 Hello and welcome to the session. In this session we will discuss square numbers. First let's see what is a square number if a natural number say m can be expressed as n square where this n is also a natural number then this m is a square number. Like for example the number 36 can be written as 6 into 6 that is 6 square so this 36 is a square number. These square numbers are also called perfect squares that is we can say that the number 36 is a perfect square. Now we shall discuss some properties of square numbers. The first property says that all square numbers end with 0, 1, 4, 5, 6 or 9 at units place. Like for example if you consider the number 169 it has 9 at the units place so we say that 169 is a square number moreover 169 is written as 13 square. Then we have if a number 1 or 9 in the units place then its square ends in 1. For example let's consider the number 11 now 11 square as you know is 121 so the number 11 has 1 in the units place and thus its square also ends in 1 that is its square also has 1 in the units place. Then we have if a number has 4 or 6 in the units place then its square ends in 6. Consider the number 14 now 14 square is equal to 196 so in the number 14 its units places 4 and the square of 14 ends in 6 that is its units places 6. Next we say the square numbers can only have even number of zeros at the end. Now like 400 is a square number it has two zeros that is it has even number of zeros at the end we have that 400 is equal to 20 square. We have one more thing if a number has 2 or 8 in the units place then its square ends in 4. Let's consider the number 12 now 12 square is 144 so this number 12 has 2 in its units place and its square has 4 in its units place. Then again if a number has 3 or 7 in the units place then its square ends in 9. For example consider number 13 it has 3 in its units place now 13 square is 169 so its square as you can see has 9 in its units place. And if a number has 5 in the units place then its square ends in 5. Consider the number 15 it has 5 in its units place now 15 square is 225 so as you can see its square also contains 5 in the units place. Now let's discuss some more interesting patterns first let's discuss adding triangular numbers. We know that triangular numbers are those whose dot patterns can be arranged as triangles like the number 1 is a triangular number as its dot pattern can be arranged as triangles. Now if you consider number 3 its dot patterns can also be arranged as triangles in this way. Now if we combine two consecutive triangular numbers we get a square number like let's consider the triangular numbers 1 and 3 itself. When we combine these two we get this pattern that is 1 plus 3 now this is equal to 4 which is equal to 2 square. On combining two consecutive triangular numbers we get a square number. Next we have numbers between square numbers we say that there are two n non-perfect square numbers between the squares of the numbers n and n plus 1. Like if you consider the numbers 3 and 4 now between the squares of these numbers they would lie 2 into 3 that is 6 non-perfect square numbers. Next we discuss adding odd numbers we have that the sum of first n odd natural numbers is n square. Now we can also say if a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1 then it is not a perfect square. So using this result we can easily find out whether a given number is a perfect square or not. Like let's consider the number 49 let's check if this is a perfect square or not. Now 49 can be expressed as 1 plus 3 plus 5 plus 7 plus 9 plus 11 plus 13 that is it is expressed as a sum of successive odd natural numbers starting with 1. So we say that 49 is a perfect square. Next we have a sum of consecutive natural numbers we can express the square of any odd number as the sum of two consecutive positive integers. Like 3 square which is 9 can be written as 4 plus 5. So the square of 3 is written as the sum of two consecutive positive integers 4 and 5. Next we have product of two consecutive even or odd natural numbers. We have that a plus 1 multiplied by a minus 1 is equal to a square minus 1. This is how we write the product of two consecutive even or odd natural numbers. Next is sum more patterns in square numbers. Observe the squares of the numbers 1, 11 and 111 they give us a very beautiful pattern which goes on. Observe this pattern in this you can see that in the first row we have 7 square equal to 49. Then in the next we have 67 square is equal to 49 that is 4 and 9 and we have inserted 4 and 8 between 4 and 9. Since we have 1, 6 before 7 so we have taken just 1, 4 and 1, 8. Now in the next row as you can see that we have 2, 6 before 7 so we have inserted 2, 4 and 2, 8 between the numbers 4 and 9. In this way this pattern goes on. Next we discuss finding the square of a number. We can easily find out the square of the small numbers but when we need to find the square of big numbers then we use some technique. Like if you consider the number 32 we need to find the square of 32. Now this 32 can be written as 30 plus 2 then we have 32 square would be equal to 30 plus 2 the whole square. Now this further is written as 30 into 30 plus 2 plus 2 into 30 plus 2. Now this gives us 30 square plus 30 into 2 plus 2 into 30 plus 2 square. Which further is equal to 30 square would be 900 plus 30 into 2 is 60 plus 2 into 30 is 60 plus 4 and this would be equal to 1024. So we have 32 square is equal to 1024. Now let's consider a number with unit digit 5 that is let the number be a5. Now we need to find its square so we have a5 square is equal to 10a plus 5 the whole square. Now this could be further written as 10a into 10a plus 5 plus 5 into 10a plus 5 which further is equal to 10a into 10a which is 100a square plus 10a into 5 is 50a plus 5 into 10a is 50a plus 5 into 5 is 25. Now this is 100a square plus 100a plus 25 that is we write this as 100a into a plus 1 plus 25 that is we say that a5 square is equal to a into a plus 1 100 plus 25. Consider the number 35 it has 5 units in its place let's try and find out the square of 35. Now when you compare this number with a5 you see that a would be equal to 3 and we know that a5 square is equal to a into a plus 1 100 plus 25. So we have 35 square would be equal to 3 into 3 plus 100 plus 25 which is equal to 1200 plus 25 which is equal to 1225 so we get 35 square is equal to 1225. Next we have Pythagorean triplets let's see what is a Pythagorean triplet. Now for any natural number m greater than 1 we have 2n square plus n square minus 1 the whole square equal to n square plus 1 the whole square. So we say that 2m then m square minus 1 and m square plus 1 these 3 numbers form a Pythagorean triplet. We need to write a Pythagorean triplet whose 1 member is given as 6 we consider 2m to be equal to 6. So from here we get m is equal to 3. Now the other members of the triplet are given by m square minus 1 that is 3 square minus 1 which is equal to 8 and also n square plus 1 which is equal to 3 square plus 1 and that is equal to 10. And according to the Pythagorean triplet we have 2n square that is 6 square plus n square minus 1 the whole square that is 8 square is equal to n square plus 1 the whole square that is 10 square. Now if you want to check here you can check this also 6 square is 36 plus 8 square is 64 and this is equal to 10 square that is 100. Now 36 plus 64 is 100 is equal to 100 which is true hence we say that 6, 8 and 10 form a Pythagorean triplet. This completes the session hope you have understood the concept of square numbers.