There is a way to calculate the perfect squares in mind without Make even one multiplication or a power, but only with sums and subtractions in a very fast way.
For example, calculating the square of 31 in mind, it might seem impossible, yet with this “magic” we can do it in a few seconds: the square of 30 is 900, and this is known. Now just add up first 30 – the previous number – and then 31 – the number itself – and 961 is obtained which is the square of 31.
How to calculate the perfect squares quickly without making multiplications
Perfect squares can be calculated quickly thanks to a Mathematical trick which requires only one account. This rule teaches us that The squares are all “concatenates” between them and we can go from to each other through a simple sum.
Let’s see the rule through a example easy. Let’s imagine we want to calculate the square of number 3:
- I consider the previous number at 3, that is the 2, and I consider the square 22 = 4
- at this point summary at 4 ours protagonist number3, and the previous numberon 2
- And here is the result 32 = 4 + 3 + 2
And this rule always applies! The square of any number is equal to the square of the previous number, N-1, added to the number itself and the previous number. Written in mathematicianthe rule is this:
In this way All squares can be calculatedone behind the other, without even making a multiplication:
- The square of 4 will be 9 + 3 + 4 = 16
- The square of 5 will be 16 + 4 + 3 = 25
- The square of 6 will be given by 25 + 5 + 6 = 36
Now, as long as we talk about small numbers the trick is of little use: anyone who remembers the tables can say on the fly that the square of 6, “You are for six”, He does 36. But with the biggest numbers, things get complicated and this trick can help us. For example, not everyone knows how to say how much the square of 11 is, but everyone knows that 10 × 10 does 100, so to calculate the square of 11 Just do 100 + 10 + 11 = 121easy no?
In practice, if we remember some squares by heart, or we know how to calculate some on the fly because it is easier, then all the squares immediately following we can calculate them on the fly with this makeup. We, in fact, started with the number 31 Precisely because it is known enough that 900 is the square of 30, or in any case it is easy to calculate (just do 30 × 10 that makes 300 and then multiply by 3), so we can calculate the following squares with our magic makeup:
- 900 + 30 + 31 = 961 which is the square of 31
- 961 + 31 + 32 = 1024 which is the square of 32
- 1024 + 32 + 33 = 1089 which is the square of 33
But it is not over here, the makeup can also be used on the contrary using the subtraction instead of the addition, try it with the square of 29:
- Let’s start from the square of the number we knowfor example the square of 30 And 900.
- Let’s subtract the number we left and its previous onein our case we subtract 30 And 29 to 900 obtaining 900-30-29 = 841 which is precisely the square of 29
You can have fun continuing to calculate the square of 28, 27 and so on.
Why the calculation works: the figured numbers
But why does this trick work? To understand it we can remind us that The squares of the numbers can be seen just like… squares. It is the graphic representation of the numbers, also called figured numbers. Let’s do it with the easiest example, the square of 2 is 4 and can be represented as 4 balls willing to form a 2 × 2 square, as we did in the figure below.
If we add a row with two balls above the square we get a 3 × 2 rectangle consisting of 6 balls. On the other hand, 6, as a number is not a perfect square, but if we add a column with 3 balls to our design, we get a 3 × 3 square with 9 balls, or the square of 3!
The same procedure can be replicated by adding a row of 3 balls first and then a column of 4 balls obtaining a 4 × 4 square, and so on: as you can see it is not about magic, but mathematics!
