It seems trivial but .. why the high numbers at the second – such as 4, which is 22 – They are called squares (or Perfect squares when it comes to whole numbers)? The answer must be sought in the “form” of these objects. In fact, the numbers can be represented graphically and in the case of natural numbers, for example, we can use points: • It is 1, •• is 2, ••• is 3 and so on. Here, the square numbers are so called precisely because they can be designed like gods squares on the side equal to your root! And these squares are called perfect squares when they are the square election of an integer.
And there are not only square numbers, but also rectangular, triangular and even hexagonal!
We see it in this article, together with some numerical curiosities that the figured numbers manage to show … Drawing!
Why are the second numbers called squares?
To many it may seem a trivial question, but many are not able to find an answer on two feet: because i numbers high second they are called squares? Well yes, the shape of the square has to do with it and now we will understand why. First of all we remember that any number if elevated to the second is said square, while i Perfect squares are the election to the second of the only whole numbers, that is, of those numbers that do not present the comma.
THE numbers they help us count the objects, in fact they are something abstract, of the symbols that indicate precisely the amount of elements that we want to count the fact that they indicate one amount, However, it suggests that the numbers themselves can be represented through gods objects, as for example of Points! This is how we learned them when we went to elementary school.
Here, if we try to draw i Numbers to the second with gods dots – that is, the square numbers – we will realize that they are precisely .. of Square! Each perfect square in fact, that is, a square obtained by raising an entire number to the second, can be designed as a set of equidistant points willing to form a geometric square with the side equal to its root. For example, 4 (22) is a square on the side 2, 9 (32) is a square on the side 3, 16 (42) is a square on the side 4 and so on.
Not only squares: what triangular and rectangular numbers are
What we have seen are the so -called figured numbers, that is natural numbers that can be represented through one regular geometric scheme Of points. And there are not only squares! But also i triangles, the rectangles, even the Fees!
All natural numbers are rectangles
As for rectangles, it must be said that any natural number n It can be represented as a rectangle Of sides equal to 1 And n. There are numbers that can be represented as more rectangles different ways: In fact, it depends on your number of dividers. If we take for example the number 6can be represented as a ran of 6 points or 2 files from three points. And we can also note that a number can have More figurations: The four, for example, is both rectangular (1 × 4) that square (2 × 2).
Square numbers are sums of consecutive triangular numbers
Then there are i triangular numbers which, as can be seen in the figure below, can be obtained by providing regularly the points to form one “pyramid”. We can see that triangular numbers are obtained by adding natural numbers in succession starting from 1: 1, 3 (1+2), 6 (1+2+3), 10 (1+2+3+4) and so on.
There is one particularity which makes triangular numbers fascinating: when We add two consecutive triangular numbers, we always get a square number. For example, if we add 10 and 15, we get 25 which is just 5 to the second. This property can be sensed graphically, as we see in the figure below.
The figured numbers help to discover mathematical truths
Draw Numbers are not a simple fun, but it also helps us to discover interesting Mathematical intuitions. For example, if we draw i odd numbers one next to the other as rectangles, we can easily notice that adding the odd numbers consecutive starting from 1, we get Always a square number As we can see in the figure below. And it’s true!
There sum first n odd numbers it is always the same as a square, and specifically it is the same just a n2.
