 Hi and welcome to the session. Today we will learn about using brackets. Let us learn how we can write expressions using brackets. Suppose we want to write the expression for 4 multiplied by the sum of say 9. So we will write 4 multiplied by the sum of 6 and 9. We will put 6 plus 9 in the bracket. Now first of all turn everything inside the bracket into a single number. That is this will be equal to 4 into 6 plus 9 will be equal to 15. After this do the operation outside which in this case is to multiply by 4. So 4 into 15 will be equal to 60. Now let's see how we can use brackets to follow the procedure systematically. Suppose we want to multiply 102 by 106. So here we can write 102 as 100 plus 2 and we will write it in bracket. Again 106 can be written as 100 plus 6. We will also write it in bracket. Now we can write it as 100 plus 2 within bracket into 100 plus 100 plus 2 within bracket into 6. Further on expanding this bracket we will get 100 into 100 plus 2 into 100 plus Now we will expand this bracket and we will get 100 into 6 plus 2 into 6. So this will be equal to 10,000 plus 200 plus 600 plus and this gives us 10,812. Now our next topic is Roman numerals. In Roman numerals we express 1 like this that is capital I. We express 2 like this, we make capital I twice, 3 like this, this is repeated 3 times. Now we express 4 as this, capital I then V, capital V 5 like this. Now 6 like this, 7 then 9 and 10 like this. Remember few Roman numerals and then you can write any number in Roman numerals using these. One is expressed like this that is capital I then comes 5. 5 is represented by capital V 10 is represented by capital X then 50 is represented by capital N 100 is represented by capital C then 5 on height is represented by capital D and 1000 is represented by capital M. Now there are few rules for the numeral system, first is if a symbol is repeated its value is added that is if we are given this that is capital X is repeated 2 times and capital X represents 10 so we will add the value of capital X 2 times that is 10 plus 10 gives us 20. Now second rule is a symbol is not repeated more than 3 times also the symbols capital V capital N written D are never repeated. Now third rule is if a symbol of smaller value is written to the right of a symbol of greater value then its value gets added to the value of greater symbol. For example in this N represents 50 and X represents 10 so the symbol of a smaller value that is X is written to the right of the symbol of a greater value that is L. So now we will add the value of X to the value of L that is we will add 10 to 50 and we will get 60 and now the fourth rule is if a symbol of a smaller value is written to the left of a symbol of greater value then its value is subtracted from the value of the greater symbol. For example X represents 10 and N represents 50 so the symbol of a smaller value that is X is written to the left of a symbol of greater value that is L. So here the value of X will be subtracted from the value of L that is we will subtract 10 from 50 and we will get 40 and as a fifth rule we have the symbols V, N and D are never written to the left of a symbol greater value that is symbol V, L and D are never subtracted. Also the symbol can be subtracted from V and X only and lastly the symbol X can be subtracted from L only. Now let us try to write 78 in Roman numerals. Now 78 can be written as 70 plus 8. Now we don't have any symbol for 70 but we do have symbols for 50 and 10 so 70 can be written as 50 plus 10 plus 10 plus 8. Now this we can represent as NXX plus 8 will be represented by this so this will be equal to this. So this represents 78. Now this represents 96 in Roman numerals 90 plus 6 which will be equal to 100 minus 10 plus 6. Now 100 minus 10 will be represented by is written on the left of C so we subtract 10 from 100 and we get 90 plus so this becomes with this we finish this session hope you must have understood all the concepts. Goodbye take care and have a nice day.