Algebraic Identities Chart :

FACTORIZATION :

 Factorization, is breaking down a number into smaller numbers, that on multiplication gives the original number. The smaller numbers are called Factors or Divisors of that number.

Let the number be 12,

the number 12 is written as 2 times 6 = 2*6 , the factors are 2 and 6.

            (or)

 3 times 4 = 3*4 , the factors are 3and 4.

 (or)

 1 times 12 = 1*12, the factors are 1 and 12.

 

THE ALGEBRAIC IDENTITIES:

1. Square of a Binomial Identities:

i) (a + b)² = a² + 2 _* a _* b +  b² = a² + 2 ab + b² = (a + b)( a + b) .

                           (or)

    (- a – b )² = a² + 2 _* a _* b + b^2. = a²  + 2 ab + b²  =  (a + b) ( a + b).

ii) (a – b)² = a² – 2_* a _* b + b²  =  a² – 2ab + b²  = ( a – b ) ( a – b ).  

2. Difference of Squares Identity :

  a² – b² = (a – b )(a + b)

3. Cube of a Binomial Identity:

 (a + b)³  = a³ + 3a²b + 3ab² + b³.

 ( a – b )³ = a³ – 3a²b + 3ab² – b^3.

 

4. Sum of Cubes Identity:

a³ + b³  = (a + b) ( a²   - ab + b² ) 

              = (a + b)³ – 3ab( a + b )

5. Difference of Cubes Identity:

a³ – b³ = ( a – b )(a² + ab + b² ).

            = ( a – b )³ + 3ab ( a – b )

6. Product of Two Binomials Identity:

( x + a)( x + b ) = x² + (a + b) x + ab.

7. Squares of Trinomial Identities :

 i) ( a + b + c )²  = a² + b² + c² + 2 ab + 2 bc + 2 ca .

ii) ( a + b – c )²   = a² + b² + c² + 2 ab  - 2 bc – 2 ca .

iii) ( a – b – c )²  =  a² + b² + c²  - 2 ab + 2 bc – 2 ac.

iv) ( - a + b + c )² =  a² + b² + c² -  2ab  + 2 bc – 2 ac.

v) (  a – b  +  c )²  =  a² + b² + c² -  2ab  - 2 bc + 2 ac.

 

8. Cubes of Trinomial Identity :

 a³ + b³ + c³ = ( a + b + c )( a² + b² + c² – ab – bc – ca ) +  3abc  .

 If  ( a + b + c ) = 0 ; then a³ + b³ + c³ = 3 abc .

Forming Greatest and Smallest Number with Non-Repetitive Digits.

FORMING GREATEST AND SMALLEST NUMBER WITH THE GIVEN DIGITS:

Forming Greatest Number:

In Forming a greatest number with the given digits, we need to check the given digits:

(i) The number of digits given, by which the greatest place value occupied by the greatest digit can be found.

(ii)  The digits are written in Descending order.

The greater digit in the given digits occupies greatest place value ,the next greater digit occupies the next place value and so on.

 

1) Digits : 4, 0 , 8, 7

Answer:

(i) The number of digits = 4. Therefore, the greatest place value is THOUSANDS (Th).

Place value chart:

Thousands (Th)

Hundreds (H)

Tens (T)

Ones ( O)

 (ii) The Descending order of the given digits i.e; 8 > 7 > 4 > 0.

Therefore, the greatest number formed by the digits is : 8740.

 

2) Digits : 0, 9 , 3

Answer:

(i) The number of digits = 3 . Therefore, the greatest place value is HUNDREDS(H).

Place value chart:

Hundreds  (H)

Tens  (T)

Ones (O)

(ii) The Descending order of the given digits: 9 > 3 > 0.

Therefore, the greatest number formed by the digits is 930.

 

3) Digits : 1,3,5,8,6,9

Answer:

(i) The number of digits = 6. Therefore, the greatest place value is Lakhs(L).

