Number System

Numebr System | IndianTechnoEra

A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems.

Two types of number systems are:

Non-positional number systems
Positional number systems

1. NON-POSITIONAL NUMBER SYSTEMS

Characteristics

Use symbols such as i for 1, ii for 2, iii for 3, iiii for 4, iiiii for 5, etc.
Each symbol represents the same value regardless of its position in the number.
The symbols are simply added to find out the value of a particular number.

Difficulty

It is difficult to perform arithmetic with such a number system.

2. POSITIONAL NUMBER SYSTEMS

The value of each digit is determined by:

The digit itself
The position of the digit in the number
The base of the number system

(base = total number of digits in the number system)

The maximum value of a single digit is always equal to one less than the value of the base

Characteristics

Use only a few symbols called digits.
These symbols represent different values depending on the position they occupy in the number.

DECIMAL NUMBER SYSTEM

Characteristics

A positional number system has 10 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Hence, its base = 10.
The maximum value of a single digit is 9 (one less than the value of the base).
Each position of a digit represents a specific power of the base (10)
We use this number system in our day-to-day life.

Example

2586₁₀= (2 x 10³) + (5 x 10²) + (8 x 10¹) + (6 x 10⁰)

= 2000 + 500 + 80 + 6

BINARY NUMBER SYSTEM

Characteristics

A positional number system has only 2 symbols or digits (0 and 1). Hence its base = 2.
The maximum value of a single digit is 1 (one less than the value of the base).
Each position of a digit represents a specific power of the base (2).
This number system is used in computers.

Example

10101₂= (1 x 2⁴) + (0 x 2³) + (1 x 2²) + (0 x 2¹) x (1 x 2⁰)

= 16 + 0 + 4 + 0 + 1

= 21₁₀

Representing numbers in different number systems

In order to be specific about which number system we are referring to, it is a common practice to indicate the base as a subscript. Thus, we write:

10101₂= 21₁₀

BIT (Binary digiT)

Bit stands for binary digit.
A bit in computer terminology means either a 0 or a 1.
A binary number consisting of n bits is called an n-bit number.

Characteristics

A positional number system has total 8 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7). Hence, its base = 8.
The maximum value of a single digit is 7 (one less than the value of the base.
Each position of a digit represents a specific power of The base (8).

OCTAL NUMBER SYSTEMS

Since there are only 8 digits, 3 bits (2³= 8) are sufficient to represent any octal number in binary

Example

2057₈

= (2 x 8³) + (0 x 8²) + (5 x 8¹) + (7 x 8⁰)

= 1024 + 0 + 40 + 7

= 1071₁₀

HEXADECIMAL NUMBER SYSTEM

Characteristics

A positional number system has total 16 symbols or digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f). Hence its base = 16.
The symbols a, b, c, d, e and f represent the decimal values 10, 11, 12, 13, 14 and 15 respectively.
The maximum value of a single digit is 15 (one less than the value of the base).
Each position of a digit represents a specific power of the base (16).
Since there are only 16 digits, 4 bits (2⁴ = 16) are sufficient to represent any hexadecimal number in binary.

Example

1af₁₆= (1 x 16²) + (a x 16¹) + (f x 16⁰)

= 1 x 256 + 10 x 16 + 15 x 1

= 256 + 160 + 15

= 431₁₀

CONVERT A NUMBER'S BASE:

Another base number to decimal base

Method

Step 1: determine the column (positional) value of each digit

Step 2: multiply the obtained column values by the digits in the corresponding columns

Step 3: calculate the sum of these products

Example

4706₈= ?₁₀

Common values multiplied by the corresponding digits

4706₈ = 4 x 8³ + 7 x 8² + 0 x 8¹ + 6 x 8⁰

= 4 x 512 + 7 x 64 + 0 + 6 x 1

= 2048 + 448 + 0 + 6

Sum of these products

= 2502₁₀

Base

Decimal base to another base

Division-remainder method

Step 1: divide the decimal number to be converted by the value of the new base
Step 2: record the remainder from step 1 as the right most digit (least significant digit) of the new base number.
Step 3: divide the quotient of the previous divide by the new bas
Step 4: record the remainder from step 3 as the next digit (to the left) of the new base number

