Base Conversion of Decimal Numbers

Convert decimal 7562.45 to octal

Integer part/8	Quotient	Remainder
7562/8	945	2 (LSB)
945/8	118	1
118/8	14	6
14/8	1	6
1/8	0	1 (MSB)

                                    (7562)₁₀ = (16612)₈

	Integer	Remainder
.45 X 8 =	3(MSB)	.6
.6 X 8 =	4	.8
.8 X 8 =	6	.4
.4 X 8 =	3	.2
.2 X 8 =	1	.6
.6 X 8	4(LSB)	.8

                               (.45)₁₀ = (346314..)₈

(7562.45)₁₀ = (16612.346314..)₈

  

Convert decimal 4562.25 to hexadecimal

Integer part/16	Quotient	Remainder
4562/16	285	2 (LSB)
285/16	17	13
17/16	1	1
1/16	0	1 (MSB)

                                    (4562)₁₀ = (2D11)₁₆

	Integer	Remainder
.25 X 16 =	4	.0

                               (.25)₁₀ = (4)₁₆

(4652.25)₁₀ = (2D11.4)₁₆

Convert decimal 175.175 to binary

Integer part/2	Quotient	Remainder
175/2	87	1 (LSB)
87/2	48	1
48/2	24	0
24/2	12	0
12/2	6	0
6/2	3	0
3/2	1	1
1/2	0	1 (MSB)
0

                                    (175)₁₀ = (11000011)₂

	Integer	Remainder
.175 X 2 =	0(MSB)	.35
.35 X 2	0	.7
.7 X 2 =	1	.4
.4 X 2 =	0	.8
.8 X 2 =	1	.6
.6 X 2	1	.2
.2 X 2	0	.4
.4 X 2	0	.8

                               (.175)₁₀ = (00101100..)₂

(175.175)₁₀ = (11000011.00101100..)₂

Base Conversion of Decimal Fraction Part

Decimal to Binary Conversion 

Base 10 -> Base 2

(.234375)₁₀ = (?)₂

	Integer	Remainder
.234375 X 2 =	0 (MSB)	.46875
.46875 X 2 =	0	.9375
.9375 X 2 =	1	.875
.875 X 2 =	1	.75
.75 X 2 =	1	.5
.5 X 2 =	1 (LSB)	0

                                                                                               Answer: (001111)₂

Decimal to Octal Conversion

Base 10 -> Base 8

(.513)₁₀ = (?)₈

	Integer	Remainder
.513 X 8 =	4(MSB)	.104
.104 X 8 =	0	.832
.832 X 8 =	6	.656
.656 X 8 =	5	.248
.248 X 8 =	1	.984
.984 X 8	7(LSB)	.872

                                                                              Answer: (406517....)₈

Decimal to Hexadecimal Conversion

Base 10 -> Base 16

(.257)₁₀ = (?)₁₆

	Integer	Remainder
.257 X 16 =	4(MSB)	.112
.112 X 16 =	1	.792
.792 X 16 =	12 -> C	.672
.656 X 8 =	10 -> A(LSB)	.752

                                                                                             Answer: (41CA...)₁₆

To convert a decimal fraction part to any base-r system, multiplication is done by r.

Base Conversion of Decimal Integer Part

Decimal to Binary Conversion 

Base 10 -> Base 2

(51)₁₀ = (?)₂

Integer part/2	Quotient	Remainder
51/2	25	1 (LSB)
25/2	12	1
12/2	6	0
6/2	3	0
3/2	1	1
1/2	0	1 (MSB)

                                                                                               Answer: (110011)₂

Decimal to Octal Conversion

Base 10 -> Base 8

(151)₁₀ = (?)₈

Integer part/8	Quotient	Remainder
151/8	18	7 (LSB)
18/8	2	2
2/8	0	2 (MSB)

                                                                                               Answer: (227)₈

Decimal to Hexadecimal Conversion

Base 10 -> Base 16

(1938)₁₀ = (?)₁₆

Integer part/16	Quotient	Remainder
1938/16	121	2 (LSB)
121/16	7	9
7/16	0	7 (MSB)

                                                                                               Answer: (792)₁₆

To convert a decimal integer part to any base-r system, division is done by r.

Number Base Conversions

For a number expressed in the form

Base conversion can be divided into two parts:

1. Base conversion of Integer part,
2. Base conversion of Fraction part

Number System - Base r

General Number in base r is expressed as:

where,

• Allowed values of a_i (digits) are 0, 1, 2, ......., r-1 ONLY
• a_n is the MSD (Most Significant Digit)
• a_-m is the LSD (Least Significant Digit)

A number with base r with form/shape as

is equal to

 Example:

(1234.5)₁₀ = 1X10³ + 2X10² + 3X10³ + 4X10⁴ + 5X10^-1

Table 1 : Numbers with Different Bases

 

Note: Numbers in base r can use only r allowable digits. 

Example:

• 10 allowable digits (0, 1, 2, ....., 9) for decimal numbers (base 10)
• 2 allowable digits (0 and 1) for binary numbers (base 2)
• 8 allowable digits (0, 1, 2, ......., 7) for octal numbers (base 8)
• 16 allowable digits (0, 1, 2, ......, 9, A, B, ....., F) for hexadecimal numbers (base 16)

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