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Class 10 Maths notes for Real Numbers


Flashback of IX real Number's


Natural and Whole Number


What is Natural Numbers?
Set of counting numbers is called the Natural Numbers
N = {1,2,3,4,5,...}
What is whole number?
Set of Natural numbers plus Zero is called the Whole Numbers
W= {0,1,2,3,4,5,....}
Note:
So all natural Number are whole number but all whole numbers are not natural numbers

Integers


What are Integers Numbers
Integers is the set of all the whole number plus the negative of Natural Numbers
Z={..,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...}
Note
1) So integers contains all the whole number plus negative of all the natural numbers
2)the natural numbers without zero are commonly referred to as positive integers
3)The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer
4)natural numbers with zero are referred to as non-negative integers
5) The natural numbers form a subset of the integers.

Rational and Irrational Numbers


Rational Number

: A number is called rational if it can be expressed in the form p/q where p and q are integers ( q> 0).
Example : 1/2, 4/3 ,5/7 ,1 etc.
Important Points to Note
  • every integers, natural and whole number is a rational number as they can be expressed in termsof p/q
  • There are infinite rational number between two rational number
  • They either have termination decimal expression or repeating non terminating decimal expression
  • The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.


Irrational Number

: A number is called rational if it cannot be expressed in the form p/q where p and q are integers ( q> 0).
Example : √3,√2,√5,p etc
Important Points to Note
  • Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem we can represent the irrational numbers on the number line.
  • They have non terminating and non repeating decimal expression
  • The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division

Real Numbers:


  • All rational and all irrational number makes the collection of real number. It is denoted by the letter R
  • We can represent real numbers on the number line. The square root of any positive real number exists and that also can be represented on number line
  • The sum or difference of a rational number and an irrational number is an irrational number.
  • The product or division of a rational number with an irrational number is an irrational number.
  • This process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification

Real numbers satisfy the commutative, associative and distributive laws. These can be stated as :
Commutative Law of Addition:
a+b= b+a
Commutative Law of Multiplication:
a X b=b X a
Associative Law of Addition:
a + (b+c)=(a+b) +c
Associative Law of Multiplication:
a X (b X c)=(a X b) X c
Distributive Law:
a X (b + c)=(a X b) + (a X c)
or
(a + b) X c=(a X c) + (b X c)


Laws of exponents:



Let a > 0 be a real number and p and q be rational numbers. Then, we have
1) ap.aq=a(p+q)
2) ap/aq =a(p-q)
3) (ap)q=apq
4) ap.bp=abp

What is Number Line


A number line is a line which represent all the number. A number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point
We most shows the integers as specially-marked points evenly spaced on the line. but the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers. The number on the right side are greater than number on the left side
Each of the number explained above can be represented on the Number Line.
1) Natural Number,whole Number and integers can be easily located on the number line as we picture as per them
2) Now Real number can be either decimal expression or number explained in point 1. It is easy to located the latter one. For decimal expression, we need to use the process of successive Magnification
3) Number like (3)1/2 can be represent on number like using pythogorus theorem

What is process of successive Magnification


Suppose we need to locate the decimal 3.36 on the Number line. Now we know for sure the number is between 3 and 4 on the number line. Now lets divide the portion between 3 and 4 into 10 equal part.Then it will represent 3.1,3.2...3.9 . Now we know that 3.36 lies between 3.3 and 3.4.Now lets divide the portion between 3.3 and 3.4 into 10 equal parts. Then these will represent 3.31,3.32,3.33,3.34,3.35,3.36...3.39. So we have located the desired number on the Number line. This process is called the Process of successive Magnification


Euclid's Division Lemma

For a and b any two positive integer, we can always find unique integer q and r such that
a=bq + r , 0 ≤ r < b

It is basic concept and it is restatement of division

a is called dividend
b is called divisor
q is called quotient
r is called remainder.

If r =0, then b is divisor of a.

Proof of Euclid's Division Lemma

Here we need to argue that q and r are no unique
Let us assume q and r are not unique i.e. let there exists another pair q0 and r0 i.e. a = bq0 + r0, where 0 ≤ r0 < b

=> bq + r = bq0 + r0
=> b(q - q0) = r - r0 ................ (I)

Since 0 ≤ r < b and 0 ≤ r0 < b, thus 0 ≤ r - r0 < b ......... (II)

The above eq (I) tells that b divides (r - r0) and (r - r0) is an integer less than b. This means (r - r0) must be 0.
=> r - r0 = 0
=> r = r0

Eq (I) will be, b(q - q0) = 0
Since b > 0, => (q - q0) = 0
=> q = q0

Since r = r0 and q = q0, Therefore q and r are unique.

