Monday, 12 September 2022

Binary Number System||Bit System

 

Binary Number System – Definition, Conversion, Examples

 

Number  System||Numeral System

The number system or the numeral system is the system of naming or representing numbers in various formats. We know that a number is a numerical value that helps to count or measure objects and it helps in performing various mathematical calculations. There are different types of number systems in Maths but we categories number system in four major category- decimal number system, binary number system, octal number system, and hexadecimal number system.

Binary Number System||Bit System

 

  binary number system with base-2 is one of the four types of number systems. Binary number system played historical role in development of computer history.

 0 and 1 are two digits represents a  binary system .  In the word “binary”, “bi” means “two”.  The base-2 numeral system is used to represent binary numbers. For example, (1010)2 is a binary number where 2 is the radix. Each digit in the binary number system is called to be a “bit”. 

 

Role of binary system in Computer development

 

This number system is widely used in computers. Binary number are used in digital electronic circuitry using logic gates. Since computer only understand binary information (0’s and 1’s). Any  inputs given to a computer are decoded by it into a series of 0’s or 1’s before being processed . It is simple to convert a decimal number into a binary number and vice-versa. The notations for decimal numbers and binary numbers are different. For example, a decimal is represented as (15)10 where 10 is the base of the decimal number, and the corresponding binary number is represented as (1111)2 where 2 is the base of a binary number

     All the coding and languages(C, C++, Java , any other languages or any other coding platform ) in computers use binary digits 0 and 1 to write a program

 

 

Reminder which represent binary number from bottom to top

 

2

8

 

2

4

0

 

2

2

0

 

2

1

0

 

 

0

1

 

 

 

 

 

Binary conversion method

 

Decimal Number

Binary Number

Decimal Number

Binary Number

1

001

11

1011

2

010

12

1100

3

011

13

1101

4

100

14

1110

5

101

15

1111

6

110

16

10000

7

111

17

10001

8

1000

18

10010

9

1001

19

10011

10

1010

20

10100

Binary decimal number table

 


Monday, 5 September 2022

What is number System

 

Number System||Numeral System

The number system or the numeral system is the system of naming or representing numbers in various formats. We know that a number is a numerical value that helps to count or measure objects and it helps in performing various mathematical calculations. There are different types of number systems in Maths but we categories number system in four major category- decimal number system, binary number system, octal number system, and hexadecimal number system.

Definition of Number System.

A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction and division.

The value of any digit in a number can be determined by:

  • The digit
  • Its position in the number
  • The base of the number system

Types of Number System

There are various types of number systems in mathematics. The four most common number system types are:

·         Decimal number system            (Base : 10)

·         Binary number system              (Base : 2)

·         Octal number system                (Base : 8)

·         Hexadecimal number system  (Base :16)

Decimal Number System (Base 10)

The decimal number system has a base of 10 because it uses ten digits from 0 to 9. In the decimal number system, the positions successive to the left of the decimal point represent units, tens, hundreds, thousands and so on..Every position shows a particular power of the base (10).

Example:

The decimal number 1873 consists of the digit 3 in the unit position, 7 in the tens place, 8 in the hundreds position, and 1 in the thousands place whose value can be written as: Base 10 Number System

ð  (1×103) + (8×102) + (7×101) + (3×100)

ð  (1×1000) + (8×100) + (7×10) + (3×1)       :X0 =1

ð  1000 + 800 + 70 + 3

ð  1873

Binary Number System (Base 2 )

The number with base 2 is also known as the binary number system. only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 is a radix of 2. The figures described under this system are known as binary numbers which are the combination of 0 and 1. For example, 010101 is a binary number.

We can convert any system into binary and vice versa. Base 2 Number System

Example :

Write (15)10 as a binary number.

Solution:

2

15

 

2

7

1

2

3

1

 

1

1

 

 

Base 2 Number System Example

(15)10 = 11112

Octal Number System (Base 8 Number System)

The number with base 8 is known as the Octal Number System and it uses numbers from 0 to 7 to represent Octal Numbers System. Octal numbers are commonly used in computer applications. Converting an octal number to decimal is the same as decimal conversion and is explained below using an example.

Example: Convert 18738 into decimal.

Solution:

18738 = 1 × 83 + 8 × 82 + 7 × 81 + 3 × 80

= 1 × 512 +8 × 64 + 7 × 8 + 3 × 1

= 512 +512 + 56+ 3

= 108310

Hexadecimal Number System (Base 16 Number System)

The number with base 16 is known as the Hexadecimal Number System and it uses numbers from 0 to 9 then alphabet from A to F  to represent Octal Numbers System.

In the hexadecimal system, numbers are written or represented with base 16. In the hex system, the numbers are first represented just like in the decimal system, i.e. from 0 to 9. Then, the numbers are represented using the alphabet A(10), B(11), C(12),D(13),E(14) and  F(15).

Example: Convert 5B4E16 into decimal.

Solution:

5B4E16 = 5 × 163 + 11 × 162 + 4 × 161 + 14 × 160                :: B=11 and E=14

= 5 × 4096 +11 × 256 + 4 × 16 +14 × 1

= 20480 +2816 + 64+ 14

= 23274 10

FAQs on Number System

What is Number System?

The number system is simply a system to represent or express numbers.

 

What are Types of Number System ?

