Number System

The technique to represent and work with numbers in computer is called number system. In computer, there are few types of number system used like Binary number, Decimal number, octal number, hexadecimal number system. In these decimal number system is the most common number system. We use decimal number system in our day to day life whereas, a computer convert all kind of data and information in binary number system. The total number of digits used in a number system is called its Base or Radix.

Decimal Number System :-

Decimal number system is a base 10 number system having 10 digits from 0 to 9. This means that any numerical quantity can be represented using these 10 digit digits. Digital number system is is also a positional value system. This means that the value of digits will depend upon its position for example :- 734, 971 and 207. In these three numbers the value of 7 is different.

1) In 734, the value of 7 is 700 or 7 into hundred or 7*10².

2) In 971, the value of 7 is 70 or 7*10 or 7*10¹.

3) In 207, the value of 7 is 7 units or 7 or 7*1 or 7 * 10⁰

The weighted of each position can be represented as follows

10⁵

10⁴

10³

10²

10¹

10⁰

In distance systems, instruction are given through electric signals and variation is done by wearing the voltage of the signal. Having 10 different voltage to implement decimal number system in digital equipment is difficult. So many number system that are easier to implement digitally have been developed. Let’s have a look in detail.

Binary Number System:-

The easiest way to vary instruction through electric signal is two state system on and off. On represent 1 and off represent zero is not actually no signal but 1 signal at a low voltage. The number system having just these two digits 0 and 1 is called binary number system.

Each band in digits is also called a bit. Bani number system is also potential value system where each digit has a value expressed in the power of 2 as follows.

2⁵

2⁴

2³

2²

2¹

2⁰

In any violin number, the right most digit is called least significant bit (LSB) and the left most digit is called most significant bit (MSB).

1 (MSB)

0 (LSB)

MSB LSB

And decimal equivalence of this number is the sum of product of each digit with its potential value,

11010₂= 1 * 2⁴+ 1* 2³ + 1 * 2² + 1 * 2¹ + 1 * 2⁰

Octal Number System:-

^{Octal number systems has eight digit 0,1,2,3,4,5,6 and 7. Octal
number system is also a positional value system, where each digit has its value
expressed in the powers of 8. As given}

8⁵

8⁴

8³

8²

8¹

8⁰

^{Decimal equivalence of any octal number is sum of product of
each digit with its potential value.}

345₈ = 3 *8²+ 4 * 8¹ + 5 * 8⁰

= 3 * 64 + 4 * 8 + 5 *1

= 192 + 32 +5

= 229₁₀

Hexadecimal Number System:-

_{Hexadecimal number system has 16 symbols - 0 to 9 and A to}where E is equal to 10B is equal to eleven and so on till F please stop hexadecimal number system is also a potential value system with where each digit has its value expressed in the power of 16 as given

16⁵

16⁴

16³

16²

16¹

16⁰

_{Decimal equivalence of any hexadecimal number is sum of
product of each digit with its potential value.}

18AD₁₆= 1 *16³+ 8 *16²+ A *16¹+ D*16⁰

Where A is equal to 10 and D is equal to 13

= 1 * 4096 + 8 * 256 + 10 * 16 + 13*1

= 4096 + 2048 + 160 + 13

= 6317

Conversion of decimal number to other number system

1) 1) Convert decimal number to binary number system :-

Decimal number can be converted to by repeated division of the number by two while recording the remainder. for example :-

= 43 / 2 = 21 quotient and remainder is 1

= 21/ 2 = 10 quotient and remainder is 1

= 10/2 = 5 quotient and remainder is 0

= 5/2 = 2 quotient and remainder is 1

= 2/2 = 1 quotient and remainder is 0

Now the reminders are to be read read from bottom to top to obtain the equivalent.

43₁₀ = 101011₂

1) 2) Convert decimal number to octal number system:-

Decimal number can be converted to octal by repeated division of the number by 8 while recording the remainders. for example:-

=843 / 8 = 105 quotient and remainder is 3

= 105/ 8 = 13 quotient and remainder is 1

= 13/8 = 1 quotient and remainder is 5

Now the reminders are to be read read from bottom to top to obtain the equivalent.

843₁₀ = 513₈

Conversion of binary number to other number system

1) Convert binary number to Octal number system :-

Let's use the example from the image:

Binary number:

110101101.01101₂

Step 1: Divide the binary digits into groups of 3

Start grouping from the binary point.

For the whole-number part, move from right to left:

110 101 101

For the fractional part, move from left to right:

011 010

So we get:

110 101 101 . 011 010

If the first group does not have 3 digits, add 0s to the left.
If the last fractional group does not have 3 digits, add 0s to the right.

