Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

NUMBER SYSTEM - CBSE MATHEMATICS


NUMBER SYSTEM - CBSE MATHEMATICS
Different types of numbers  in mathematics and their practical examples, Natural Number, Whole Number, Integer, Rational Number, Irrational Number, Real Number........ number system for class 9 and 10


Syllabus For Class 9th Chapter 1 : Real Numbers

Syllabus For Class 10th Chapter 1 : Real Numbers
Explanation
1. NATURAL NUMBERS :- Counting numbers are called natural numbers.       
Eg:-  N =  {1,2,3,4,…………}
2. WHOLE NUMBERS : Natural numbers along with zero are called whole numbers.   Eg:-  W = {0,1,2,3,4,…….}  [N ⊂ W]
3. INTEGERS – All whole number and natural numbers with negative sign are called integers.  Eg :-  Z = {….-3, -2, -1, 0, 1, 2, 3…..}  [N ⊂ W ⊂ Z]
4. RATIONAL NUMBERS – The number which can be put in the form of P/q   (Where q ≠ 0) are called rational numbers.
Eg :-  Q = -5, 0, 7, 3/5, -7/8 etc.  [N ⊂ W ⊂ Z ⊂ Q

5. FRACTIONS – Positive rational number are called fractions.
  (OR)
Fraction is the ratio of two natural numbers. Eg:-  7/5, 1/2,  5/7 ….etc.

6. IRRATIONAL NUMBERS – The number which cannot be put in the form of  p/q are called irrational numbers. \[For Example:\sqrt{2},\sqrt{3}, \sqrt{5}, \sqrt{7}\; all\; are\; irrational\; numbers.\]
7. REAL NUMBERS – All rational  and irrational numbers are called real numbers. 
                                R =   S
8. PRIME NUMBERS:- The numbers which has only two factors one and itself are called prime numbers.  eg  2, 3, 5, 7, 11, 13

9. CO-PRIMES :- If H.C.F of two  numbers is 1 then numbers are called   co-prime numbers.       Eg:- (5, 7), (13, 27), (15, 16)…….

10. TWIN PRIMES :- Consecutive prime numbers which are differ by 2 are called twin primes.     
Eg :-  (11, 13),  (17, 21), (29, 31)…….etc.

11. EVEN NUMBERS:- Natural numbers which are divisible by two are called even numbers. eg  2, 4, 6, 8, 10 ,..........

12. ODD NUMBERS:- Natural numbers which are not divisible by 2 are called odd numbers.   Eg:- 1, 3, 5, 7, 9 ,........

13. COMPOSITE NUMBERS:- The numbers which has more than two factors are called composite numbers.

Note:-  "1" is neither a prime number nor a composite number. It is a unit.

14. IMAGINARY NUMBER or Non-Real Number   :- Negative square root of a natural number is called Imaginary number Or  non-real numbers.  \[Eg:-\; \; \sqrt{- 3},\sqrt{- 5}, \sqrt{- 7}....etc.\]


DECIMALS
Decimals are of three types
a. Terminating Decimal :-
 If prime factors of denominator of a rational number is either the power of 2 or 5 or both (2m x 5n )then the decimal form is known as terminating decimal.      
  Example: 2.5, 3.6,  7.895

 b. Non-Terminating But Repeating Decimal:- If prime factors of denominator of a rational number contain the factors other than power of 2 or 5 or both then the decimal form is called non terminating but repeating decimal. 
                     Eg :-   3.6767….            14.367367…....

c. Non-Terminating Non-Repeating Decimal :–
\[Decimal\; expansion\: of\: the\: numbers\: like\: \sqrt{3},\; \sqrt{5},\; \sqrt{7}\; is\; called\]\[ non-terminating\; non-repeating\; decimals.\]
Eg:-  1.1010010001......,  2.5050050005......,  3.6060060006.......

12. RATIONAL DECIMALS :-  
Terminating or non- terminating but repeating decimals are  called rational decimals.

13. IRRATIONAL DECIMALS:-  
Non-terminating, Non-Repeating decimals are called Irrational decimals

EUCLID DIVISION LEMMA :–
For given positive integers a and b there exist  unique integer q and r such that 
a = bq + r ,     ................    where 0 ≤ r <b. 
  Lemma – It is the proven statement used for proving another statement.

ALGORITHM – A series of well defined steps which gives a procedure for solving a type of problem.

 FUNDAMENTAL THEOREM OF ARITHMETIC – Every composite number can be expressed as a product of primes, and this factorization is unique irrespective of their order.                                                    
 HCF – Product of common factors with smallest power. 
 LCM – Product of all factors with greatest power.  
 When two numbers are given then      
 HCF x LCM   =  Product of two numbers
***************************************************
Solutions for some important questions for 10th Standard

Class 10    :    Exercise 1.1 

Q No. 4:   Prove that the square of any positive integer is of the form 3m or 3m+1

Solution: For two given positive integers a and b there exist unique integers q and r such that
a = bq + r, where 0 ≤ r < b
Let b = 3 
 a = 3q + r
Possible values of r = 0, 1, 2
Possible values of  a = 3q + 0, 3q + 1, 3q + 2

Now taking  a = 3q

Squaring on both side we get     

(a)2 = (3q)2      a2 = 9q2       a2 = 3(3q2 )      a2 = 3m1   … .......Where m1 = 3q2 

