Bit & Byte
Computers use the binary system. Any physical system that can
exist in two distinct states (e.g., 0-1, on-off, hi-lo, yes-no,
up-down, north-south, etc.) has the potential of being used to
represent numbers or characters.
A binary digit is called a bit. There are two possible
states in a bit, usually expressed as 0 and 1.
A series of eight bits strung together makes a byte,
much as 12 makes a dozen. With 8 bits, or 8 binary digits, there
exist 2^8=256 possible combinations. The following
table shows some of these combinations. (The number enclosed in
parentheses represents the decimal equivalent.)
00000000 ( 0) 00010000 ( 16) 00100000 ( 32) ... 01110000 (112)
00000001 ( 1) 00010001 ( 17) 00100001 ( 33) ... 01110001 (113)
00000010 ( 2) 00010010 ( 18) 00100010 ( 34) ... 01110010 (114)
00000011 ( 3) 00010011 ( 19) 00100011 ( 35) ... 01110011 (115)
00000100 ( 4) 00010100 ( 20) 00100100 ( 36) ... 01110100 (116)
00000101 ( 5) 00010101 ( 21) 00100101 ( 37) ... 01110101 (117)
00000110 ( 6) 00010110 ( 22) 00100110 ( 38) ... 01110110 (118)
00000111 ( 7) 00010111 ( 23) 00100111 ( 39) ... 01110111 (119)
00001000 ( 8) 00011000 ( 24) 00101000 ( 40) ... 01111000 (120)
00001001 ( 9) 00011001 ( 25) 00101001 ( 41) ... 01111001 (121)
00001010 ( 10) 00011010 ( 26) 00101010 ( 42) ... 01111010 (122)
00001011 ( 11) 00011011 ( 27) 00101011 ( 43) ... 01111011 (123)
00001100 ( 12) 00011100 ( 28) 00101100 ( 44) ... 01111100 (124)
00001101 ( 13) 00011101 ( 29) 00101101 ( 45) ... 01111101 (125)
00001110 ( 14) 00011110 ( 30) 00101110 ( 46) ... 01111110 (126)
00001111 ( 15) 00011111 ( 31) 00101111 ( 47) ... 01111111 (127)
10000000 (128) 10010000 (144) 10100000 (160) ... 11110000 (240)
10000001 (129) 10010001 (145) 10100001 (161) ... 11110001 (241)
10000010 (130) 10010010 (146) 10100010 (162) ... 11110010 (242)
10000011 (131) 10010011 (147) 10100011 (163) ... 11110011 (243)
10000100 (132) 10010100 (148) 10100100 (164) ... 11110100 (244)
10000101 (133) 10010101 (149) 10100101 (165) ... 11110101 (245)
10000110 (134) 10010110 (150) 10100110 (166) ... 11110110 (246)
10000111 (135) 10010111 (151) 10100111 (167) ... 11110111 (247)
10001000 (136) 10011000 (152) 10101000 (168) ... 11111000 (248)
10001001 (137) 10011001 (153) 10101001 (169) ... 11111001 (249)
10001010 (138) 10011010 (154) 10101010 (170) ... 11111010 (250)
10001011 (139) 10011011 (155) 10101011 (171) ... 11111011 (251)
10001100 (140) 10011100 (156) 10101100 (172) ... 11111100 (252)
10001101 (141) 10011101 (157) 10101101 (173) ... 11111101 (253)
10001110 (142) 10011110 (158) 10101110 (174) ... 11111110 (254)
10001111 (143) 10011111 (159) 10101111 (175) ... 11111111 (255)
Number System
You may regard each digit as a box that can hold a number. In
the binary system, there can be only two choices for this number
-- either a "0" or a "1". In the octal system, there can be eight
possibilities:
"0", "1", "2", "3", "4", "5", "6", "7".
In the decimal system, there are ten different numbers that can
enter the digit box:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9".
In the hexadecimal system, we allow 16 numbers:
"0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "A", "B",
"C", "D", "E", and "F".
As demonstrated by the following table, there is a direct
correspondence between the binary system and the octal system,
with three binary digits corresponding to one octal digit.
Likewise, four binary digits translate directly into one
hexadecimal digit. In computer usage, hexadecimal notation is
especially common because it easily replaces the binary notation,
which is too long and human mistakes in transcribing the binary
numbers are too easily made.
