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. 
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• 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. 

Square roots of numbers

Square roots of numbers

          It is customary not to write ‘+’ sign before a positive number. For example, +10 = 10. 

We shall, henceforth, observe this convention in all discussions.

6 ×  = 36     ∴ 6 is the square root of 36.

Similarly, (-6) × (-6) = 36       ∴ (-6) too is a square root of 36.

          
Thus, 6 and -6 are the two square roots of 36.

         

study the following video lessons:

 In the same way, work out in your mind the pairs of numbers of which 4, 49 and 121 are the squares and write statements like the above.

          What do we learn by observing the numbers and their square roots in the above examples?
we see that –

* Every positive number has two square roots.

* These square roots are opposite numbers of each other.

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Lowest Common Multiple (LCM)

READ THE LESSON.

GO THROUGH EXAMPLE VIDEO AND

 LET US FIND ANSWER THE QUESTION BELOW

POST YOUR ANSWER BY COMMENTS.

Lowest Common Multiple 

(LCM)

Carefully consider the following numbers that are divisible by 8 and by 12.

Numbers divisible by 8 (i. e. multiples of 8) :

8, 16, 24, 32, 40, 48, 56, 64, 72, …

Numbers divisible by 12 (i. e. multiples of 12) :

12, 24, 36, 48, 60, 72, 84, ….

Here the numbers 24, 48, 72, .. are the common multiples of 8 and 12. 24 is the smallest or the lowest of them all. 

Thus, 24 is the smallest or the lowest common multiple or LCM of  8 and 12.

·       Note that the LCM of relative prime numbers is the product of those numbers. 

Examples:

  Find the LCM :

 65, 39

Greatest Common Divisor (GCD)

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GO THROUGH EXAMPLE VIDEO AND

 LET US FIND ANSWER THE QUESTION BELOW

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Greatest Common Divisor (GCD)

let us consider the divisors of 84 and 48.

Divisors of 84 : 1 , 2 , 3 , 4 , 6 , 7 , 12 , 14 , 21 , 28 , 42, 84

Divisors of 48 : 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

We find that the underlined factors 1, 2, 3, 4, 6, 12 are present in both groups. They are called common divisors.

Of these common divisors 1, 2, 3, 4, 6, 12 the biggest divisor is 12.

 It is called the greatest common divisor or the GCD for short. It is also called highest common factor or HCF.

view the examples:

Find the GCD :

 120, 96