Place value chart:

Lakhs    (L)

TenThousands       (TTh)

Thousands       (Th)

Hundreds      (H)

Tens   (T)

Ones   (O)

   

(ii) The Descending order of the given digits : 9 > 8 > 6 > 5 > 3 > 1.

Therefore, the greatest number formed by the digits is  9,86,531.

 

Forming Smallest Number:

In Forming a smallest number with the given digits, we need to check the given digits:

(i) The number of digits given, by which the greatest place value occupied by the smallest digit can be found.

(ii)  The digits are written in Ascending order.

The smallest digit in the given digits occupies greatest place value ,the next smaller digit occupies the next place value and so on.

In forming the smallest number,if Zero is given in the digits , then that digit Zero always occupies ONES(O) place value.

 

1) Digits : 6,2,4,1,0

Answer:

(i) The number of digits = 5 . The greatest place value is TENTHOUSANDS(TTh)

Place value chart:

TenThousands       (TTh)

Thousands      (Th)

Hundreds     (H)

Tens   (T)

Ones   (O)

  

(ii) The Ascending order of the digits: 0 < 1 < 2 < 4 < 6.

Therefore, the Smallest number is : 10246.

In the given digits 6,2,4,1,0 ,Digit Zero is their which is smallest of all the digits .Eventhough it is smallest digit, Zero doesn’t occupy greatest place value. So , we need to check for the next smaller digit i.e ; 1 occupies the greatest place value(TTh), which then followed by digit Zero occupies (Th)  and then the next smaller digit 2, occupies(H) and so on.

 

2) Digits : 8, 9 ,3, 0, 2,7

Answer:

(i) The number of digits = 6. The greatest place value is TENTHOUSANDS (TTh).

Place value chart:

Lakhs    (L)

TenThousands       (TTh)

Thousands     (Th)

Hundreds     (H)

Tens   (T)

Ones   (O)

 

(ii) The Ascending order of the digits = 0 < 2 < 3 < 7< 8 < 9.

Here, also Digit Zero is their which is the smallest of all the digits.The next smaller digit is 2, which occupies the greatest place value Lakhs(L).The next place value TenThousands(Th) is occupied by digit Zero.The next smaller in the remaining digits 8, 9,3,7 is 3 ,occupies Thousands(Th) place and so on.

Therefore, the Smallest number formed: 2,03,789 .

 

3) Digits : 5 , 9 ,4, 8

Answer:

(i) The number of digits = 4.  The greatest place value is THOUSANDS(Th).

Place value chart:

Thousands     (Th)

Hundreds     (H)

Tens   (T)

Ones   (O)

(ii) The Ascending order of the digits: 4 < 5 < 8 < 9.

Therefore, the Smallest number formed  : 4589.

 

The Table shows Greatest and Smallest number with different digits:

S.No	DIGITS	SMALLEST NUMBER	GREATEST NUMBER
1	3,9,2,1	1,239	9,321
2	6,7,8,3	3,678	8,763
3	5,0,3,1,9	10,359	95,310
4	8,0,3,6,4	30,468	86,430

Decimal Number- Successor and Predecessor

DECIMAL NUMBER – SUCCESSOR AND PREDECESSOR:

Successor of Decimal number:

The number that comes just after the given decimal number is called Successor of the Decimal number.

Decimal number Successor is obtained by adding the Decimal number digits place value to the given the given decimal number.

The Successor of the decimal number need not be a Decimal number. Sometimes  we get a Whole number also.If the decimal part digits number are Nine, then the Successor of a Decimal Number is a Whole Number.   

Example:                 

If the Decimal number is 1.9, then Successor of the number = 1.9 + 0.1 = 2.0 

= 2 (whole number).

If the Decimal number is 99.99 , then its Successor = 99.99 + 0.01 = 100.00 

= 100 (whole number).

