Converting Between Denary & Hexadecimal (OCR GCSE Computer Science)

What is hexadecimal?

• Hexadecimal is a number system that is made up of 16 digits, 10 numbers (0-9) and 6 letters (A-F)

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

• Hexadecimal is referred to as a Base-16 number system
• Each digit has a weight factor of 16 raised to a power, the rightmost digit is 1s (16^0), the next digit to the left 16s (16^1)
• In GCSE you are required to work with up to and including 2 digit hexadecimal values

16s	1s
1	3
1 x16	3 x 1	= 19

• A quick comparison table demonstrates a relationship between hexadecimal and a binary nibble 
• One hexadecimal digit can represent four bits of binary data

Denary	Binary	Hexadecimal
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	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

Why is hexadecimal used?

• In Computer Science hexadecimal is often preferred when working with large values
• It takes fewer digits to represent a given value in hexadecimal than in binary
• It is beneficial to use hexadecimal over binary because:
  • The more bits there are in a binary number, the harder it is to read
  • Numbers with more bits are more prone to errors when being copied
• Examples of where hexadecimal can be seen:
  • MAC addresses

• • Colour values

How do you convert denary to hexadecimal?

Method 1 (denary to binary to hexadecimal) 

• To convert the denary number 28 to hexadecimal, start by converting the denary number to binary

128	64	32	16	8	4	2	1
0	0	0	1	1	1	0	0

• Split the 8 bit binary number into two nibbles as shown below

8	4	2	1		8	4	2	1
0	0	0	1		1	1	0	0

• Convert each nibble to its denary value
• 0001 = 1 and 1100 = 12
• Using the comparison table, the denary value 1 is also 1 in hexadecimal whereas denary value 12 is represented in hexadecimal as C
• Denary 28 is 1C in hexadecimal

Method 2 (divide by 16)

• To convert the denary number 163 to hexadecimal, start by dividing the denary value by 16 and recording the whole times the number goes in and the remainder
• 163 ➗16 = 10 remainder 3
• In hexadecimal the whole number = digit 1 and the remainder = digit 2
• Digit 1 = 10 (A)
• Digit 2 = 3
• Denary 163 is A3 in hexadecimal

How do you convert hexadecimal to denary?

Method 1 (hexadecimal to binary to denary)

• To convert the hexadecimal number B9 to denary, take each hexadecimal value and convert it as denary to 4 bit binary

B (11)					9
8	4	2	1		8	4	2	1
1	0	1	1		1	0	0	1

• Join the two nibbles to make an 8 bit number (byte)
• Convert from binary to denary

128	64	32	16	8	4	2	1
1	0	1	1	1	0	0	1

• (1 x 128) + (1 x 32) + (1 x 16) + (1 x 8) + (1 x 1) = 185
• Hexadecimal B9 is 185 in denary

Method 2 (multiply by 16)

• To convert the hexadecimal number 79 to denary, start by multiplying the first hexadecimal digit by 16
• 7 ✖ 16 = 112
• Add digit 2 to the result
• 112 + 9 = 121
• Hexadecimal 79 is 121 in denary