HCF (Highest common factor)

Before starting on this topic, we need to prove a important therom which will be used in finding HCF between two numbers

Theorem

If a and b are positive integers such that a = bq + r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.

Proof : Let m be a common divisor of a and b. Then,
m| a => a = mx for some integer x
m| b => b = mq2 for some integer y
Now, a = bq + r
=> r = a - bq
=> r = mx - my q
=> r = m( x - yq)
=> m| r
=> m| r and m | b
=> m is a common divisor of b and r.
Hence, a common divisor of a and b is a common divisor of b and r.

How to find HCF (Highest common factor)

Now HCF of two positive integers can be find using the Euclid's Division Lemma algorithm and above stated therom

We know that for any two integers a,b. we can write following expression
a=bq + r , 0 ≤ r < b
If r=0 ,then
HCF( a,b) =b
If r >0 , then
HCF ( a,b) = HCF ( b,r) as already proved from above theorem

Again expressing the integer b,r in Euclid’s Division Lemma, we get
b=pr + r1
HCF ( b,r)=HCF ( r,r1)
Similarly successive Euclid ‘s division can be written until we get the remainder zero, the divisor at that point is called the HCF of the a and b

Examples

Use Euclid's algorithm to find the 65 and 117.

Solution :
Step:1 Since 117 > 65 we apply the division lemma to 117 and 65 to get ,
117 = 65 x 1 + 52
Step:2 Since 52 > 0 , we apply the division lemma to 65 and 52 to get
65 = 52 x 1 + 13
Step:3 Since 13 > 0 , we apply the division lemma to 52 and 13 to get
52 = 13 x 4 + 0
The remainder has now become zero, so our procedure stops. Since the divisor at this Step is 13, the HCF of 117 and 52 is 13.

Some other points to remember
1)The HCF of three numbers can be calculated by first calculating the HCF of first two numbers ,then the calculating the HCF of the HCF of previous two numbers and third number.
2)If HCF ( a,b) =1 ,the a and b are co primes.

What is Prime Numbers

A Prime Number is a number that cannot be evenly divided by any other number (except 1 or itself).

Examples 1,2,3,5,7,11,13,17,19,23,29.......
Prime numbers are interesting blocks in Mathematics.They are highly used in cryptograpy field

Prime Number help
PRIME NUMBERS to 100 ={2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}
How to find Prime Number Rule

Rule Number 1:
Since even numbers are always divisible by 2, no even number (other than 2) can be a prime number because they'll always have more than two factors. For example, the factor pairs for 6 are: 1,2,3,6 So 6 has four factors and therefore is not a prime number.
Rule Number 2:
Numbers that end in 5 or 0 are always divisible by 5. So no numbers divisible by 5 (other than 5) can be prime numbers because they'll always have more than two factors. They'll have 1 and themselves, and they'll also have 5 and some other factor.
Rule Number 3:
0 and 1 are not prime numbers. It may seem at first glance that they are, but remember that a prime number has only 1 and itself as factors. But 0 has an infinite number of factors because 0 * 1 = 0, and 0 * 2 = 0, and 0 * 3 = 0, and so on. So 0 is far from prime.
In the case of 1, it doesn't have two distinct factors because it's only factor is itself: 1 * 1 = 1. If you change one of those factors to any other integer, you no longer get a product of 1. Since it has only one factor, it's not prime.
Rule Number 4:
Add all of the digits composing the number. If the sum of the digits is divisible by 3, then so is the original number. Example: If the number is 111, add 1 + 1 + 1 = 3. The sum of the digits (3) is divisible by 3, so 111 is also divisible by 3.
Rule Number 5:
Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7 also
Rule Number 6:
If you are checking if a number is a prime number,then you dont need to check all the prime number below it. you need to check run for small primes(2,3,5,7) and you're not done factoring, then keep trying bigger and bigger primes (11, 13, 17, 19, 23, etc) until you find something that works — or until you reach primes whose squares are bigger than what you're dividing into. if you dont find then it is a prime Number Reason: If your prime doesn't divide in, then the only potential divisors are bigger primes. Since the square of your prime is bigger than the number, then a bigger prime must have as its remainder a smaller number than your prime. The only smaller number left, since all the smaller primes have been eliminated, is 1. So the number left must be prime, and you're done.