If the decimal number is 185.999, then its Successor = 185.999 + 0.001 = 186.000 = 186 (whole number)

 

 

1) Successor of 24.8

Answer:

In the number 24.8, the number of digits in the decimal part  is one. So, we need to add Tenths(t)  place value i.e: 0.1 to the given number. Therefore,

Successor of 24.8 = 24.8 + 0.1 = 24.9

 

2) Successor of 354.9

Answer:

In this number 354.9, the number of digits in the decimal part is 1. So, we add Tenths (t) place value i.e; 0.1 to the given decimal number. Therefore,

Successor of 354.9 = 354.9 + 0.1 = 355.

 

3) Successor of 1467.58

Answer:

Here, the number of digits in the decimal part are Two. So, we add Hundredths(h)  place value i.e; 0.01 to the given decimal number.

Successor of 1467.58 = 1467.58 + 0.01 = 1467.59.

 

4) Successor of 999.989

Answer:

Here, the number of digits in the decimal part are Three. So, we need to add Thousandths(th) place value i.e; 0.001 to the given decimal number.

Successor of 999.989 = 999.989 + 0.001 = 999.990.

 

Predecessor:

The number that comes just before the given decimal number is called its Predecessor.

The Predecessor of decimal number is obtained by Subtracting Decimal number digits place value from the given decimal number.

 

1) Predecessor of 76.6

Answer:

Here, the number of decimal part digit is One. So, we need to Subtract Tenths(t) digit place value i.e: 0.1 from the decimal number 76.6

Predecessor of 76.6 = 76.6 – 0.1 = 76.5.

 

2) Predecessor of 10.99

Answer:

Here, the number of decimal digits are Two. So, we need to Subtract Hundredths( h ) digit place value i.e; 0.01 from the given decimal number.

Predecessor of 10.99 = 10.99 – 0.01 = 10.08

 

3) Predecessor of 456.987

Answer:

In this decimal number, the number of decimal digits are Three. So,we need to Subtract Thousandths(th) i.e; 0.001  from the given decimal number.

Predecessor of 456.987 = 456.987 – 0.001 = 456.986.

 

Successor and Predecessor for Decimal Numbers:

S.No.	PREDECESSOR	DECIMAL NUMBER	SUCCESSOR
1	0.1 - 0.1 = 0	0.1	0.1 + 0.1 = 0.2
2	13.25 - 0.01 =13.24	13.25	13.25 + 0.01= 13.26
3	375.98 – 0.01 = 375.97	375.98	375.98 + 0.01 = 375.97
4	99.999 – 0.001 = 99.998	99.999	99.999 + 0.001 = 100

Successor and Predecessor of Numbers

SUCCESSOR AND PREDECESSOR

Successor:

The number that comes just after a given number is its called Successor.

The successor of a number is 1 more than the given number (after number).

                                    (or)

The Successor of a number is obtained by adding 1 to the given number.

 

1) Successor of 4633

Answer:

The Successor of the number 4633 is 1 more than the 4633 = 4634.

                          (or)

The successor of the number 4633 is, we have to add 1 to the number 4633

 4633 + 1 = 4634.

 

2) Successor of 400

Answer:

 Successor of 400 = 1 more than 400 = 401.

                 (or)

 Successor of 400 = add 1 to 400 = 400 + 1 =401.

 

3) Successor of 204800

Answer:

 Successor of 204800 = 1 more than 204800 = 204801.

                    (or)

 Successor of 204800 = add 1 to 20488 = 204800 + 1 = 204801.

 

4) Successor of 399.

Answer:

 Successor of 399 = 1 more than 399 = 400.

                        (or)     

 Successor of 399 = add 1 to 399 = 399 + 1 = 400.

 

Predecessor:

The number that comes just before a given number is called its Predecessor.

The Predecessor of a number is 1 less than the given number (before number).

                        (or)     

The Predecessor of a number is obtained by subtracting 1 from the given number.

 

1) Predecessor of 6000

Answer:

Predecessor of 6000 = 1 less than the 6000 = 5999.