Step 2: Convert each group of 3 bits

Use this simple table:

Binary	Octal
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

Now convert:

110 → 6

101 → 5

After the decimal point:

011 → 3

010 → 2

Therefore:

110 101 101 . 011 010

becomes

6 5 5 . 3 2

Final answer:

(110101101.01101)₂ = (655.32)₈

Easy rule to remember:

Binary → Octal → Groups of 3

110 | 101 | 101 . 011 | 010

↓ ↓ ↓ ↓ ↓

6 5 5 3 2

Answer = 655.32₈

2) Convert binary number to decimal number system :-

_{Step-by-Step Method}

_{Write
down the binary number.}

_{Starting
from the right (least significant bit), assign each digit a positional
value based on powers of 2.}

_{Multiply
each binary digit by its corresponding power of 2.}

_{Sum
all the results to get the decimal equivalent.}
The easiest rule is:

Binary → Decimal = multiply each binary digit by its power of 2, then add the results for example

Convert `(1011)₂` into decimal.

Step 1: Write the powers of 2

Start from the right side with $2^0$ , then increase the power as you move left.


Binary:     1     0     1     1
            ↓     ↓     ↓     ↓ 
Power:     2³     2²    2¹    2⁰

Now calculate the powers:

2³ = 8

2² = 4

2¹ = 2

2⁰ = 1

Binary: 1 0 1 1

Value: 8 4 2 1

Step 2: Multiply each binary digit by its place value

1 × 8 = 8

0 × 4 = 0

1 × 2 = 2

1 × 1 = 1

Step 3: Add the results

8 + 0 + 2 + 1 = 11

Therefore:

(1011)₂ = (11)₁₀

3) Convert binary number to Hexadecimal number system :-
Now let's look at the second image.

The example is:

1011011101101₂

For hexadecimal, make groups of 4 bits, starting from the right.

We have:

1011011101101

Starting from the right:

1 | 0110 | 1110 | 1101

The first group has only one digit.

So we add zeros to the left:

0001 | 0110 | 1110 | 1101

Step 2: Convert each group of 4 bits

Here is the important table:

Binary Hexadecimal
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

Now convert each group:

0001 → 1

0110 → 6

1110 → E

1101 → D

So:

0001 | 0110 | 1110 | 1101

becomes:

1 | 6 | E | D

Final answer:

(1011011101101)₂ = (16ED)₁₆

Why do we use 3 and 4?

This is the most important concept for students.

Binary → Octal

Octal has 8 possible values:
0, 1, 2, 3, 4, 5, 6, 7
And:
2³ = 8
Therefore:
3 binary bits = 1 octal digit
Example:
101₂ = 5₈

Binary → Hexadecimal

Hexadecimal has 16 possible values:
0–9, A, B, C, D, E, F
And:
2⁴ = 16
Therefore:
4 binary bits = 1 hexadecimal digit
Example:
1101₂ = D₁₆

🧠 Super-Easy Memory Trick

Tell students to remember this:

OCTAL = 3

Octal → 3 bits

HEXA = 4

Hexadecimal → 4 bits

So:


             BINARY
                │
        ┌───────┴────────┐
        ↓                ↓
      OCTAL          HEXADECIMAL
        │                │
     Groups of 3      Groups of 4
        │                │
        ↓                ↓
     0 – 7           0 – 9, A – F

One Important Difference

For octal, students only need to remember:

000–111 → 0–7

For hexadecimal, after 9 we use letters:


10 = A
11 = B
12 = C
13 = D
14 = E
15 = F

So students should not get confused when they see letters in a hexadecimal answer.

For example:

1010₂ = 10₁₀ = A₁₆

and

1111₂ = 15₁₀ = F₁₆

🔥 Compare All Three Methods

Conversion What to do
Binary → Decimal Use powers of 2 and add
Binary → Octal Group binary digits in 3s
Binary → Hexadecimal Group binary digits in 4s

Final shortcut

Conversion Group binary into Example
Binary → Octal 3 bits `101 → 5`
Binary → Hexadecimal 4 bits `1101 → D`

:

Number System - Computer

Number System

Conversion of binary number to other number system

Step 1: Divide the binary digits into groups of 3

Step 2: Convert each group of 3 bits

Final answer:

Easy rule to remember:

Convert `(1011)₂` into decimal.

Step 1: Write the powers of 2

(1011)₂ = (11)₁₀

3) Convert binary number to Hexadecimal number system :-
Now let's look at the second image.

The example is:

1011011101101₂

For hexadecimal, make groups of 4 bits, starting from the right.

We have:

1011011101101

Starting from the right:

1 | 0110 | 1110 | 1101

The first group has only one digit.

So we add zeros to the left:

0001 | 0110 | 1110 | 1101

Step 2: Convert each group of 4 bits

Here is the important table:

Binary Hexadecimal
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

Now convert each group:

0001 → 1

0110 → 6

1110 → E

1101 → D

So:

0001 | 0110 | 1110 | 1101

becomes:

1 | 6 | E | D

Final answer:

(1011011101101)₂ = (16ED)₁₆

Why do we use 3 and 4?

This is the most important concept for students.