Base Conversion Table
BIN OCT HEX DEC
----------------------
0000 00 0 0
0001 01 1 1
0010 02 2 2
0011 03 3 3
0100 04 4 4
0101 05 5 5
0110 06 6 6
0111 07 7 7
----------------------
1000 10 8 8
1001 11 9 9
1010 12 A 10
1011 13 B 11
1100 14 C 12
1101 15 D 13
1110 16 E 14
1111 17 F 15
Convert From Any Base To Decimal
Let's think more carefully what a decimal number means. For
example, 1234 means that there are four boxes (digits); and there
are 4 one's in the right-most box (least significant digit), 3
ten's in the next box, 2 hundred's in the next box, and finally 1
thousand's in the left-most box (most significant digit). The
total is 1234:
Original Number: 1 2 3 4
| | | |
How Many Tokens: 1 2 3 4
Digit/Token Value: 1000 100 10 1
Value: 1000 + 200 + 30 + 4 = 1234
or simply, 1*1000 + 2*100 + 3*10 + 4*1 = 1234
Thus, each digit has a value: 10^0=1 for the least
significant digit, increasing to 10^1=10, 10^2=100,
10^3=1000, and so forth.
Likewise, the least significant digit in a hexadecimal number
has a value of 16^0=1 for the least significant digit,
increasing to 16^1=16 for the next digit, 16^2=256
for the next, 16^3=4096 for the next, and so forth.
Thus, 1234 means that there are four boxes (digits); and there are
4 one's in the right-most box (least significant digit), 3
sixteen's in the next box, 2 256's in the next, and 1 4096's in
the left-most box (most significant digit). The total is:
1*4096 + 2*256 + 3*16 + 4*1 = 4660
Example. Convert the hexadecimal number 4B3 to decimal
notation. What about the decimal equivalent of the hexadecimal
number 4B3.3?
Solution:
Original Number: 4 B 3 . 3
| | | |
How Many Tokens: 4 11 3 3
Digit/Token Value: 256 16 1 0.0625
Value: 1024 +176 + 3 + 0.1875 = 1203.1875
Example. Convert 234.14 expressed in an octal notation
to decimal.
Solution:
Original Number: 2 3 4 . 1 4
| | | | |
How Many Tokens: 2 3 4 1 4
Digit/Token Value: 64 8 1 0.125 0.015625
Value: 128 + 24 + 4 + 0.125 + 0.0625 = 156.1875
Another way is to think of a cash register with different
slots, each holding bills of a different denomination.
Convert From Decimal to Any Base
Again, let's think about what you do to obtain each digit. As
an example, let's start with a decimal number 1234 and convert it
to decimal notation. To extract the last digit, you move the
decimal point left by one digit, which means that you divide the
given number by its base 10.
1234/10 = 123 + 4/10
The remainder of 4 is the last digit. To extract the next last
digit, you again move the decimal point left by one digit and see
what drops out.
123/10 = 12 + 3/10
The remainder of 3 is the next last digit. You repeat this
process until there is nothing left. Then you stop. In summary,
you do the following:
Quotient Remainder
-----------------------------
1234/10 = 123 4 --------+
123/10 = 12 3 ------+ |
12/10 = 1 2 ----+ | |
1/10 = 0 1 --+ | | | (Stop when the quotient is 0.
1 2 3 4 (Base 10)
Now, let's try a nontrivial example. Let's express a decimal
number 1341 in binary notation. Note that the desired base is 2,
so we repeatedly divide the given decimal number by 2.
Quotient Remainder
-----------------------------
1341/2 = 670 1 ----------------------+
670/2 = 335 0 --------------------+ |
335/2 = 167 1 ------------------+ | |
167/2 = 83 1 ----------------+ | | |
83/2 = 41 1 --------------+ | | | |
41/2 = 20 1 ------------+ | | | | |
20/2 = 10 0 ----------+ | | | | | |
10/2 = 5 0 --------+ | | | | | | |
5/2 = 2 1 ------+ | | | | | | | |
2/2 = 1 0 ----+ | | | | | | | | |
1/2 = 0 1 --+ | | | | | | | | | | (Stop when the quotient is 0)
| | | | | | | | | | |
1 0 1 0 0 1 1 1 1 0 1 (BIN; Base 2)
Let's express the same decimal number 1341 in octal notation.
Quotient Remainder
-----------------------------
1341/8 = 167 5 --------+
167/8 = 20 7 ------+ |
20/8 = 2 4 ----+ | |
2/8 = 0 2 --+ | | | (Stop when the quotient is 0)
| | | |
2 4 7 5 (OCT; Base 8)
Let's express the same decimal number 1341 in hexadecimal
notation.
Quotient Remainder
-----------------------------
1341/16 = 83 13 ------+
83/16 = 5 3 ----+ |
5/16 = 0 5 --+ | | (Stop when the quotient is 0)
| | |
5 3 D (HEX; Base 16)
Example. Convert the decimal number 3315 to hexadecimal
notation. What about the hexadecimal equivalent of the decimal
number 3315.3?
