Number base convertion ( Tutorial )

Decimal To binary
Question 1: Convert 112 from decimal to binary.

112 = 64 + 32 + 16 = 1×26 + 1×25 + 1×24 + 0×23 + 0×22 + 0×21 + 0×20 = 11100002

Question 2: Convert 25673 from decimal to binary.


25673 = 16384 + 8192 + 1024 + 64 + 8 + 1 = 1×214 + 1×213 + 0×212 + 0×211 + 1×210 + 0×29 + 0×28 + 0×27 + 1×26 + 0×25 + 0×24 + 1×23 + 0×22 + 0×21 + 1×20 = 1100100010010012

Question 3: Convert 555 from decimal to binary.

555 = 512 + 32 + 8 + 2 + 1 = 1×29 + 0×28 + 0×27 + 0×26 + 1×25 + 0×24 + 1×23 + 0×22 + 1×21 + 1×20 = 10001010112

Question 4: Convert 100 from decimal to binary.

100 = 64 + 32 + 4 = 1×26 + 1×25 + 0×24 + 0×23 + 1×22 + 0×21 + 0×20 = 11001002

Question 5: Convert 12999 from decimal to binary.

12999 = 8192 + 1024 + 64 + 8 + 1 = 1×213 + 1×212 + 0×211 + 0×210 + 1×29 + 0×28 + 1×27 + 1×26 + 0×25 + 0×24 + 0×23 + 1×22 + 1×21 + 1×20 = 110010110001112


Decimal To Octal 

Question 1: Convert 128 to octal form.

Solution:
Given decimal number is 128
Start the division process
8 | 128
8 | 16 -- 0
8 | 2 -- 0
8 | 0 --2

Correct answer is The equivalent octal number for 128 is 2008

Question 2: Convert 58 to octal form
Solution:
Given decimal number is 58
Start the division process
8 | 58
8 | 7 -- 2
8 | 0 --7

Correct answer is the equivalent octal number for 58 is 728

Question 3: Convert 451 to octal form

Solution:
Given decimal number is 451
Start the division process
8 | 451
8 | 56 -- 3
8 | 7 -- 0
8 | 0 -- 7

Correct answer is the equivalent octal number for 128 is 7038

Question 4: Convert 4 to octal form?

Solution:
Given decimal number is 4
Since 4 is less than 7

Correct answer is the equivalent octal number for 4 is 48

Question 5: Convert 287 to octal form.

Solution:
Given decimal number is 287
Start the division process
8 | 287
8 | 35 -- 7
8 | 4 -- 3
8 | 0 --4

Correct answer is the equivalent octal number for 287 is 4378


Decimal To Hexadecimal

Question 1: Convert 56 from decimal to hexadecimal.

5610 = 1×25 + 1×24 + 1×23 + 0×22 + 0×21 + 0×20 = 1110002 = 0011 10002 = 38H

Question 2: Convert 273 from decimal to hexadecimal.

27310 = 1×28 + 0×27 + 0×26 + 0×25 + 1×24 + 0×23 + 0×22 + 0×21 + 1×20 = 1000100012 = 0001 0001 00012 = 111H

Question 3: Convert 105 from decimal to hexadecimal.

10510 = 1×26 + 1×25 + 0×24 + 1×23 + 0×22 + 0×21 + 1×20 = 11010012 = 0110 10012 = 69H

Question 4: Convert 158 from decimal to hexadecimal.

15810 = 1×27 + 0×26 + 0×25 + 1×24 + 1×23 + 1×22 + 1×21 + 0×20 = 100111102 = 1001 11102 = 9EH

Question 5: Convert 171 from decimal to hexadecimal.

17110 = 1×27 + 0×26 + 1×25 + 0×24 + 1×23 + 0×22 + 1×21 + 1×20 = 101010112 = 1010 10112 = ABH



Binary To  Decimal

Convert the following binary numbers to decimal:
a) 101110
b) 1110101

Solution:
Binary numbers: base (radix) = 2
a) 1*25 + 0*2 + 1*23 + 1*2 + 1*2 + 0*20 = 32 + 0 + 8 + 4 + 2 + 0 = 46
b) 1*26+1*25+1*24+0*23+1*22+0*21+1*2= 64 + 32 + 16 + 0 + 4 + 0 + 1 = 117

Convert the binary numbers into decimal numbers
1. 11
2. 101
3. 1111
4. 110111011
5. 1111100011110011

Solution:
1. 11 = 1*21 + 1*20 = 2 + 1 = 3
2. 101 = 1*22 + 0*21 + 1*20 = 4 + 1 = 5
3. 1111 = 1*23 + 1*22 + 1*21 + 1*20 = 8 + 4 + 2 + 1 = 15
4. 110111011 = 1*28 + 1*27 + 0*26 + 1*25 + 1*24 + 1*23 + 0*22 + 1*21 + 1*20 = 256 + 128 + 32 + 16 + 8 + 2 + 1 = 443
5. 1111100011110011 = 1*215 + 1*214 + 1*213 + 1*212 + 1*211 + 0*210 + 0*29 + 0*28 + 1*27 + 1*26 + 1*25 + 1*24 + 0*23 + 0*22 + 1*21 + 1*20
= 32768 + 16384 + 8192 + 4096 + 2048 + 128 + 64 + 32 + 16 + 2 + 1 = 63731

Binary to Octal

Convert the binary number 10101012 to octal

Solution:
Given binary number is 10101012
First we convert given binary to decimal

10101012 = (1 * 26) + (0 * 25 ) + (1 * 24) + (0 * 23) + (1 * 22) + (0 * 21) + (1 * 20)
= 64 + 0 + 16 + 0 + 4 + 0 + 1
= 64 + 21
= 85  (Decimal form)

Now we will convert this decimal to octal form
8 | 85
-------
8 | 10 -- 5
-------
8 | 1 -- 2
-------
8 | 0 --1

Correct answer is equivalent octal form is 1258



Binary to Hexadecimal

Question 1: Convert 10010001 from unsigned binary to hexadecimal.

