Number System Conversion

CHAPTER 2: NUMBER SYSTEM CONVERSION

The binary number system is a positional notation numbering system, but in this case, the base is not ten, but is instead two. Each digit position in a binary number represents by power of two. So, when we write a binary number, each binary digit is multiplied by an appropriate power of 2 based on the position in the number.

In the decimal number system, there are ten possible values that can appear in each digit position, and so there are ten numerals required to represent the quantity in each digit position. The decimal numerals are the familiar zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

In addition to binary, another number base that is commonly used in digital systems is base 16. This number system is called hexadecimal, and each digit position represents a power of 16. For

any number base greater than ten, a problem occurs because there are more than ten symbols needed to represent the numerals for that number base. It is customary in these cases to use the ten decimal numerals followed by the letters of the alphabet beginning with A to provide the needed numerals.(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)

In this section, the number system conversion between decimal, binary and hexadecimal number will be shown out.

Decimal	Binary	Hexadecimal
0	0	0
1	1	1
2	10	2
3	11	3
4	100	4
5	101	5
6	110	6
7	111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

                                                Table 1: Number system conversion

1.1 Conversion from decimal number to binary number.

Example: Convert decimal number 101.6875 to binary number.

Weight	26	25	24	23	22	21	20	2-1	2-2	2-3	2-4
Value Represented	64	32	16	8	4	2	1	0.5	0.25	0.125	0.0625
Binary	1	1	0	0	1	0	1	1	0	1	1

For the non-decimal point,

101-64 = 37  →  37-32 = 5  →  5-4 = 1  →  1-1 = 0

For the decimal point,

0.6875-0.5 = 0.1875  →  0.1875-0.125 = 0.0625  →  0.0625-0.0625 = 0

Then, fill in the binary row with 1 if we have used the value represented and fill in binary column with 0 if we did not use corresponding value represented.

Therefore, 101.6875₁₀ = 1100101.1011₂

*Note: Take the closer number of value represented to do subtraction.

1.2 Conversion from binary number to decimal number.

For example: Convert 1010110₂ to decimal.

1 0 1 0 1 1 0₂

  \    \     \   \________1 x 2¹ = 2

\    \     \_________1 x 2² = 4

  \    \___________1 x  2⁴ = 16

       \_____________1 x 2⁶ = 64

Therefore, 1010110₂ = (2+4+16+64) = 86₁₀

2.1 Conversion from decimal to hexadecimal number.

Example: Convert 3456.125₁₀ to hexadecimal number.

Weight	163	162	161	160	16-1	16-2
Value Represented	4096	256	16	1	0.0625	0.00390625
Hexadecimal		D(13)	8	0	2

For the non-decimal point,

3456 – (256 x 13) = 3456 -3328    →    128 – (16 x 8) = 128 - 128

                              = 128                                            = 0

For the decimal point,

0.125 – (0.0625 x 2) = 0

Therefore, 3456.125₁₀ = D80.2  (Refer to Table 1, D = 13 in hexadecimal number.)

*Note: Take the closer number of value represented to do subtraction.

2.2 Conversion from hexadecimal number to decimal number.

Example: Convert E23C₁₆ to decimal number.

E23C₁₆ = (14 x 16³)  + (2 x 16²) + (3 x 16¹) + (12x16⁰)

             = 57344+512+48+12

             = 57916₁₀

Refer to Table 1, E=14₁₀ and C=12₁₀ in hexadecimal number system.

Therefore, E23C₁₆ = 57916₁₀

3.1 Conversion from hexadecimal number to binary number.

Example: Convert the hexadecimal number 543F into binary number.

  5    4    3    F

 0101 0100 0011 1111

Therefore, 543F = 0101010000111111₂

Note: In this case, divide the hex number into 4 digits and convert it into binary number according to the Table 1.

3.2 Conversion from binary number to hexadecimal number.

Example: Convert 101101110001₂ into hexadecimal number.

  1011 0111 0001

   B    7    1

Therefore, 101101110001₂ = B71₁₆

Note: Divide the binary number into few parts and each part contain 4 numbers only and convert it into hex number (refer to the Table 1).

The reason for the common use of hexadecimal numbers is the relationship between the numbers 2 and 16. Sixteen is a power of 2 (16 = 2⁴). Because of this relationship, four digits in a binary number can be represented with a single hexadecimal digit. This makes conversion between binary and hexadecimal numbers very easy, and hexadecimal can be used to write large binary numbers with much fewer digits.

 Furthermore, working with large digital systems, such as computers, it is common to find binary numbers with 8, 16 and even 32 digits.

Last but not least, the numbers can be written with fewer digits and much less likelihood of error by using hexadecimal number.

Written by: Sim Fu Cheng (B031210069)