What is Composite Numbers

A whole number that can be divided evenly by numbers other than 1 or itself

Examples
4,6,8,9,10.....

Since 4 is divided by 2,6 is divided by 2 and 3

Fundamental Theorom of Arithmetic

Every composite number can be written as the product of power of primes and this factorization is unique
Composite number = Product of primes

Example

16=2X2x2X2
This is unique, We cannot express it in any other prime numbers

Few things to take note
1) The prime Number can be repeated in the factorization
2) The order does not matter

How to Factorize the Composite Numbers


1) Few things we need to remember,
If the number is even,then it will be divisible by 2
If the sums of its digits is divisible by 3,then it is divisible by 3
if the end of the number is 0 or 5,then it is divisible by 5
2) We have start with the small prime number with the rules given in step 1. Once we find the quotient, repeat the same process for the quotient. The last quotient will be a prime number itself

Example

Suppose the composite Number is 168
1) Now 168 is even number,so we know it will get divided by 2
168 / 2 = 84
2) Again 84 is even number,so we know it will get divided by 2
84 / 2 = 42
3) Again 42 is even number,so we know it will get divided by 2
42 / 2 = 21
4) Now sum of digits of 21 is 3,so we know we can divide it by 3
21 / 3 = 7
5)7 is a prime Number

So prime factors = 2 X 2 X 2 X 3 X 7

Here are the prime factors of the composite numbers between 1 and 30.
4 = 2 X 2
6 = 3 X 2
8 = 2 X 2 X 2
9 = 3 X 3
10 = 5 X 2
12 = 3 X 2 X 2
14 = 7 X 2
15 = 5 X 3
16 = 2 X 2 X 2 X 2
18 = 3 X 3 X 2
20 = 5 X 2 X 2
21= 3 X 7
22=2 X 11
24= 2 X 2 X2 X3
25= 5 X 5
26= 2 X 13
27= 3 X 3 X 3
28= 2 X 2 X7
30= 2 X 3 X 5

HCF and LCM by prime factorization method


We can find HCF and LCM between two number by the prime factorization method also
HCF (Highest Common factor) = Product of the smallest power of each common factor in the numbers

Example: Suppose the number are 14,24
1) Lets do prime factorization method for both the numbers
14=2 X 7
24= 2 X2 X2 X3
2) Now as per method
HCF =2

LCM( Lowest Common Multiple) = Product of the greatest power of each prime factor involved in the number
Example: Suppose the number are 14,24
1) Lets do prime factorization method for both the numbers
14=2 X 7
24= 2 X2 X2 X3
2) Now as per method
HCF =2X2X2X7X3=168
Also important Formula to remember
If a and b are two number,then
HCF(a,b) X LCM (a,b) =a X b

Irrational Numbers

A number r is called irrational number if it cannot be expressed in the form p/q where p and q are integer and q≠ 0

Important Theorem

Let p be a prime number ,if p divides a2 ,the p divide a where a is positive number

Rational numbers

A number r is called rational number if it can be expressed in the form p/q where p and q are integer and q≠ 0
Decimal expressions of rational number are of two types
a) Terminating decimal expression
b) Non terminating repeating decimal expression

Terminating decimal expression can be written in the form
p/2n5m
Non terminating repeating decimal expression can not be expressed in this form
p/2n5m

How to prove the irrational numbers or Rational numbers

1) We know the defination of each of them,so first try to express the number in the form p/q.If it is not possible then proceed to next step
2) If the proof is irrational number which will be the case always,assume the number to rational in the form p/q and then try to prove it wrong

Solved Examples


Prove that √2 is irrational number
Solution: Since we cannot clearly express in p/q form,it is difficult to say,So lets us assume this is rational nunber
then
√2=p/q

where p and q are co primes.
or
q√2=p
squaring both sides
2q2=p2

So 2 divides p2,from theorem we know that,
2 will divide p also. p=2c

2q2=4c2
or
q2=2c2
So q divided by 2 also

So both p and q are divided by 2 which is contradiction from we assumed
So √2 is irrational number

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