) (12 + 12) = 22 + 0.
The general formula is illustrated below:
if n = (a2 + b2), m = c2 + d2, 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 10th power.
The last digit of a 10th power of an integer is the same as the last digit of the square.
For example: 210 = 1024 and 22 = 4;
For another one, 710 = 282475249, which has the last digit 9, whereas it’s known that 72 = 49 too.
In this post we study “5” again – but with reference specifically to Graphs and Shapes, to which number 5 illustrate many interesting properties.
> Do you know complete graph? Mark any n points, then connect each pair of points, you will have a complete graph. Here is a graph with 4 points.
It’s not necessary to connect all points with straight lines: you can connect them with an arc or a curve – like in the figure shown in this paragraph. What do you note: no connecting line intersect with any other lines. Such a graph is called a planar graph.
What about a complete graph with 5 points? We draw it as a pentagon plus a 5-star (also called pentagram). No matter how you try it, some two lines of all connecting lines must intersect somewhere. There is no way to avoid or escape from that.
Put it simple:
A complete graph with five points is not planar.
(if you want avoid crossing each other in connecting lines), you have to raise one point so that it does not share the same plane with the other four points.
> An equilateral triangle is called a golden triangle if
(I) the ratio of its base to (any of) its sides is 1.618 .. — called golden triangle of type I; [that is: BC : AB = 1.618.. ]
(II) the ratio of (any of) its sides to base is 1.618 .. — called golden triangle of type II.
The number 1.618 .. is called a golden ratio.
> What’s so special about the shapes with golden ratio? It’s visually pleasing to the eye – shown by some experiments. Not only in triangles. In a rectangle, if the ratio of longer side to shorter side is 1.618 .. It also looks pleasing to a human eye.
> In the title we mentioned pentagon-pentagram. A regular pentagon is a polygon with 5 sides, with all the sides being equal and all the interior angles of this pentagon being equal. A pentagram is constructed by connecting straight lines between the vertices — usually being referred to as the (regular) five-star.
In the diagram of regular pentagon-pentagram, we can observe many of these golden ratios.
There are many many golden triangles.
Can you find some of them?
This is the first puzzle we have posted. can you find?
Let us end this post with a verse in Christmas’s popular song,
“In the fifth day of Christmas,
my true love gives me
FIVE GOLDEN RINGS .. .. ..
How about the five golden rings all of which contains the golden ratio?
> 5 is a factor of 10, which links to our decimal system (carry one to the next higher place whenever 10 is reached).
All multiples of 5 ends in digit 0 or 5.
> The square of any integer ending in 5 will also have 5 as the last digit (units place).
–> We can even go one step further: for the square of any integer ending in “5”, the last two digits of the results must be 25. For example: 25 × 25 = 625, and 45 × 45 = 2025.
> When a fraction m/n (both m,n are integers) is converted to a decimal, then:
If n = 5 => the decimal has only one digit after the decimal point
If m/n, after converting to decimals, has only one digit after the decimal point, then n = 2 or n = 5
> 5 is a prime number: 5 = 1 × 5
5 = 221 +1 (this type of numbers are called Fermat numbers)
The next Format number is 17 = 222 +1.
> [this item is a bit hard] The numbers in Fermat’s number sequence increases very rapidly: as
221 +1, 222 +1, 223 +1, 224 +1, … …
We calculate these numbers as:
5, 17, 257, 65537, ..
So the number increase very rapidly. Interesting enough, the first four prime numbers are all prime numbers, yet the fifth one (225 +1) is a composite number, with the smaller divisor being 641.
And once started, it keeps going this way: it has been known that from the fifth Fermat number to the thirty-second Fermat number (which is 2232 +1), all of them are composite numbers. It remains open whether there is any prime number in this sequence down the way.
There are lots of facts regarding prime 5: let’s be simple not to mention everything.
One is the beginning of everything.
