Square, can be said to be the most decent, unremarkable graphics. Almost everyone knows the shape of a square, and they also know how the square area and perimeter are calculated. But do you know how many magical digital secrets are hidden by this unremarkable square? Today we will take a look at what secrets are hidden in the square.
First, let's take a look at the pictures of the squares below.
We found that the above squares are composed of small squares, and the number of small squares is square, which is the square of 1, the square of 2, the square of 3, and the square of 4.
Next, let's make a simple change, making a change to the color of the square.
Another expression of a square number
By changing the color of the small square, we find that the number of small squares can be written in the form of 1+3+5+7...+n (n is odd).
By comparison we find that the square of 1 + 3 = 2 1 + 3 + 5 = 3 square of 1 + 3 + 5 + 7 = 4 square
Through the law, we find the square of 1+3+5+7...+n=(1+n)/2.
Applying this law, we can find the sum of all odd numbers up to 100:
1+3+5+7+……+97+99=50^2=2500
Then all the sums of even numbers within 100 can also be obtained:
2+4+6+8+……+98+100=（1+3+5+7+……+97+99）+50=2550
So we will also calculate 1+2+3+...+99+100=5050.
This calculation method is very simple after we learn the equal difference series, but we do not use the equal difference series now, and we can calculate it through the law of squares.
Next, the wonderful thing will happen again, and we will make another change in the color of the small square:
Another expression of a square number
We find that the number of squares can be calculated in a different way, can be written in the form of 1 + 2 + 3 + 4 + 3 + 2 + 1, then how much is this equation equal, obviously, equal to the square of 4.
So 1+2+3+4+... 99+100+99+...... 4 + 3 + 2 + 1 is equal to how much, through the law, we find that the middle number is how much, is equal to how much square, so this equation is equal to 100 square 10000, is not very magical.
Above, we discovered two important digital secrets by piecing small squares into large squares, and then, let's look at another magical secret in a different way.
Let's start with a topic:
1/2+1/4+1/8+1/16=?
1/2+1/4+1/8+1/16+……+1/128=？
For all one equation, we may not have a big problem directly, but the number of children of the second equation has increased, and it is troublesome to calculate, is there a simple way?
We find the answer by breaking down the squares.
We continuously divided the square with an area of 1 into 2 equal parts and found that:
1/2=1-1/2
1/2+1/4=1-1/4
1/2+1/4+1/8=1-1/8
1/2+1/4+1/8=1/16=1-1/16
Squares can be infinitely divided, and this law can be carried out indefinitely. So we can figure out the answer to the above question in a second.
1/2+1/4+1/8+1/16+……+1/128=1-1/128=127/128.
Isn't that amazing?
There are number laws in the figures, which is the magic of the combination of numbers and shapes. What are the secrets of the combination of numbers and shapes, welcome to share them in the comments section.