There are various types of number systems in mathematics. The four most common number system types are:

1. Decimal number system (Base- 10)

2. Binary number system (Base- 2)

3. Octal number system (Base-8)

  4. Hexadecimal number system (Base- 16)



Monday, 16 January 2017

Whole Numbers: Definition, Properties, Examples and pictures

Whole Numbers: Definition, Properties, and Examples

Whole Numbers Symbol : W

 

Numbers System: We use numbers to count things. There are various kinds of numbers in a number system, such as; natural numbers, whole numbers, real numbers, odd and even numbers, etc.
Whole numbers and natural number are real numbers that do not contain fractions, decimals, or negative values .We make use of numbers in our everyday life for telling the cost of items, telling time, counting objects, representing or exchanging money, measuring the temperature, etc.

Whole Numbers Definition

Whole Numbers are the set of natural numbers and 0. The set of whole numbers is written as W= {0,1,2,3,…}.

Properties of Whole Numbers

The basic operations on whole numbers are addition, subtraction, multiplication, and division, which further leads to four main properties of whole numbers that are listed below:

 

Closure Property

When two whole numbers W1 and W2 are added or multiplied, the result (W1+W2) or W1.W2 will always be a whole number.

Example: 2+3=5 OR 3+2=5 (whole number), 2×3=6 OR 3×2=6 (whole number)

The closure property is not true in the subtraction of whole numbers.

Example: Subtraction 5–3= 2 (whole number), 3–5= –2 (not a whole number )

Commutative Property

The sum or the product of two whole numbers  remain the same even after interchanging the order of the numbers.

Let W1 and W2 be two whole numbers, according to the commutative property W1+W2=W2+W1 or W1.W2=W2.W1

Example: Sum, 8+3=11 and 3+8=11.

Product, 8×3=24 and 3×8=24

This property does not hold good for subtraction or division.

Example: Subtraction 5–3= 2 (whole number), 3–5= –2 (not a whole number )

Division, 8/4=2 and 4/8=1/2 which is not a whole number.

Associative Property

The sum or product of three whole numbers remains unchanged by grouping the numbers in any order.

Let W1,W2 and W3 be three whole numbers, according to the associative property (W1+W2)+W3=W1+(W2+W3) or W1×(W2×W3)=(W1×W2)×W3

Example: When we add three numbers, Let 3,4,5 we get the same sum:

3+(4+5) = 12 or (3+4)+5=12

Similarly, when we multiply any three numbers 3,4,5, we get the same product no matter how the numbers are grouped:

3×(4×5)=60, (3×4)×5=60, (5x3)x4=60

The associative property is not true for subtraction and division.

Distributive Property

This property states that the multiplication of a whole number is distributed over the sum of the whole numbers.

Suppose W1 and W2 are multiplied with the whole number W3 and the products are added, then W3 can be multiplied with the sum of W1 and W2 to get the same answer. This situation can be represented as: W3×(W1+W2)=(W1×W3)+(W3×W2).

Example: W1=2,W2=4 and W3=6 : 6×(2+4)=36 and (6×2)+(6×4)=36

The distributive property is true for subtraction as well.

Example: W1=2,W2=4 and W3=6 : 6×(4-2)=12 and (6×4)-(6×2)=12

 

 

Special Properties of Whole Numbers

There are some special properties of whole numbers other than the main properties discussed in the above section,

  Additive identity

When a whole number is added to 0, its value itself whole number i.e., if W1 is a whole number, then W1+0 =0+W1 =W1

Example: 8+0=8=0+8.4+0=4=0+4.

The number 0 is called the additive identity for the whole number.

 Multiplicative identity

Any whole number multiplied by 1 gives the same value of the whole number, i.e., if W1 is a whole number, then W1×1=W1=1×W1

Example: 8×1=1×8=8.

The number 1 is called the multiplicative identity for the whole number.

   Multiplication by zero

Any whole number multiplied by 0 always gives 0, i.e., W1×0=0=0×W1

Example: 8×0=0×8=0.

 Division by zero

Division of a whole number by 0 is not defined, i.e., if W1 is a whole number, then W1x0 is not defined.

 

FAQs on Whole Numbers

Following are some common questions which candidates may have in their mind regarding Whole Numbers:

Q1. Define whole numbers, or what are whole numbers?
Ans: Whole numbers in Math is the set of positive integers and 
0. In other words, it is a set of natural numbers, including 0.Decimals, fractions, negative integers are not part of whole numbers.

Q2. What are the four properties of whole numbers?
Ans: The four properties of whole numbers are:
1. Closure property.
2. Associative property.
3. Commutative property.
4. Distributive property

Q3. What is the use of whole numbers?
Ans: These are numbers that we are the most used to working with, including zero. We see whole numbers on nutrition labels or signs on the highway telling us how many miles are to the next city. 

Q4. What is the smallest whole number?
Ans: Zero is the smallest whole number.

Q5. Which numbers are not whole numbers?
Ans: A negative integer, fractions, part of rational numbers and decimals do not belong to whole numbers.

Q6. Which is the largest whole number?
Ans: There is no largest whole number. Every whole number has an immediate successor or a number that comes after. So the whole numbers are infinite to count, and thus, there is nothing such largest whole number.

 

Whole Numbers on Number Line

 All the integers starting from 0 represent the Whole numbers .Whole numbers on the number line.


 

Refer to the following Reference Books for Maths

 

1.      Mathematics by Chand Publishing

2.      NCERT Textbooks  

3.      Quantitative Aptitude for competitive examinations by RS Aggarwal

4.      For Entrance Exam by ES Ramasamy 

 

Binary Number System||Bit System

  Binary Number System – Definition, Conversion, Examples   Number   System||Numeral System The number system or the numeral system is...