10012 = 910 and 00012 = 110.
100100012 = 9110

Question 2: Convert 101010 from unsigned binary to hexadecimal.

102 = 210 and 10102 = A10.
1010102 = 2A10

Question 3: Convert 100110101 from unsigned binary to hexadecimal.

12 = 110, 00112 = 310 and 01012 = 510.
1001101012 = 13510

Question 4: Convert 1010101001 from unsigned binary to hexadecimal.

102 = 210, 10102 = A10 and 10012 = 910.
10101010012 = 2A910

Question 5: Convert 1100111 from unsigned binary to hexadecimal.

1102 = 610 and 01112 = 710.
11001112 = 6710


Octal To Binary

Question 1: Convert 418 to a binary number.

Solution:

Given number is 418
418 = (4 * 81) + (1 * 80)
= 4 * 8 + 1 * 1
= 32+1
= 33(Decimal number)

Now convert this decimal number to a binary number.

2 | 33
2 | 16 -- 1
2 | 8  -- 0
2 | 4  --0
2 | 2  -- 0
      1 -- 0

The binary number is 1000012

Question 2: Convert 278 to a binary number.

Solution:

Given number is 278
278 = (2 * 81) + (7 * 80)
= 2 * 8 + 7 * 1
= 16 + 7
= 23(Decimal number)

Now convert this decimal number to a binary number.

2 | 23
2 | 11 -- 1
2 | 5   -- 1
2 | 2   --1
2 | 1   -- 0
2 | 0   -- 1

The binary number is 101112

Question 3: Convert 108 to a binary number.
Solution:

Given number is 108
108 = (1 * 81) + (0 * 80)
= 1 * 8 + 0 * 1
= 8 + 0
= 8(Decimal number)

Now convert this decimal number to a binary number.

2 | 8 -- 0
2 | 4 --0
2 | 2 -- 0
     1

The binary number is 10002



Octal To Decimal


Question 1: Convert 57468 to decimal number

Solution:
The given number is 57468

57468 = (5 * 8^3)+ (7 * 8^2) + (4 * 8^1) + (6 * 8^0)
=5 * 512 + 7 * 64 + 4 * 8 + 6 * 1
= 2560+448+32+6
= 3046

The equivalent decimal number for 57468 is 3046

Question 2: Convert 2018 to decimal number

Solution:
The given number is 2018

2018 = (2 * 8^2) + (0 * 8^1) + (1 * 8^0)
= 2 * 64 + 0 * 8 + 1 * 1
= 128+0+1
= 129

The equivalent decimal number for 2018 is 129


Question 3: Convert 548 to decimal number

Solution:

The given number is 548
548 = (5 * 8^1) + (4 * 8^0)
=5 * 8 + 4 * 1
= 40 + 4
= 44

The equivalent decimal number for 548 is 44



Question 4: Convert 108 to decimal number

Solution:
The given number is 108

108 = (1 * 8^1) + (0 * 8^0)
= 1 * 8 + 0 * 1
= 8+0
= 8

The equivalent decimal number for 108 is 8



Octal To Hexadecimal

Question 1: Convert 10028 to hexadecimal

Solution:
The given number is 10028

10028 = (1 * 83)+ (0 * 82) + (0 * 81) + (2 * 80)
=1 * 512 + 0 * 64 + 0 * 8 + 2 * 1
= 512 + 0 + 0+ 2
= 514(decimal number)

Now we convert the above decimal to hexadecimal

16 | 514
16 | 32   --2
        2 -- 0

The hexadecimal number is 202

Question 2: Convert 1008 to hexadecimal

Solution:
The given number is 1008

10028 = (1 * 82) + (0 * 81) + (0 * 80)
= 1 * 64 + 0 * 8 + 0 * 1
= 64 + 0 + 0
= 64(decimal number)

Now we convert the above decimal to hexadecimal

16 | 64
16 | 4   --0
16 | 0   -- 4

The hexadecimal number is 40

Question 3: Convert 158 to hexadecimal

Solution:
The given number is 158

158 =(1 * 81) + (5 * 80)
= 1 * 8 + 5 * 1
= 8 + 5
= 13(decimal number)

Now we convert the above decimal to hexadecimal
13 is less than 16 so the equivalent hexadecimal number is D
The hexadecimal number is D

158 = D16


Hexadecimal To Decimal

Question 1: Convert 35 from hexadecimal to decimal.

3×16 + 5 = 53.

Question 2: Convert 1AA from hexadecimal to decimal.

1×16^2 + 10×16 + 10 = 426.

Question 3: Convert 12 from hexadecimal to decimal.

1×16 + 2 = 18.

Question 4: Convert 4B from hexadecimal to decimal.

4×16 + B = 75.

Question 5: Convert 89 from hexadecimal to decimal.

8×16 + 9 = 137.


Hexadecimal To Octal



Example: Convert the Hexadecimal (B 5 A)₁₆ to its octal equivalent


The equivalent octal number is (5 5 3 2)₈



Octal To Hexadecimal

Example: Convert octal number (7 5 2)₈ to its equivalent hex number


The hex number equivalent is (1 D 9)₈

Conversion Table

Decimal	Binary	Octal	Hexadecimal
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F