Repeat steps 3 and 4, recording remainders from right to left, until the quotient becomes zero in step 3 note that the last remainder thus obtained will be the most significant digit (msd) of the new base number

Example

952₁₀= ?₈

Solution:

8	952	Reminders
	119	0
	14	7
	1	6
	0	1

Hence, 952₁₀ = 1670₈

Some based to another based

Step 1: convert the original number to a decimal number (base 10)
Step 2: convert the decimal number so obtained to the new base number

Example

545₆ = ?₄

Solution:

Step 1: convert from base 6 to base 10

545₆ = 5 x 6² + 4 x 6¹ + 5 x 6⁰

= 5 x 36 + 4 x 6 + 5 x 1

= 180 + 24 + 5

= 209₁₀

Step 2: convert 209₁₀ to base 4

4	209	Reminders
	52	1
	13	0
	3	1
	0	3

Hence, 209₁₀ = 3101₄

So, 545₆ = 209₁₀= 3101₄

Thus, 545₆ = 3101₄

Shortcut method for converting:
Binary number to its equivalent octal number

Step 1: divide the digits into groups of three starting from the right.
Step 2: convert each group of three binary digits to one octal digit using the method of binary to decimal conversion.

Example

1101010₂ = ?₈

Step 1: divide the binary digits into groups of 3 starting from right

001 101 010

Step 2: convert each group into one octal digit

001₂ = 0 x 2² + 0 x 2¹ + 1 x 2⁰ = 1

101₂ = 1 x 2² + 0 x 2¹ + 1 x 2⁰ = 5

010₂ = 0 x 2² + 1 x 2¹ + 0 x 2⁰ = 2

Hence, 1101010₂= 152₈

Octal number to its equivalent binary number

Step 1: convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
Step 2: combine all the resulting binary groups (of 3 digits each) into a single binary number.

Example

562₈ = ?₂

Step 1: convert each octal digit to 3 binary digits

5₈= 101₂, 6₈= 110₂, 2₈= 010₂

Step 2: combine the binary groups

562₈ = 101/5 110/6 010/2

Hence, 562₈ = 101110010₂

Binary number to hexadecimal number

Step 1: divide the binary digits into groups of four starting from the right
Step 2: combine each group of four binary digits to one hexadecimal digit

Example

111101₂ = ?₁₆

Step 1: divide the binary digits into groups of four starting from the right

0011 1101

Step 2: convert each group into a hexadecimal digit

0011₂ = 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰= 3₁₀ = 3₁₆

1101₂ = 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2⁰= 3₁₀ = d₁₆

Hence, 111101₂= 3d₁₆

Hexadecimal to its equivalent binary number

Step 1: convert the decimal equivalent of each hexadecimal digit to a 4 digit binary number.
Step 2: combine all the resulting binary groups (of 4 digits each) in a single binary number.

Example

2ab₁₆ = ?₂

Step 1: convert each hexadecimal digit to a 4 digit binary number

2₁₆ = 2₁₀= 0010₂

A₁₆ = 10₁₀ = 1010₂

B₁₆ = 11₁₀ = 1011₂

Step 2: combine the binary groups

2ab₁₆ = 0010/2 1010/a 1011/b

Hence, 2ab₁₆ = 001010101011₂ractional numbers

10. FRACTIONAL NUMBERS

Fractional numbers are formed same way as decimal number system

In general, a number in a number system with base b Would be written as:

A_na_n-1… a₀. A_-1a_-2… a_-m

And would be interpreted to mean:

A_n x bⁿ + a_n-1 x b^n-1 + … + a₀ x b⁰ + a_-1 x b^-1 + a_-2 x b^-2 + … + a_-m x b^-m

The symbols a_n, a_n-1, …, a_-m in above representation should be one of the b symbols allowed in the number system

Formation of fractional & binary number system

Binary point

Position 4 3 2 1 0 . -1 -2 -3 -4

Position value

2⁴2³2²2¹2⁰2^-12^-22^-32^-4

Quantity represented

16 8 4 2 1 ^½ ¹/₄¹/₈ ¹/₁₆

Example

110.101₂ = 1 x 2² + 1 x 2¹ + 0 x 2⁰ + 1 x 2^-1 + 0 x 2^-2+ 1 x 2^-3

= 4 + 2 + 0 + 0.5 + 0 + 0.125

= 6.625₁₀

Number system (example)

