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Binary
Binary Number System
Binary to Decimal
Decimal to Binary
Octal and Hexadecimal
Binary to Hexadecimal
Hexadecimal to Binary
Hexadecimal to Decimal
Any Number Base to Decimal
Decimal to Any Number Base
Binary Coded Decimal (BCD)
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A computer is a “bistable” device
A bistable device:
◦ Easy to design and build
◦ Has 2 states: 0 and 1
One Binary digit (bit) represents 2 possible
states (0, 1)
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With 2 bits, 4 states are possible (22 = 4)
Bit1 Bit0 State
Bit2 Bit1 Bit0
State
0
0
0
1
0
0
1
2
0
0
1
0
1
2
0
1
0
3
1
0
3
0
1
1
4
1
1
4
1
0
0
5
1
0
1
6
1
1
0
7
1
1
1
8
• With 3 bits, 8 states are possible (23 = 8)
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• With n bits, 2n states are possible
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From left to right, the position of the digit
indicates its magnitude (in decreasing order)
◦ E.g. in decimal, 123 is less than 321
◦ In binary, 011 is less than 100
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A subscript indicates the number’s base
◦ E.g. is 100 decimal or binary? We don’t know!
◦ But 1410 = 11102 is clear
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Binary is base 2
Example: convert 10110 (binary) to decimal
101102 = 1x24 + 0x23 + 1x22 + 1x21 +
0x20
= 1x16 + 0x8 + 1x4 + 1x2 + 0x1
= 16
+ 0 + 4 + 2 +
0
=
22
So 101102 = 2210
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Binary is base 2
Example: convert 35 (decimal) to binary
Quotient
35 / 2 = 17
17 / 2 = 8
8/2=4
4/2=2
2/2=1
1/2=0
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So 3510 = 1000112
Remainder
1
1
0
0
0
1
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It is difficult for a human to work with long
strings of 0’s and 1’s
Octal and Hexadecimal are ways to group bits
together
Octal: base 8
Hexadecimal: base 16
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With 4 bits, there are 16
possibilities
Use 0, 1, 2, 3, …9 for the
first 10 symbols
Use a, b, c, d, e, and f for
the last 6
Bit3 Bit2 Bit1 Bit0 Symbol
0
0
0
0
0
0
0
0
1
1
0
0
1
0
2
0
0
1
1
3
0
1
0 0
4
0
1
0
1
5
0
1
1
0
6
0
1
1
1
7
1
0
0
0
8
1
0
0
1
9
1
0
1 0
a
1
0
1
1
b
1
1
0
0
c
1
1
0
1
d
1
1
1
0
e
1
1
1
1
f Go
back
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01010110101100112 = ? in hex
Group into 4 bits, from the right:
0101, 0110, 1011, 00112
Now translate each (see previous table):
01012 => 5, 01102 => 6, 10112 => b,
00112 => 3
So this is 56b316
What if there are not enough bits?
◦ Pad with 0’s on the left
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• f0e516 = ? in binary
• Translate each into a group of 4 bits:
f16 => 11112, 016 => 00002, e16 => 11102, 516 => 01012
So this is 11110000111001012
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Hexadecimal is base 16
Example: convert 16 (hex) to decimal
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1616 = 1x161 + 6x160
= 1x16 + 6x1
= 16
+ 6
=
22
So 1616 = 2210
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Not surprising, since 1616 = 0001, 01102
◦ If one of the hex digits had been > 9, say c, then we would
have used 12 in its place.
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From right to left, multiply the digit of the
number-to-convert by its baseposition
Sum all results
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Take the decimal number, and divide by the
new number base
Keep track of the quotient and remainder
Repeat until quotient = 0
Read number from the bottom to the top
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Why not use 4 bits to represent decimal?
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Let
Let
Let
Let
0000
0001
0010
0011
represent
represent
represent
represent
0
1
2
3, etc.
◦ This is called BCD
◦ Only uses 10 of the 16 possibilities
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