8th Class Mathematics Squares and Square Roots Pythagorean Triplet

         Pythagorean Triplet

A Pythagorean triplet consists of three positive integers a, b, and c, such that \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].

Pythagorean theorem states that, in any right triangle, the sum of squares of base and height is equal to the square of its hypotenuse. Pythagorean triplets describe a relation among three sides of a right triangle. For every natural number n > 1, we have the Pythagorean triplet is given by \[(2n,{{n}^{2}}-1,{{n}^{2}}+1)\]    

Let n = 3, then the corresponding Pythagorean triplet is obtained as

\[\text{2n}=\text{2}\times \text{3}=\text{6}\]

\[{{\text{n}}^{2}}-1={{\text{3}}^{2}}-1=8\]

\[{{\text{n}}^{2}}+1={{\text{3}}^{2}}+1=10\]

Hence 6, 8, 10 are Pythagorean triplets.  

            Square of Negative Numbers

Square of a negative number is always positive. Some example of square of negative numbers are given below:

\[{{(-a)}^{2}}=-a\times -a={{a}^{2}}\]  

            Numbers between Square Numbers

In general there are 2n non-perfect square numbers between the squares of any two numbers n and n + 1.

For example between 5 and 6 the number of numbers is \[{{6}^{2}}-{{5}^{2}}=36-25=11\]

Thus there are 10 non square numbers which is one less than the difference of the square of two numbers which is equal to 2n i.e. \[\text{2}\times \text{5}=\text{1}0\].