# Number Systems - Binary and Hexadecimal Systems ```Number Systems – Binary, Octal, Decimal, and Hexadecimal
Basic Number Systems and Conversions
Binary
 Base or radix 2 number system
 Binary digit is called a bit.
 Numbers are 0 and 1 only.
 Numbers are expressed as powers of 2.
 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64, 27 = 128, 28 = 256, 29 =
512, 210 = 1024, 211 = 2048, 212 = 4096, 212 = 8192, …
Conversion of binary to decimal ( base 2 to base 10)
Example: convert (110011)2 to decimal
= (1 x 25) + (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
= 32 + 16 + 0 + 0 + 2 + 1
= (51)10
Conversion of decimal to binary (base 10 to base 2)
Example: convert (51)10 to binary
51  2 = 25 remainder is 1
25  2 = 12 remainder is 1
12  2 = 6 remainder is 0
6  2 = 3 remainder is 0
3  2 = 1 remainder is 1
1  2 = 0 remainder is 1
Answer = 1 1 0 0 1 1
Note: the answer is read from bottom (MSB) to top (LSB) as 1100112
Octal
 Base or radix 8 number system
 1 octal digit is equivalent to 3 bits.
 Numbers are 0-7.
 Numbers are expressed as powers of 8. 80 = 1, 81 = 8, 82 = 64, 83 = 512, 84
= 4096.
Number Systems - 1
Conversion of octal to decimal ( base 8 to base 10)
Example: convert (632)8 to decimal
= (6 x 82) + (3 x 81) + (2 x 80)
= (6 x 64) + (3 x 8) + (2 x 1)
= 384 + 24 + 2
= (410)10
Conversion of decimal to octal (base 10 to base 8)
Example: convert (177)10 to octal
177  8 = 22 remainder is 1
22  8 = 2 remainder is 6
2  8 = 0 remainder is 2
Note: the answer is read from bottom to top as (261)8, the same as with
the binary case.
 Base or radix 16 number system
 1 hex digit is equivalent to 4 bits.
 Numbers are 0-9, A, B, C, D, E, and F.
 (A)16 = (10)10, (B)16 = (11)10, (C)16 = (12)10, (D)16 = (13)10, (E)16 = (14)10, (F)16 =
(15)10
 Numbers are expressed as powers of 16.
 160 = 1, 161 = 16, 162 = 256, 163 = 4096, 164 = 65536, …
Conversion of hexadecimal to decimal ( base 16 to base 10)
Example: convert (F4C)16 to decimal
= (F x 162) + (4 x 161) + (C x 160)
= (15 x 256) + (4 x 16) + (12 x 1)
= 3840 + 64 + 12
= (3916)10
Conversion of decimal to hex (base 10 to base 16)
Example: convert (77)10 to hex
77  16 = 4 remainder is D
4  16 = 0 remainder is 4
Note: the answer is read from bottom to top as (4D)16, the same as with
the binary case.
Number Systems - 2
decimal
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
binary
decimal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
10
11
12
13
14
15
16
17
18
19
1A
1B
1C
1D
1E
1F
binary
00010000
00010001
00010010
00010011
00010100
00010101
00010110
00010111
00011000
00011001
00011010
00011011
00011100
00011101
00011110
00011111
Table 1: Decimal, hexadecimal and binary numbers
Conversion of Octal and Hex to Binary
Conversion of octal and hex numbers to binary is based upon the the bit
patterns shown in the table above and is straight forward. For octal
numbers, only three bits are required. Thus 68 = 1102, and 3458 =
111001012. For hex numbers, four bits are required. Thus E16 = 11102,
and 47D16 = 100011111012.
Conversion of Binary to Octal and Hex
Conversion of binary numbers to octal and hex simply requires grouping
bits in the binary numbers into groups of three bits for conversion to octal
and into groups of four bits for conversion to hex. Groups are formed
beginning with the LSB and progressing to the MSB. Thus, 111001112 =
3478 and 111000101010100100012 = 70252218. Similarly, 111001112 =
E716 and 110001010100001112 = 18A8716.