Solution:
Quotient Remainder
-----------------------------
3315/16 = 207 3 ------+
207/16 = 12 15 ----+ |
12/16 = 0 12 --+ | | (Stop when the quotient is 0)
| | |
C F 3 (HEX; Base 16)
(HEX; Base 16)
Product Integer Part 0.4 C C C ...
-------------------------------- | | | |
0.3*16 = 4.8 4 ----+ | | | | |
0.8*16 = 12.8 12 ------+ | | | |
0.8*16 = 12.8 12 --------+ | | |
0.9872*2 = 12.8 12 ----------+ | |
: ---------------------+
:
Thus, 3315.3 (DEC) --> CF3.4CCC... (HEX)
Note that from the Base Conversion Table, you can easily get
the binary notation from the hexadecimal number by grouping four
binary digits per hexadecimal digit, or from or the octal number
by grouping three binary digits per octal digit, and vice versa.
HEX 5 3 D
BIN 0101 0011 1101
OCT 2 4 7 5
BIN 010 100 111 101
Finally, the fractional part is a decimal number can also be
converted to any base by repeatedly multiplying the given number
by the target base. Example: Convert a decimal number 0.1234 to
binary notation
(BIN; Base 2)
Product Integer Part 0.0 0 0 1 1 1 1 1 1 0 0 1 ...
-------------------------------- | | | | | | | | | | | | |
0.1234*2 = 0.2468 0 ----+ | | | | | | | | | | | |
0.2468*2 = 0.4936 0 ------+ | | | | | | | | | | |
0.4936*2 = 0.9872 0 --------+ | | | | | | | | | |
0.9872*2 = 1.9744 1 ----------+ | | | | | | | | |
0.9744*2 = 1.9488 1 ------------+ | | | | | | | |
0.9488*2 = 1.8976 1 --------------+ | | | | | | |
0.8976*2 = 1.7952 1 ----------------+ | | | | | |
0.7952*2 = 1.5904 1 ------------------+ | | | | |
0.5904*2 = 1.1808 1 --------------------+ | | | |
0.1808*2 = 0.3616 0 ----------------------+ | | |
0.3616*2 = 0.7232 0 ------------------------+ | |
0.7232*2 = 1.4464 1 --------------------------+ |
: ----------------------------+
:
Additon and Multiplication Tables
You generate the addition tables in bases other then 10 by
following the same rule you do in base 10. The resulting tables
have the appearance of shifting the columns to the left by one in
each subsequent rows. Note how simple the addition and
multiplication tables are for the binary system; addition
operation is simply the bit-wise XOR operation with carry, and
multiplication is simply the logical AND operation.
Decimal Addition Table:
| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 2 3 4 5 6 7 8 9 10
2 | 2 3 4 5 6 7 8 9 10 11
3 | 3 4 5 6 7 8 9 10 11 12
4 | 4 5 6 7 8 9 10 11 12 13
5 | 5 6 7 8 9 10 11 12 13 14
6 | 6 7 8 9 10 11 12 13 14 15
7 | 7 8 9 10 11 12 13 14 15 16
8 | 8 9 10 11 12 13 14 15 16 17
9 | 9 10 11 12 13 14 15 16 17 18
Binary Addition Table:
| 0 1
---+-----
0 | 0 1
1 | 1 10
Octal Addition Table:
| 0 1 2 3 4 5 6 7
---+-----------------------
0 | 0 1 2 3 4 5 6 7
1 | 1 2 3 4 5 6 7 10
2 | 2 3 4 5 6 7 10 11
3 | 3 4 5 6 7 10 11 12
4 | 4 5 6 7 10 11 12 13
5 | 5 6 7 10 11 12 13 14
6 | 6 7 10 11 12 13 14 15
7 | 7 10 11 12 13 14 15 16
Hexadecimal Addition Table:
| 0 1 2 3 4 5 6 7 8 9 A B C D E F
---+-----------------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9 A B C D E F
1 | 1 2 3 4 5 6 7 8 9 A B C D E F 10
2 | 2 3 4 5 6 7 8 9 A B C D E F 10 11
3 | 3 4 5 6 7 8 9 A B C D E F 10 11 12
4 | 4 5 6 7 8 9 A B C D E F 10 11 12 13
5 | 5 6 7 8 9 A B C D E F 10 11 12 13 14
6 | 6 7 8 9 A B C D E F 10 11 12 13 14 15
7 | 7 8 9 A B C D E F 10 11 12 13 14 15 16
8 | 8 9 A B C D E F 10 11 12 13 14 15 16 17
9 | 9 A B C D E F 10 11 12 13 14 15 16 17 18
A | A B C D E F 10 11 12 13 14 15 16 17 18 19
B | B C D E F 10 11 12 13 14 15 16 17 18 19 1A
C | C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B
D | D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
E | E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D
F | F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E
You can also generate multiplication tables in bases other than
10 by following the same rule you do in base 10.