Now taking  a = 3q +1

    Squaring on both side we get     

(a)2 = (3q + 1)2     a2 = 9q2 + 1 + 6q     a2 = 9q2 + 6q + 1      a2 = 3(3q2 + 2q) + 1

 a2 = 3m2 +1                                                       ...........Where m2 = 3q2 + 2q

Now taking  a = 3q + 2 

Squaring on both side we get     

(a)2 = (3q + 2)2      a2 = 9q2 + 4 + 12q        a2 = 9q2 + 12q + 4         9q2 + 12q + 3 + 1 

  a2 = 3(3q2 + 4q + 1) + 1       a2 = 3m3 + 1        ............Where m3 = 3q2 + 4q + 1

Here we prove that the square of any positive integer is of the form 3m or 3m + 1

Q No. 5:   Prove that the cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8

Solution: For two given positive integers a and b there exist unique integers q and r such that
a = bq + r, where 0 ≤ r < b
Let b = 3 
 a = 3q + r
Possible values of r = 0, 1, 2
Possible values of  a = 3q + 0, 3q + 1, 3q + 2

Now taking  a = 3q

Cubing on both sides we get     

(a)3 = (3q)3      a3 = 27q3       a3 = 9(3q3 )      a3 = 9m1   … .......Where m1 = 3q3 

Now taking  a = 3q +1

Cubing on both side we get     

(a)3 = (3q + 1)3     a3 = 27q3 + 1 + 27q2 + 9q    a3  = 27q3  + 27q2 + 9q +1   

 a3  = 9(3q3  + 3q2 + q) +1   a3 = 9m2 +1                ...........Where m2 = 3q3  + 3q2 + q

Now taking  a = 3q +2

 (a)3 = (3q + 2)3     a3 = 27q3 + 8 + 54q2 + 36q    a3  = 27q3  + 54q2 + 36q +8   

 a3  = 9(3q3  + 6q2 + 4q) +8   a3 = 3m3 +8          ...........Where m3 = 3q3  + 6q2 + 4q

Here we prove that the cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8

 *****************************************************

Class 10  Chapter 1 Exercise 1.2

Q. 5  Check whether 6n  can ends with  digit 0 for any natural number n.

Solution: If 6n ends with 0 then its prime factorization must contain the power of 2 and 5. Prime factors of 6 are 2 x 3 and by the fundamental theorem of arithmetic this factorization is unique. This shows that prime factors of 6n does not contain 5.

Hence 6n never ends with zero for any value of  n

**************************************

Class 10  :   Exercise 1.3

Question 1) Prove that √5 is an irrational number

Solution:

Let √5 is a rational number

Then there exist co-prime a and b such that

√5 = a/b

Cross multiply here we get

√5b = a 

Now squaring on both side we get

5b2 = a2

  b2 = a2/5  ……………..(1)

 5 divides a2    5 divides a ………(2)

Let a = 5c ,  Squaring on both sides, we get

a2 = 25c2 ………………(3)

Putting  eqn.(3) in equation (1) we get

 b2 = 25c2/5 , 

 b2 = 5c2   

 b2/5  =  c2

 5 divides b2  5 divides b…..(4)

From (2) and (4)

5 divides a and b both

 a  and b are not co-prime

But it is given in the starting that a and b are co-prime numbers.

So there is a contradiction and our assumption is wrong

Hence  √5  is an irrational number

Question: Prove that  6 - 2√5 is an irrational number

Solution:  
Let 6 - 2√5 is a rational number.
Then there exist unique integers a and b such that \[6-2\sqrt{5}=\frac{a}{b}\]\[-2\sqrt{5}=\frac{a}{b}-6\]\[-2\sqrt{5}=\frac{a-6b}{b}\]\[\sqrt{5}=\frac{a-6b}{-2b}\]
a, b, 2 and 6 all are integers \[\Rightarrow \frac{a-6b}{-2b}\: \: is\: \: a\: \: rational\: \: number\] But √5 is an irrational number.
So there is a contradiction and our assumption is wrong
Hence 6 - 2√5  is an irrational number.

Famous Mathematicians 

THALES:-
Thales was a famous Greek Mathematician during the period  640 BC to 546 BC. He was given the first proof in mathematics. This proof was of the statement that a circle is bisected by the diameter.

PYTHAGORAS:-
Pythagoras was a famous Greek Mathematician and he was the famous pupil of Thales. Pythagoras and his group discovered many geometric properties and developed a theory of geometry to a great extent.

EUCLID:-
Euclid was a teacher of Mathematics at Alexandria in Egypt during the period of 325 BC to 265 BC. He collected all the known work in Mathematics and arranged it in his famous treatise. This treatise is called " Elements". He divided the "Elements" into thirteen chapters, each called a book. These books influenced the whole world's understanding for geometry.

EUCLID'S AXIOMS AND POSTULATES:-
Euclid assumed certain properties, which were not to be proved. These assumptions are called the universal truths in Mathematics. He divided them into two types: axioms and postulates.

AXIOMS:- 
Common notions in Mathematics which are specifically used throughout the whole Mathematics are called axioms.

POSTULATES:-
Common notions in Mathematics which are specifically used in geometry are called postulates.


NUMBER SYSTEM - CBSE MATHEMATICS

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