Binary → Octal

Octal has 8 possible values:
0, 1, 2, 3, 4, 5, 6, 7
And:
2³ = 8
Therefore:
3 binary bits = 1 octal digit
Example:
101₂ = 5₈

Binary → Hexadecimal

Hexadecimal has 16 possible values:
0–9, A, B, C, D, E, F
And:
2⁴ = 16
Therefore:
4 binary bits = 1 hexadecimal digit
Example:
1101₂ = D₁₆

🧠 Super-Easy Memory Trick

Tell students to remember this:

OCTAL = 3

Octal → 3 bits

HEXA = 4

Hexadecimal → 4 bits
So:
`BINARY │ ┌───────┴────────┐ ↓ ↓ OCTAL HEXADECIMAL │ │ Groups of 3 Groups of 4 │ │ ↓ ↓ 0 – 7 0 – 9, A – F`

One Important Difference

🔥 Compare All Three Methods

Conversion What to do
Binary → Decimal Use powers of 2 and add
Binary → Octal Group binary digits in 3s
Binary → Hexadecimal Group binary digits in 4s

Final shortcut

Conversion Group binary into Example
Binary → Octal 3 bits `101 → 5`
Binary → Hexadecimal 4 bits `1101 → D`

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Number System - Computer

Number System

Conversion of binary number to other number system

Step 1: Divide the binary digits into groups of 3

Step 2: Convert each group of 3 bits

Final answer:

Easy rule to remember:

Convert (1011)₂ into decimal.

Step 1: Write the powers of 2

(1011)₂ = (11)₁₀

3) Convert binary number to Hexadecimal number system :-Now let's look at the second image. The example is: 1011011101101₂

For hexadecimal, make groups of 4 bits, starting from the right. We have: 1011011101101 Starting from the right: 1 | 0110 | 1110 | 1101 The first group has only one digit. So we add zeros to the left: 0001 | 0110 | 1110 | 1101

Step 2: Convert each group of 4 bits

Here is the important table: BinaryHexadecimal000000001100102001130100401015011060111710008100191010A1011B1100C1101D1110E1111F Now convert each group: 0001 → 1 0110 → 6 1110 → E 1101 → D So: 0001 | 0110 | 1110 | 1101 becomes: 1 | 6 | E | D

Final answer:

(1011011101101)₂ = (16ED)₁₆

Why do we use 3 and 4?

This is the most important concept for students.

Binary → Octal

Octal has 8 possible values:0, 1, 2, 3, 4, 5, 6, 7And:2³ = 8Therefore:3 binary bits = 1 octal digitExample:101₂ = 5₈

Binary → Hexadecimal

Hexadecimal has 16 possible values:0–9, A, B, C, D, E, FAnd:2⁴ = 16Therefore:4 binary bits = 1 hexadecimal digitExample:1101₂ = D₁₆

🧠 Super-Easy Memory Trick

Tell students to remember this:

OCTAL = 3

Octal → 3 bits

HEXA = 4

Hexadecimal → 4 bitsSo: BINARY │ ┌───────┴────────┐ ↓ ↓ OCTAL HEXADECIMAL │ │ Groups of 3 Groups of 4 │ │ ↓ ↓ 0 – 7 0 – 9, A – F

One Important Difference

For octal, students only need to remember:000–111 → 0–7For hexadecimal, after 9 we use letters:10 = A 11 = B 12 = C 13 = D 14 = E 15 = FSo students should not get confused when they see letters in a hexadecimal answer.For example:1010₂ = 10₁₀ = A₁₆and1111₂ = 15₁₀ = F₁₆

🔥 Compare All Three Methods

Conversion What to do Binary → Decimal Use powers of 2 and add Binary → OctalGroup binary digits in 3s Binary → HexadecimalGroup binary digits in 4s

Final shortcut

ConversionGroup binary intoExampleBinary → Octal3 bits101 → 5Binary → Hexadecimal4 bits1101 → D

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Convert `(1011)₂` into decimal.

3) Convert binary number to Hexadecimal number system :-
Now let's look at the second image.

The example is:

1011011101101₂

For hexadecimal, make groups of 4 bits, starting from the right.

We have:

1011011101101

Starting from the right:

1 | 0110 | 1110 | 1101

The first group has only one digit.

So we add zeros to the left:

0001 | 0110 | 1110 | 1101

Here is the important table:

Binary Hexadecimal
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F

Now convert each group:

0001 → 1

0110 → 6

1110 → E

1101 → D

So:

0001 | 0110 | 1110 | 1101

becomes:

1 | 6 | E | D

Octal has 8 possible values:
0, 1, 2, 3, 4, 5, 6, 7
And:
2³ = 8
Therefore:
3 binary bits = 1 octal digit
Example:
101₂ = 5₈

Hexadecimal has 16 possible values:
0–9, A, B, C, D, E, F
And:
2⁴ = 16
Therefore:
4 binary bits = 1 hexadecimal digit
Example:
1101₂ = D₁₆

Hexadecimal → 4 bits
So:
`BINARY │ ┌───────┴────────┐ ↓ ↓ OCTAL HEXADECIMAL │ │ Groups of 3 Groups of 4 │ │ ↓ ↓ 0 – 7 0 – 9, A – F`

Conversion What to do
Binary → Decimal Use powers of 2 and add
Binary → Octal Group binary digits in 3s
Binary → Hexadecimal Group binary digits in 4s

Conversion Group binary into Example
Binary → Octal 3 bits `101 → 5`
Binary → Hexadecimal 4 bits `1101 → D`