Ch - Number Work (Part 1)

Different Types of Numbers

  1. Natural numbers
  2. Whole numbers
  3. Integers
  4. Rational numbers
  5. Irrational numbers
  6. Real numbers

N=Natural Numbers

  • Non-negative counting numbers excluding zero are known as natural numbers.
  • 1, 2, 3, 4 ......etc are all natural numbers.
  • 1 is the smallest Natural Number.

W=Whole numbers 

  • All non-negative counting numbers including zero are known as whole numbers.
  • 0, 1, 2, 3, 4.....etc are all Whole Numbers.
  • 0 is the smallest Whole Number.


I=Integers

  • All negative and non-negative numbers including zero altogether known as integers.
  • ............– 3, – 2, – 1, 0, 1, 2, 3, 4, ………….. are integers.

 

Q=Rational Numbers

  • Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q≠0. 
         Example -1/2, 4/5, 1,0,−3  and so on.
  •  
     

    What is meant by Standard Form of a Rational Number ?

    If p/q is a rational number,and p and q have no common factors other than 1(t hat is, p and q are co-prime) then we say that the rational number p/q is in standard form or in its lowest terms.
     
    Rational numbers can be written in decimal form also which could be either terminating or non-terminating. Example - 5/2 = 2.5 (terminating) and

    (non-terminating).

Q’=Irrational Numbers

  • In Number system, a number is called Irrational number if it cannot be expressed in the form p/q, where p and q are integers ( q> 0).
  • There are infinitely many irrational numbers too.

Example : etc

If we do the decimal expansion of an irrational number then it would be non –terminating non-recurring and vice-versa. i. e. the remainder does not become zero and also not repeated.

Example:

π = 3.141592653589793238……

Note :

  • If p is a prime number and p divides a2 , then p is one of the prime factors of a2 which divides a, where a is a positive integer.

  • If p is a positive number and not a perfect square, then √n is definitely an irrational number.

  • If p is a prime number, then √p is also an irrational number.

R=Real Numbers

  • Real numbers constitute the union of all rational and irrational numbers.
  • Any real number can be plotted on the number line.

     Place Value Chart

    Place Value Chart according to Indian System of Numeration.

    • In Indian system we start grouping the number from right in group of 3 and further in group of 2.
    • The place value chart have been separated into groups called periods i.e. ones, thousands, lakh and crores.

    Reading of number according to Indian System of Numeration

    In Indian System of Numeration periods such as ones, thousands, lakh, crores, etc. are used so that number can be easily read.

    Let’s read the below number according to Indian System of Numeration

    11, 54, 08, 453

    First put the number in their respective places.



    This number is read as eleven crore fifty four lakh eight thousand four hundred fifty three.

    Marking periods according to Indian System of Numeration

    • Different periods like ones, thousands, lakh and crores are separated by comma (,) starting from the right to differentiate the periods.
    • We start with the first period, named as ones period, consist the first three digits of the given number.
    • The second period (i.e. thousands period), consist of the next two digits of the given number.
    • The third period (i.e. lakh period), consist of the next two digits of the given number. The fourth period (i.e. crores period), consist of the next two digits of the given number.

    Let’s differentiate periods by placing commas in between.

    Example: 457833228

    This number is separated by comma as 45, 78, 33, 228

    Place Value and Face Value

    • Place Value of a digit is the value of the digit depending on it's position in the number.
    • Place value of a digit is obtained by multiplying the face value of the digit with its corresponding value of the place in that number.
    • Face Value of the digit is the digit itself.
    Example: 88, 45, 11, 009

    Expanded Form and Standard Form

    When each digit of a given number is written with its place value, then the number is written as the sum of place value of each digit in the number. Then we get the expanded form of that number.

    Example: 34, 16, 97, 832

    We can expand any given number in three ways:
    • 3 ten crore + 4 crore + 1 ten lakh + 6 lakh + 9 ten thousand + 7 thousand + 8 hundred + 3 ten + 2 one
    Or
    • 3 x 10,00,00,000 + 4 x 1,00,00,000 + 1 x 10,00,000 + 6 x 1,00,000 + 9 x 10,000 + 7 x 1,000 + 8 x 100 + 3 x 10 + 2 x 1
    Or
    • 30,00,00,000 + 4,00,00,000 + 10,00,000 + 6,00,000 + 90,000 + 7,000 + 800 + 30 + 2

    Successor and Predecessor

    • To find the successor of a given number, we add 1 to the number.
    • To find predecessor of a given number, we subtract 1 from  the number.
    Example:
    • Successor of 99,99,999 is
    99,99,999 + 1 = 1,00,00,000
    • Predecessor of 4,00,00,000 is
    4,00,00,000 – 1 = 3,99,99,999

    Ascending order

    The arrangement of numbers from the smallest to the greatest is called ascending order. This is also called as increasing order.
    Example: 88,88,870 < 7,54,34,108 < 67,65,76,676 < 67,76,78,676

    Descending order

    The arrangement of numbers from the greatest to the smallest is called descending order. This is also called as decreasing order.
    Example: 67,76,78,676 > 67,65,76,676 > 7,54,34,108 > 88,88,870

    Forming greatest and smallest number from given digits.

    Numbers can be formed using the given digits with or without repetition of digits.
    Example: 0,1,3,5,6,7,8,and 9
  • Greatest 8 digit number from above digits is 9,87,65,210
  • Smallest 8 digit number from above digits is 1,02,56,789

Place Value Chart according to the International System of Numeration.