Decimal Multiplication Table:
| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------
0 | 0 0 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7 8 9
2 | 0 2 4 6 8 10 12 14 16 18
3 | 0 3 6 9 12 15 18 21 24 27
4 | 0 4 8 12 16 20 24 28 32 36
5 | 0 5 10 15 20 25 30 35 40 45
6 | 0 6 12 18 24 30 36 42 48 54
7 | 0 7 14 21 28 35 42 49 56 63
8 | 0 8 16 24 32 40 48 56 64 72
9 | 0 9 18 27 36 45 54 63 72 81
Binary Multiplication Table:
| 0 1
---+-----
0 | 0 0
1 | 0 1
Octal Multiplication Table:
| 0 1 2 3 4 5 6 7
---+-----------------------
0 | 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7
2 | 0 2 4 6 10 12 14 16
3 | 0 3 6 11 14 17 22 25
4 | 0 4 10 14 20 24 30 34
5 | 0 5 12 17 24 31 36 43
6 | 0 6 14 22 30 36 44 52
7 | 0 7 16 25 34 43 52 61
Hexadecimal Multiplication Table:
| 0 1 2 3 4 5 6 7 8 9 A B C D E F
---+-----------------------------------------------
0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7 8 9 A B C D E F
2 | 0 2 4 6 8 A C E 10 12 14 16 18 1A 1C 1E
3 | 0 3 6 9 C F 12 15 18 1B 1E 21 24 27 2A 2D
4 | 0 4 8 C 10 14 18 1C 20 24 28 2C 30 34 38 3C
5 | 0 5 A F 14 19 1E 23 28 2D 32 37 3C 41 46 4B
6 | 0 6 C 12 18 1E 24 2A 30 36 3C 42 48 4E 54 5A
7 | 0 7 E 15 1C 23 2A 31 38 3F 46 4D 54 5B 62 69
8 | 0 8 10 18 20 28 30 38 40 48 50 58 60 68 70 78
9 | 0 9 12 1B 24 2D 36 3F 48 51 5A 63 6C 75 7E 87
A | 0 A 14 1E 28 32 3C 46 50 5A 64 6E 78 82 8C 96
B | 0 B 16 21 2C 37 42 4D 58 63 6E 79 84 8F 9A A5
C | 0 C 18 24 30 3C 48 54 60 6C 78 84 90 9C A8 B4
D | 0 D 1A 27 34 41 4E 5B 68 75 82 8F 9C A9 B6 C3
E | 0 E 1C 2A 38 46 54 62 70 7E 8C 9A A8 B6 C4 D2
F | 0 F 1E 2D 3C 4B 5A 69 78 87 96 A5 B4 C3 D2 E1
Arithmetic Operations
You do arithematic with hexadecimal numbers or numbers in any
base in exactly the same way you do with decimal numbers, except
that the addition and multiplcation tables you employ to base your
calculations are a bit different. Substraction is equivalent to
adding a negative number, and division is equivalent to
multiplying by the inverse.
Example. Find the sum of two hexadecimal integers 123
and DEF.
Solution:
From the above hexadecimal addition table, we see that:
3+F=12, 2+E=10, and 1+D=E
123
+ DEF
-----
carry 11
E02
-----
sum F12
Example. Find the product of two hexadecimal integers
123 and DEF.
Solution:
Step 1: We break down the second multiplier into single digits.
123*DEF = 123*(D00+E0+F)
= (123*D)*100 + (123*E)*10 + (123*F)
Step 2: We find the product in parentheses.
From the above hexadecimal multiplication table, we see that:
1*D=D, 2*D=1A, 3*D=27; thus,
123*D = (100+20+3)*D
= 1*D*100 + 2*D*10 + 3*D
= D*100 + 1A*10 + 27
= D00 + 1A0 + 27
= EC7
Likewise,
123*E = (100+20+3)*E
= 1*E*100 + 2*E*10 + 3*E
= E*100 + 1C*10 + 2A
= E00 + 1C0 + 2A
= FEA
123*F = (100+20+3)*F
= 1*F*100 + 2*F*10 + 3*F
= F*100 + 1E*10 + 2D
= F00 + 1E0 + 2D
= 110D
Or, in elementary school style:
123 123 123
x D x E x F
----- ----- -----
27 2A 2D
1A 1C 1E
D E F
----- ----- -----
EC7 FEA 110D
Step 3: We sum up the individual products.
123*DEF = (123*D)*100 + (123*E)*10 + (123*F)
= EC7*100 + FEA*10 + 110D
= EC700 + FEA0 + 110D
= FD6AD
Or, in elementary school style:
123
x DEF
-----
110D
FEA
EC7
-----
FD6AD