Octal point

Position 3 2 1 0 . -1 -2 -3

Position value

8³8²8¹8^{0 .}8^-18^-28^-3

Quantity represented

512 64 8 1 ¹/₈¹/₆₄¹/₅₁₂

Example

127.54₈

= 1 x 8² + 2 x 8¹ + 7 x 8⁰ + 5 x 8^-1 + 4 x 8^-2

= 64 + 16 + 7 + ⁵/₈ + ⁴/₆₄

= 87 + 0.625 + 0.0625

= 87.6875₁₀

Key words/phrases

Base
Least significant digit (lsd)
Binary number system
Memory dump
Binary point
Most significant digit (msd)
Bit
Non-positional number
Decimal number system
System
Division-remainder technique
Number system
Fractional numbers
Octal number system
Hexadecimal number system
Positional number system

Questions from this unit:

What is number system?
How many types of number system?
What is non-positional number system?
What are characteristic of non-positional number system?
What are difficulties of non-positional number system?
What is positional number system?
What are characteristic of positional number system?
What are difficulties of positional number system?
How many types of representation of number system?
What decimal number system?
What is binary number system?
What is octal number system?
What is hexadecimal number system?
What is ASCII code?
What is BIT?
What is full form of bit?
What is binary digit?
What is the characteristic of binary digit?
How to convert number base?
How to convert another number base into decimal base?
How to convert decimal number base into another base?
How to convert some based to another based?
How to convert binary number to its equivalent octal number?
How to convert octal number to its equivalent binary number?
How to convert binary number to hexadecimal number?
How to convert hexadecimal to its equivalent binary number?
What is fractional numbers?
What is formation of fractional & binary number system?

2 comments

Anonymous 29 September 2021 at 11:59

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{
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printf("Enter any small letter; \n");
scanf("%d",&letter);

switch(letter)
{
case 1:
printf("Averngers. \n");
break;

case 2:
printf("Batsman. \n");
break;

case 3:
printf("chaio. \n");
break;

case 4:
printf("Dum. \n");
break;
case 5:
printf("Ester. \n");
break;

case 6:
printf("Fear. \n");
break;

case 7:
printf("Good Father. \n");
break;

case 8:
printf("Hitman. \n");
break;
case 9:
printf("Iron Man. \n");
break;

case 10:
printf("Jaguarrr. \n");
break;
default: printf("Pleas enter the number between 1 to 10!");

}

ret…
Anonymous 11 November 2021 at 10:43

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then follow the given steps:

1. open cmmand prompt at your existing java file
2. run 'javac Hello.java'
3. then run 'java Hello'

Feel free to ask your query...

IndianTechnoEra

Number System - Computer Fundamental - IndianTechnoEra

NUMBER SYSTEM

1. NON-POSITIONAL NUMBER SYSTEMS

2. POSITIONAL NUMBER SYSTEMS

DECIMAL NUMBER SYSTEM

BIT (Binary digiT)

OCTAL NUMBER SYSTEMS

HEXADECIMAL NUMBER SYSTEM

CONVERT A NUMBER'S BASE:

Another base number to decimal base

Decimal base to another base

Some based to another based

Shortcut method for converting:
Binary number to its equivalent octal number

Octal number to its equivalent binary number

Binary number to hexadecimal number

Hexadecimal to its equivalent binary number

10. FRACTIONAL NUMBERS

Formation of fractional & binary number system

Key words/phrases

Questions from this unit:

2 comments

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Number System - Computer Fundamental - IndianTechnoEra

NUMBER SYSTEM

1. NON-POSITIONAL NUMBER SYSTEMS

2. POSITIONAL NUMBER SYSTEMS

DECIMAL NUMBER SYSTEM

BIT (Binary digiT)

OCTAL NUMBER SYSTEMS

HEXADECIMAL NUMBER SYSTEM

CONVERT A NUMBER'S BASE:

Another base number to decimal base

Decimal base to another base

Some based to another based

Shortcut method for converting: Binary number to its equivalent octal number

Octal number to its equivalent binary number

Binary number to hexadecimal number

Hexadecimal to its equivalent binary number

10. FRACTIONAL NUMBERS

Formation of fractional & binary number system

Key words/phrases

Questions from this unit:

2 comments

Shortcut method for converting:
Binary number to its equivalent octal number