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Octal and Hexadecimal Representation
Assuming the reader is familiar with binary notation, binary to decimal to binary conversion and
binary arithmetic we consider base 8 (octal) and base 16 (hexadecimal) representations. Note that
8 and 16 are power of 2 (23 and 24 respectively) which is why they are used.
Binary representation is awkward since many bits (binary digits) are needed to represent even
moderately sized quantities. Octal and hexadecimal notation is more compact and as we shall see
conversion between octal/hexadecimal and binary is easily done (See Grouping by Four and
Grouping by Three below)
The digits for octal representation are 0, 1, 2, 3, 4, 5, 6, and 7. Since 10 octal is 8 decimal, we
need to go no further. There is a convention that we will follow where octal notation is denoted
by a leading 0. So for example 010 is 10 base 8 while 10 is simply ten in decimal.
While octal only needs the eight digits 0 thru 7, hexadecimal requires sixteen digits. So to
represent that decimal quantities ten thru fifteen we use the letters A thru F. Thus the
hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (or a) for ten decimal, B (or b) for eleven, C
(or c) for twelve, D (or d) for thirteen, E (or e) for fourteen and F (or f) for fifteen. Thus 10
hexadecimal is sixteen decimal. There is a convention where hexadecimal numbers are denoted
by a leading 0x. So 0x10 is equal to 16 decimal.
Number Conversions between Bases 2, 8 and 16
Base 10 Number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Base 2 Number
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Base 8 Number
0
01
02
03
04
05
06
07
010
011
012
013
014
015
016
017
Base 16 Number
0
0x1
0x2
0x3
0x4
0x5
0x6
0x7
0x8
0x9
0xA
0xB
0xC
0xD
0xE
0xF
From this point we will concentrate on hexadecimal notation since any technique involving
hexadecimal notation is easily applied to octal. Moreover use of octal notation is not a
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widespread as hexadecimal. Note that the contents of any byte can be represented by two
hexadecimal digits.
Direct Hexadecimal to Decimal Conversions
The binary to decimal technique of expanding by power of 2 is easily extended to hexadecimal to
decimal conversions by expanding by powers of 16. Thus
0xA2C = (10) × 16 2 + 2 × 161 + (12) × 160 = 10 × 256 + 2 × 16 + 12 × 1 = 2604
This requires knowing the powers of 16 (the powers of 2 are more easily obtained by doubling)
and conversion the hexadecimal digits 0xA thru 0xF to 10 thru 15.
Horner’s Method also works (?).
10
2
12
0 160 2592
---------------16 × | 10 162 2604
+
Direct Decimal to Hexadecimal Conversions
Again the decimal to binary technique of subtracting powers of 2 is not so easily expended to
subtracting out multiple powers of 16.
4096
256
16
1
0
----
A
----
2
----
C
----
2604
10 × 256: -2560
---44
2 × 16:
-32
---12
12 × 12:
12
---0
The Integer Division method for decimal to hexadecimal conversions also works except integer
division by 16 is more complicated and you get remainders between 0 and 0xF.
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Hexadecimal – Binary Conversions: Grouping by Four
Every hexadecimal digit can be represented by 4 binary digits and vice versa thus making
hexadecimal to binary and binary to hexadecimal conversions easy using table lookup
Binary
0000 0001 0010 0011 0100 0101 0110 0111
Hexadecimal
0
0x1
0x2
0x3
0x4
0x5
0x6
0x7
Binary
1000 1001 1010 1011 1100 1101 1110 1111
Hexadecimal 0x8
0x9
0xA
0xB
0xC
0xD
0xE
0xF
Decimal – Hexadecimal Conversions
This leads to an indirect approach to hexadecimal – binary conversions where a conversion is
make between decimal and binary then between binary and hexadecimal
Example: 0xA2C = 1010 0010 1100 (binary) = 2048 + 512 + 32 + 8 + 4 = 2604
Decimal to hexadecimal conversions reverse the process: decimal to binary then binary to
hexadecimal
Octal – Binary Conversions: Grouping by Three
Since every octal digit can be represented by 3 binary digits and vice versa, octal binary
conversions are easily done by table lookup.
Binary
Octal
000
0
001
01
010
02
011
03
100
04
101
05
110
06
111
07
Decimal – Octal Conversions
Thus the same indirect approach introduced above can be used for octal binary conversions.
Example: 2604 = 101 000 101 100 (binary) = 05054
Exercise: Can you find a simple way to convert between hexadecimal and octal?
Aug 18, 2016
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