  •  In International system we start grouping the number from right in group of 3, called period and we put comma or space after each period to make the number easily readable.

Reading number according to International System of Numeration

  •  In International System of Numeration periods such as ones, thousands, millions, billions etc. are used so that number can be easily read.
Let’s read the below number according to International System of Numeration

Example: 115,408,453

First put the number in their respective places.


This number is read as one hundred fifteen million four hundred eight thousand four hundred fifty three.

Marking periods according to International System of Numeration

  • Different periods like ones, thousands, millions and billions are separated by comma (,) starting from the right to differentiate the periods.
  • We start with the first period, named as ones period, consist the first three digits of the given number.
  •  The second period (i.e. thousands period) is consist of the next three digits of the given number.
  • The third period (i.e. millions period), consist of the next three digits of the given number.
  • The fourth period (i.e. billions period), consist of the next three digits of the given number.
Let’s differentiate periods by placing commas in between.

Example: 457833228


This number is separated by comma as 457,833,228

Place Value and Face Value

  • Place Value of a digit is the value of the digit depending on it's position in the number.
  • Place value of a digit is obtained by multiplying the face value of the digit with its corresponding value of the place in that number.
  • Face Value of the digit is the digit itself.
Example: 884,511,009

Expanded Form and Standard Form

When each digit of a given number is written with its place value, then the number is written as the sum of place value of each digit in the number. Then we get the expanded form of that number.
Example: 341,697,832

We can expand any given number in three ways:
  • 3 hundred million + 4 ten million + 1 million + 6 hundred thousand + 9 ten thousand + 7 thousand + 8 hundred + 3 ten + 2 one
Or
  • 3 x 100,000,000 + 4 x 10,000,000 + 1 x 1,000,000 + 6 x 100,000 + 9 x 10,000 + 7 x 1,000 + 8 x 100 + 3 x 10 + 2 x 1
Or
  • 300,000,000 + 40,000,000 + 1,000,000 + 600,000 + 90,000 + 7,000 + 800 + 30 + 2

Successor and Predecessor

  • To find the successor of a given number, we add 1 to the number.
  • To find predecessor of a given number, we subtract 1 from  the number.
Example:
  • Successor of 9,999,999 is
 9,999,999 + 1 = 10,000,000
  • Predecessor of 40,000,000 is
40,000,000 – 1 = 39,999,999

Ascending order

The arrangement of numbers from the smallest to the greatest is called ascending order. This is also called as increasing order.
Example: 8,888,870 < 75,434,108 < 676,576,676 < 677,678,676

Descending order

The arrangement of numbers from the greatest to the smallest is called descending order. This is also called as decreasing order.
Example: 677,678,676 > 676,576,676 > 75,434,108 > 8,888,870


Important points to Remember

  • If the digits in the units and the tens places of any number are interchanged, the differences between the number obtained and the original number is in multiples of 9.
Consider the number 4597, if the digit 7 and 9 are interchange the difference between 4597 and 4579 is 18 which is the multiple of 9. It is also 9 times of the difference between the two digits, i.e. 
 9 * ( 9 - 7) =9 * 2 = 18
 
  • If the same digit occurs in the 10th place and the units place of a number , .then the difference between the place values is 9 times that digit.
If the digit 5 is in the tens and units places, then the difference between their place values is 50 - 5 = 45 This is 9 times 5
  • If the same digit occurs in the hundreds place and the tens place of a number , then the difference between the place values 90 times that digits.
If the digit 6 is in the hundreds and the tens places. then the difference between the place values is 600 - 60 = 540 .This is 90 times 6.
  • If the same digit occurs in the hundreds place and the unit place of a number then the difference between the place value is 99 times that digit 
In the number 5474, the difference between the place values of 4 is 400 - 4 which is equal to 396 which is equal to 99 times 4
  • If the same to you takers in the thousands place and the tense place of a number then the difference between their place value is 990 times that digit 
In the number 75659 ,the difference between the place values of 5 is 5000 - 50 = 4950 = 990 times 5.
  • If the same to you takers in the thousands place and the units place of a number then the difference between their place value is 999 times that digit 
In the number 46326 the difference between the place values of 6 is 6000 - 6 = 5994 = 999 times 6

  • Given a number of a certain digits the greatest number is formed by using all its digits as 9 
 The greatest 5 digit number is 99999.
  • Given a number of certain it is smallest number is formed by using the first digit as one and the following digits are zero 
The smallest 5 digit number is 10000.
  • Given certain digits, the greatest number is formed by arranging the digits in descending order (each digit used only once).
The greatest number form by the digits 7,8,5,9,2 is 98752.
  • Given certain digit the smallest number is formed by arranging the digits in an ascending order each digit used only once however if one of the digits is zero it should be placed second from left
The smallest number formed by the digits 7,8,5,9,2 is 25789.
 
However the smallest number formed by digits 9,5,0,2,7 is 20579.
  • To form the greatest or the smallest 4 , 5 or 6 digit numbers using the given 2, 3 or 4 digits only (each digit is used at least once )
While writing this type of greatest number , arrange the given digits in a descending order and write the greatest digits for maximum times .
The greatest 6 digit number formed by using the digits 3,5,0 is 55530. 
While writing this type of smallest number, arrange the digit given digit in and ascending order and write the smallest digit for maximum times.
The smallest 5 digit numbers form by using the digit 9,2,3 is 22239 .
If one of the digit is '0',  it should be placed second from the left and taken for maximum times
 The smallest six digit number by using digit 8,0,2 is 200008.