9th CBSE MATHS – NUMBER SYSTEMS

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

Examples:
2 is Natural Number
-2 is not a Natural number

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 terms of 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

NUMBER LINE

  1. Each of the number explained above can be represented on the Number Line.
  2. Natural Number,whole Number and integers can be easily located on the number line as we picture as per them 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 Pythagoras theorem.

WHAT IS PROCESS OF SUCCESSIVE MAGNIFICATION

  1. Suppose we need to locate the decimal 3.36 on the Number line.
  2. Now we know for sure the number is between 3 and 4 on the number line.
  3. Now let’s divide the portion between 3 and 4 into 10 equal parts. Then it will represent 3.1,3.2…3.9 .
  4. Now we know that 3.36 lies between 3.3 and 3.4.
  5. Now let’s divide the portion between 3.3 and 3.4 into 10 equal parts.
  6. 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.

9th CBSE NUMBER SYSTEMS (1) 9th CBSE NUMBER SYSTEMS (2) 9th CBSE NUMBER SYSTEMS (3) 9th CBSE NUMBER SYSTEMS (4)

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