How to find the roots, squares and reciprocals of numbers using a compass and a ruler

Imagine that you do not have a calculator at hand (but you have a compass and a ruler or square) and you need to calculate the result as a line segment. The problem is solved in less than 5 easy steps.





Basic calculation formula

To begin with, we will prove one formula that will help us with a further solution.





In a right-angled triangle, ABC



draw the height h



to the side C



. By the Pythagorean theorem, we deduce:





C ^ 2 = A ^ 2 + B ^ 2 A ^ 2 = h ^ 2 + a ^ 2 B ^ 2 = h ^ 2 + b ^ 2 C = a + b

We substitute everything in the first formula:





(a + b) ^ 2 = h ^ 2 + a ^ 2 + h ^ 2 + b ^ 2

And if you expand the brackets:





a ^ 2 + 2ab + b ^ 2 = 2h ^ 2 + a ^ 2 + b ^ 2

After reduction we get:





ab = h ^ 2

.





, 1



. 1



, 10



, 100



4



. 4 . , - . - , .





:





  • , .





  • .





  • . , 4 .





X



:





a = 1, h = X, b = X ^ 2

.





.





1



X



.





1



X



.





1



. . , .





. - , X



, . . X



. ( 1



). , , .





:





a = 1, b = X, h = sqrt (X)

.





1.







X



.





1+X



O



. , , . R



.





O



, R



.





1 . X



.





:





h = 1, a = X, b = 1 / X

, a



h



.





.





X



.





X



1



.





.





X



1



. .





, , .





X



1



, . , . , 20, 16 (4 ), 4 (4 ).








All Articles