Order of Operations and the Use of Brackets

Order of Operations and

 the Use of Brackets

You have learned to carry out the operations of addition, subtraction, multiplication, and division. Let us study these operations further. Consider the following examples :

Ex (1) Simplify : 75 + 25 x 10

Addition and multiplication are the two operations in this example.

If we carry out the addition 75 + 25 first, the value of this expression will be 100 x 10 = 1000

If we carry out the multiplication 25 x 10 first, then the value of the expression w

ill be 75 + 250 = 325.

We shall have to say that the same expression has two different values. It is obvious that this will cause confusion.

Use of Brackets

It is necessary to know which operation should be carried out first in order that the expression yields a single value. Brackets are used for this purpose. The operation to be carried out first is written inside brackets.

             

For example, if the addition is to be carried out first in the expression 75 + 25 x 10, then the expression is written as                   (75 + 25) x 10.

 If the expression is 75 +(25x10), multiplication will be carried out first.

The value of the expression (75 + 25 ) x 10 is 1000.

And the value of 75 + (25 x 10 ) is 325.

Similarly, ( 48 ÷ 8 ) ÷ 2 = 3 and 48 ÷ ( 8 ÷ 2 ) = 12.

The following rules are generally

 followed to minimize the use of 

brackets in an expression.

(1)  Operations in brackets should be carried out first.

(2)  After that, multiplication and division must be carried out in the order in which they occur from left to right.

(3)  Then, addition and subtraction should be carried out in the order in which they occur from left to right.

(4) If there are more than one operations inside a bracket, they are carried out first, following rules 2 and 3.

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Operations on Numbers : Multiplication and division Multiplication

Operations on Numbers : Multiplication and division

Multiplication

To multiply numbers having upto five digits by a three-digit number.

          

Study the following multiplication.

Ex. 1. 879 × 248

In the multiplication 879 × 248 = 217992; 879 is the multiplicand, 248 is the multiplier and 217992 is the product.

1.    A pump draws 1240 litres of water per minute. How much water will be drawn in 3 hours?

place your answers by comments:

Operations on Numbers : Addition and Subtraction

READ THE NUMBERS

FIRST WATCH THE READ  NUMBERS VIDEOS AND THAN FOLLOW THE OTHER LESSONS.

SUBTRACT THE NUMBERS

Operations on Numbers : Addition and Subtraction

Addition

          

We have learnt how to add numbers having up to five digits. We can add numbers having more than five digits using the same method. Study the following examples.

Ex. 1. Add                                                 

658003 + 804017                                    

	TL	L	TTH	TH	H	T	U
+		6 8	5 0	8 4	0 0	0 1	3 7
	1		1			1
	1	4	6	2	0	2	0

Ex. 2. Add

890067 + 9989744

	1	1	1			1	1
	C	TL	L	TH	TH	H	T	U
			8	9	0	0	6	7
+		9	9	8	9	7	4	4
	1	0	8	7	9	8	1	1

Make a habit of keeping the carried over number in your mind rather than writing it down.

Numbers

1.    Numbers

The numbers we use for counting objects are called counting numbers.These numbers have names like one,two,three,and so       on. In English,they are written using numberls such as 1,2,3, -----These numberls are known as International numberls.The numberls in the Devanagari script are 1,2,3,and so on.

Numbers greater then ten lakh : reading and writing

•           In the previous standard, we learnt how to write numbers
• upto the lakhs place. 
• 
  
• The smallest six-digit number is 1,00,000 and 9,99,999 (nine lakh ninety-nine thousand nine hundred and ninety-nine) is the largest. 
• 
  
• Whenwe add I to this number,we get the next number, which is 10,00,000 or ten lakh. Ten lakh is also called a million.
•          
•  Ten lakh is the smallest seven-digit number and 99,99,999 (ninety-nine lakh ninety-nine thousand nine hundred and ninety-nine) is the largest seven-digit number. 
• 
  
• When we add I to it, we get the eight-digit number 1,00,00,000.

          C       TL      L        TTH   TH     H       T        U

          1       0       0       0       0       0       0       0

In this number, the place to the left of ten lakhs is called ‘crores’.One crore is the smallest eight-digit number.

          

The place to the left of the crores place is that of ten crores. Thus, one ten-crores and 5 crore, that is, 15 crore is written as 15,00,00,000.

          

Now, let us take the example of 42,35,78,959. The group of TC (ten crores) and C is together read as forty-two crore;

the group of TL and L is read as thirty-five lakh; the group of TTH and TH is read as seventy-eight thousand and the group of H, T and U is read as nine hundred and fifty-nine. 

Hence, the number is read and written in words as ‘ forty-two crore,thirty-five lakh seventy-eight thousand nine hundred and fifty-nine’.

         

If we add I to 99,99,99,999; that is ,to the largest nine-digit number, we get the next number 1000000000  which is the smallest ten-digit number. It is read as one hundred crore. It is also called a billion.

Expansion of (x + a) (x +b)

Expansion of (x + a) (x +b)

          Which two quantities are to be multiplied ? (x +a) and (x + b). which is the equal term in (x + a) (x + b) ?      ……….. x.

          Which are the unequal terms in the quantities (x + a) and                        (x + b) ?                                       ……. a and b

          Study the following multiplication shown in horizontal arrangement.

(x + a) (x + b)       = x × (x + b) + a × (x + b)

                             = x² + xb + ax + ab

                             = x² + ax + bx + ab

                             = x² + (a + b)x + ab

Which are the terms in the product ?    ……x², (a+b)x and ab

What is the characteristic of the first term ?

………… square of the equal term.

What is the characteristic of the second term ?

…… Sum of the two unequal terms multiplied by the equal term.

What is the characteristic of the third term ?

……..product of the unequal terms.

This can be written in words as follows.

(equal term + first unequal term) (equal term + second unequal term)

= (equal term)² + (sum of unequal terms) × (equal term) + (product of unequal terms)

This gives us the following formula.

(x+a) (x+b) = x² + (a+b)x  + ab

Irrational and Real Numbers

Irrational and Real Numbers

Revision : Rational numbers

         

 The numerators in the numbers , , ,  are integers while their denominators are non-zero integers. Hence, these are rational numbers. If p is any integer and q, any non-zero integer, then,  is a rational number

The decimal form of a rational number

          

Two rational numbers can have different decimal forms. To understand the nature of this difference, let us work out the decimal form of two rational numbers  and , and study them. What is the difference in the decimal forms ?

(1)  The decimal form of .

The decimal form of  is 7.9375

As the remainder in this example is 0, the process of division is complete. Hence, we do not get any digit after 5 in the quotient obtained.

          It means that the decimal form 7.9375 is a terminating one.