Binary Mathematics
Counting system
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There are three kinds of people in the world:
those who can count, and those who can not.
- Unknown Wisdom
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Today’s class
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Numbering system
Conversion between 10 based and 2 based
numbering system.
Binary Mathematics.
Quiz
Base 10 counting system
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We happened to use
the current counting
system, because we
happened to have ten
fingers.
If dinosaurs had ruled
the earth, they would be
happy to use a 8-based
counting system.
Numbers
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Ancient Africa
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Egyptians/Roman
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Notches on a bone.
Each magnitude is represented by a symbol.
Indian/Arabian (Modern numbering
system)
1,475,268
Base 10 (Decimal numbers)
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What does 157 mean?
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157 = 1 x 100 + 5 x 10 + 7 x 1
= 1 x 102 + 5 x 101 + 7 x 100
Binary Code
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Imagine a specie that only has two fingers.
how can they count?
A computer is such kind of two-finger specie.
0 and 1
Each place is the exponential of 2
Base 10 vs Base 2
Base 10
157
157 = 1 x 100 + 5 x 10 + 7 x 1
= 1 x 102 + 5 x 101 + 7 x 100
Base 2
1011 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20
1011 = 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1
Binary Bits and Bytes
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1 bit is a single bit of information, a 1 or 0
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1 byte is 8 bits, an 8 bit word
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Only two possible values
256 possible values from 0-255 base 10 or
00000000 to 11111111 base 2
10100110 is a single byte
Base 10 to Binary
Binary mathematics
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0+0=0
1+0=1
1+1=10
Hexadecimal (base 16)
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Binary code is too long in representation. Hex
is much shorter.
Converting a binary number to a Hex number
is relatively easy
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Every 4 bit can convert to a Hex
Problem: we are short of numbers
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A-10 B-11 C-12 D-13 E-14 F-15
Lookup table
Binary
Hex
Binary
Hex
0000
0
1000
8
0001
1
1001
9
0010
2
1010
A
0011
3
1011
B
0100
4
1100
C
0101
5
1101
D
0110
6
1110
E
0111
7
1111
F
Example
Wisdom said
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There are 10 kinds of people in the world,
those who use binary counting system, and
those who don’t.
Quiz
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No Calculators!!!!
Convert binary code to Decimal number.
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Convert Decimal number to binary code
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176 (Dec)
Convert Hexadecimal number to binary
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10100101 (Bin)
BADDEF
Add these two binary numbers
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10001101+11011100=?
Answer
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10100101 (Binary) = 165 (Decimal)
176 (Decimal)= 10110000 (Binary)
BADDEF=1011,1010,1101,1101,1110,1111
The result of summation
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101101001