Binary Mathematics

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Binary Mathematics
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Counting system

• There are three kinds of people in the world
  those who can count, and those who can not.
• 
  - Unknown Wisdom
• Todays class
• Numbering system
• Conversion between 10 based and 2 based numbering
  system.
• Binary Mathematics.
• Quiz

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Base 10 counting system

• 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.

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Numbers

• Ancient Africa
• Notches on a bone.
• Egyptians/Roman
• Each magnitude is represented by a symbol.
• Indian/Arabian (Modern numbering system)

1,475,268
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Base 10 (Decimal numbers)

• What does 157 mean?
• 157 1 x 100 5 x 10 7 x 1
• 1 x 102 5 x 101 7 x 100

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Binary Code

• 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

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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

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Binary Bits and Bytes

• 1 bit is a single bit of information, a 1 or 0
• Only two possible values
• 1 byte is 8 bits, an 8 bit word
• 256 possible values from 0-255 base 10 or
  00000000 to 11111111 base 2
• 10100110 is a single byte

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Base 10 to Binary
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Binary mathematics

• 000
• 101
• 1110

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Hexadecimal (base 16)

• Binary code is too long in representation. Hex is
  much shorter.
• Converting a binary number to a Hex number is
  relatively easy
• Every 4 bit can convert to a Hex
• Problem we are short of numbers
• A-10 B-11 C-12 D-13 E-14 F-15

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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
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Example
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Wisdom said

• There are 10 kinds of people in the world, those
  who use binary counting system, and those who
  dont.

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Quiz

• No Calculators!!!!
• Convert binary code to Decimal number.
• 10100101 (Bin)
• Convert Decimal number to binary code
• 176 (Dec)
• Convert Hexadecimal number to binary
• BADDEF
• Add these two binary numbers
• 1000110111011100?

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Answer

• 10100101 (Binary) 165 (Decimal)
• 176 (Decimal) 10110000 (Binary)
• BADDEF1011,1010,1101,1101,1110,1111
• The result of summation
• 101101001