Class 9 Number System

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Number System

Playing with numbers has always been fun. From the start of our scholastic studies to higher education, numbers have always been an integral part of a wide range of subjects. Thus, one must make sure that all your concepts about the numbers are thoroughly clear. One such topic that elucidates the details and types of numbers is the Number System. Getting a hold of this topic will help you build a sound foundation for higher studies as well as in cracking competitive exams. Here is a blog which aims to explain the insights of this topic.

What are the Types of Number System?

The number system begins by explaining the fundamentals of various types of numbers and how they are different in their nature. Let us have a look at the types and Number System examples.

Whole Numbers
The entire set of numbers including the number zero is known as whole numbers. For example: 0, 1, 13, 5, 8, 45 etc

Natural Numbers
Numbers ranging from 1 to infinity are known as Natural Numbers. For example: 1, 4, 6, 3, 7, 23, 47 etc

Integers
The set of whole numbers including their negative counterparts are known as Integers. For example: -56, -1, 0, 4, 5, 20, 67, 98 etc

Rational Numbers
Entities that can be expressed as the ratio of two natural numbers are called rational numbers. For example 5/8, 11/19, 2/8, etc

Irrational Numbers
Entities that cannot be expressed as a ratio of two numbers are known as the Irrational Numbers.

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Properties of Numbers

Now that we have understood the major types of number systems, let us have a look at some properties of these entities which will gear up our knowledge and help us understand the intricate concepts of the number system for higher studies. 

  • Number 1 is neither prime nor composite
  • The only prime number which is even is 2
  • The square of all the natural numbers can be written in the form of 3n or (3n+ 1)  as well as 4n or (4n+1) where ‘n’ is the natural number 
  • Every prime number greater than 3 can be written as (6k+ 1) or (6k-1) where ‘k’ is an integer 
  • The product of a consecutive natural number will always be divisible by a!
    Where a!= 1x2x3x4x5x6…xn

Representation of Irrational Numbers on Number Line

Let √x be an irrational number. Mentioned below are some simples steps to represent this number on the number line-

  • Take any point A. Draw a line AB = x units.
  • Extend AB to point C such that BC = 1 unit.
  • Find out the mid-point of AC and name it ‘O’. With ‘O’ as the centre draw a semi-circle with radius OC.
  • Draw a straight line from B which is perpendicular to AC, such that it intersects the semi-circle at point D.

Length of BD=√x.

Number System

Properties of Irrational Numbers

Here are some of the important properties of irrational numbers-

  • Sum of rational and irrational is irrational.
  • Product of rational and irrational is irrational.
  • These satisfy the commutative, associative and distributive laws for addition and multiplication.
  • Difference of two irrationals need not be irrational.

Example: (5 + √2) – (3 + √2) = 2

  • Sum of two irrationals need not be irrational.

Example: (2 + √3) + (4 – √3) = 6

  • Product of two irrationals need not be irrational.

Example: √3 x √3 = 3

  • The quotient of two irrationals need not be irrational.

2√3/√3 = 2

  • The quotient of rational and irrational is irrational.
  • The difference of rational and irrational number is irrational.

Rationalization

The process of converting the denominator of a given irrational number into a rational number by multiplying its denominator and numerator with a suitable number is known as Rationalization.

Tricks to Ace Number System 

Once you are through with the basic details and concepts related to the number system, learning some quick tips and tricks will help you score well in quantitative aptitude and other such sections in various competitive exams. As we have to complete a recruitment or entrance test within a stipulated time, knowing such tricks will help us solve tricky questions easily. Here are some important tips and tricks:

To Check Divisibility by 12

We know that the factors of number 12 are 2, 3, 4, 6. Now that if you want to check whether any given number is divisible by 12, you can check the divisibility of the given number by 3 and 4. As 3 and 4 both are prime factors of 12. Hence, you should the divisibility for 3 and 4 both.

To Check Divisibility by 18

The factors of 18 are 2, 3, 6, 9. Now, if you want to check whether a number is divisible by 18 or not, then, you have to check the divisibility of the given number with 2 and 9. If the number is divisible by both these numbers, then it will be divisible by 18 also.

To Check Divisibility by a number N which ends with 1, 3, 7, 9 

Here is a simple formula you can use to check this method.

Lets us multiply N by any number to get 9 as the last digit.
After this, add 1 to the result and divide it by 10.

Let us say that the result from the above step is R.

Now, let us take a number X and check whether it is divisible by N or not.

Split X as, 
X= 10y +z

X is divisible by N, only if Rz + y is divisible by N

Q. Find whether 645 is divisible by 23 or not.

N= 23, let us multiply 23 by 3 which is 69, thus,
9 is the last digit

R= (69+1)= 70
70/10= 7, R=7

X= 645, X=10y+z
645= (10* 64) + 5

Therefore, y=64 and z= 5

Putting values in formula we get,
Rz+y = (7*5) + 64= 99

99 is not divisible by 23, hence 645 is not divisible by 23

Thus, we hope that the blog on number systems has equipped you with the fundamentals of the topic. If you are planning to pursue management courses at the postgraduation level abroad then it is important to note that this topic forms an essential part of the GMAT exam. And to score well, you need to have a stronghold over every topic. Take online classes for GMAT from Leverage Live where the subject professionals, apart from equipping you with the requisite knowledge will also conduct online tests!  

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