Number Systems - 3
DECIMAL NUMBER SYSTEM
Decimal system is a BASE TEN system
It uses 10 symbols: 0,1,2,3,4,5,6,7,8,9
Expand the decimal number 6,458
6,458 = ( 6 x 103 )+ ( 4 x 102 )+ ( 5 x 101 )+ ( 8 x 100 )
Decimal place values
105
104
100,000
10,000
Hundred
Ten
Thousands
Thousands
103
1,000
102
100
101
10
100
1
Thousands
Hundreds
Tens
Ones
BINARY NUMBER SYSTEM
Binary number system is said to be a Base Two
Number of symbols—2
Symbols “0” and “1”
A bit in the binary system is like a digit in the decimal system
211
210
2048 1024
29
512
28
256
27
128
26
64
25
32
24
16
23
8
22
4
21
2
20
1
CONVERT BINARY TO DECIMAL --- Series of additions
11002 = ( 1 x 23 )+ ( 1 x 22 )+ ( 0 x 21 )+ ( 0 x 20 )
=
8 + 4
+ 0
+ 0
=
1210
100012 = (1 x 24) + ( 0 x 23 )+ ( 0 x 22 )+ ( 0 x 21 )+ ( 1 x 20 )
=
16 + 0
+ 0
+ 0 +
1
=
1710
1010112= (1 x 25) + (0 x 24) + ( 1 x 23 )+ ( 0 x 22 )+ ( 1 x 21 )+ ( 1 x 20 )
= 32 + 0
+ 8
+ 0 +
2
+
1
=
4310
1111012= (1 x 25) + (1 x 24) + ( 1 x 23 )+ ( 1 x 22 )+ ( 0 x 21 )+ ( 1 x 20 )
= 32 + 16
+ 8
+ 4 +
0
+
1
=
6110
1102
= 610
Number Systems - 4
10102
= 1010
1100002 = 4810
111112 = 3110
CONVERT DECIMAL TO BINARY --- Series of subtractions
211
2048
210
1024
29
512
28
256
27
128
26
64
25
32
24
16
23
8
710 =
3410 =
22
4
1
21
2
1
20
1
1
= 1112
1
0
0
0
1
0
= 1000102
1
0
1
1
0
0
1
= 10110012
1
0
0
1
0
1
1
= 110010112
1910 =
1
0
0
1
1
= 100112
2210 =
1
0
1
1
0
= 101102
1
1
1
1
= 11112
8910 =
20310 =
1
1510 =
3610 =
1
0
0
1
0
0
= 1001002
4810 =
1
1
0
0
0
0
= 1100002
3910 =
1
0
0
1
1
1
= 1001112
6410 =
1
0
0
0
0
0
0
= 10000002
10310 =
1
1
0
0
1
1
1
= 11001112
EXAMPLE:
710 = 1112
because 7- 4 = 3
3–2=1
1 - 1= 0
SYSTEM
Hexadecimal number system is a Base Sixteen
Number Systems - 5
Number of symbols—16
Symbols 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, F
10, 11, 12, 13, 14, 15
1611
1610
169
168
167
166
165
164
163
16,777,216 1,048,576 65,536 4,096
162
256
161
16
160
1
7216
= ( 7 x 161 )+ ( 2 x 160 )
=
112 + 2
=
11410
235916 = ( 2 x 163 )+ ( 3 x 162 )+ ( 5 x 161 )+ ( 9 x 160 )
= 8192 + 768
+ 80
+ 9
=
904910
C2916 = ( C x 162 )+ ( 2 x 161 )+ ( 9 x 160 )
= 3072 + 32
+ 9
=
311310
12AB16 = ( 1 x 163 )+ ( 2 x 162 )+ ( 10 x 161 )+ ( 11 x 160 )
= 4096 + 512
+ 160
+
11
=
477910
3AC216 = ( 3 x 163 )+ ( 10 x 162 )+ ( 12 x 161 )+ ( 2 x 160 )
= 12288 + 2560
+ 192
+
2
=
1504210
Number Systems - 6
CONVERT DECIMAL TO HEXADECIMAL --- Series of subtractions
Allowable symbols 0, 1, 2, 3, 4,5, 6, 7, 8, 9, A, B, C, D, E, F
10, 11, 12, 13, 14, 15
166
16,777,216
165
1,048,576
164
65,536
163
4096
67210 =
162
256
2
161
16
A
160
1
0
= 2A016
1,76310 =
6
E
3
= 6E316
1,32410 =
5
2
C
= 52C16
A
B
C
D
= ABCD16
2
A
B
= F2A3B16
3
1
8
= 31D816
43,98110 =
993,85110 =
F
12,76010 =
3
D
For example:
2 @ 256
10 @ 16
672
-512
160
-160
0
2A016
5 @ 256
2 @ 16
1763
6 @ 256 -1536
227
14 @ 16
-224
3
6E316
1324
-1280
44
- 32
12
52C16
10 @ 4096
11 @ 256
12 @ 16
43,981
- 40,960
3021
- 2816
205
-192
13
ABCD16
Number Systems - 7
1010 1100b
10
1110 1001 b
12
A
14
C
9
6
9
6
E
ACh
0011 1111b
0110 1011 b
11
6Bh
0001 1000b
0101 1010b
12
4
C
4Ch
3
15
1
8
5
10
3
F
1
8
5
A
18h
1100
4
B
E9h
3Fh
0100
1101 0111
13
D
5Ah
7
7
D7h
83h
57h
21h
1000 0011b
0101 0111b
0010 0001b
3Ch
0011 1100b
4Fh
0100 1111b
DBh
1101 1011b
ABh
1010 1011b
EEh
1110 1110b
Number Systems - 8
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