                        (or)

Predecessor of 6000 = subtract 1 from 6000 = 6000 – 1 = 5999.

 

2) Predecessor of  389

Answer:

Predecessor of 389 = 1 less than 389 = 388.

                        (or)

Predecessor of 389 = subtract 1 from 389 = 389 – 1 = 388.

3) Predecessor of 99999

Answer :

 Predecessor of 99999 = 1 less than 99999 = 99998.

                        (or)

Predecessor of 99999 = subtract 1 from 99999 = 99999 – 1  = 99998.

 

4) Predecessor of 4500

Answer:

 Predecessor of 4500 = 1 less than 4500 = 4499.

                        (or)

 Predecessor of 4500 =  subtract 1 from 4500 = 4500 – 1 = 4499.

 

Successor and Predecessor of the numbers.

S.No	PREDECESSOR	NUMBER	SUCCESSOR
1	5798	5799	5800
2	7999	8000	8001
3	979	980	981
4	3488	3489	3490

Decimal Numbers- Comparisions

 DECIMAL NUMBER COMPARISION:

We compare two decimal numbers to find out which is greater or smaller or equal by using >,<, = signs.

(i)  We use Equal to ( = )  sign, if the given digits place value of two decimal numbers are Equal.

(ii) In the given decimal numbers, if one of the decimal number is greater than the other decimal number, then we will use Greater than symbol (>).

(iii) In the given decimal number, if one of the decimal number is lesser than the other decimal number, then we will use Lesser than symbol (<).    

In comparision of two decimal numbers, we have to check the Whole part and the Decimal part.

Case(i)

If the Whole part digits place values in the two given decimal numbers are equal,then we have to check the decimal part.

While comparing the digits place values in the decimal part, we have to consider from Left --> Right from the Decimal point. i.e; from decimal part greatest place value: Tenths place, next greatest place value is Hundredths, next Thousandths and so on.

1) Compare 141.234 and 141.094

Answer:

In the decimal numbers, the Whole part number ,i.e; 141 is same in the two decimal numbers . So, we have to compare the decimal digits place value of the given numbers.

The decimal digits in the number are 234 and 094.When compared the two decimal digits,the decimal digit 234 is greater than decimal digit 094.Therefore, the decimal number 141.234 is greater.

 

141.234 > 141.094

2) Compare 95.136 and 95.181

Answer:

In the two decimal numbers, the Whole part digits : 95 are equal. So, we compare the decimal part digits place values.

 

95.136  <  95.181

(or)

95.181 > 95.136

 

 

3) Compare 0.283 and  0.4

Answer:

Here, also the whole part digit is same i.e; 0

The decimal part digits place value is to be compared. The two decimal part number of digits are not same. With that we can’t say that the decimal part having more number of digits is greater.

We have to make the two decimal part digits same. When we put Zeroes at the Right most part of the number, the number value doesn’t change.

i.e; 0.4 = 0.400  ,  but 0.4 # 0.04

 

0.283 < 0.4

   (or)

0.4 > 0.283

 

Case(ii)

If the Whole part digits place value in the number are not equal,we have to compare the digits places values in the Whole part . In this case there is no need to compare the digits place values in the Decimal part.

 

1) Compare 456.87 and 368.96

Answer:

In this problem, the whole part digits place value are different. So, we have to compare the Whole part number and have to find which number is greater.

There is no need to compare the Decimal part digits place value ,even though the decimal part number is greater or lesser.

Here, the decimal part digit 96 > 87.But in the decimal number the whole part digit number

456 > 368.

Based on the Whole part digits place value of the number only we have to tell the number is Greater or Lesser.

 456. 87   >  368.96

 

2) Compare  7523.923 and 7860.563

Answer:

Here, the Whole part digits places are not same,so we have to compare only Whole part.There is no need to compare Decimal part digits even though the Decimal digits of a number are greater or lesser.

7523.923 < 7860.563

       (or)

7860.563 > 7523.923