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Binary Mathematics. Counting system. There are three kinds of people in the world: those who can count, and those who can not. - Unknown Wisdom Today’s class Numbering system Conversion between 10 based and 2 based numbering system.

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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

- Today’s class
- Numbering system
- Conversion between 10 based and 2 based numbering system.
- Binary Mathematics.
- Quiz

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.

Numbers

- Ancient Africa
- Notches on a bone.

- Egyptians/Roman
- Each magnitude is represented by a symbol.

- Indian/Arabian (Modern numbering system)

1,475,268

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

- 157 = 1 x 100 + 5 x 10 + 7 x 1

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

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

- 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

Binary mathematics

- 0+0=0
- 1+0=1
- 1+1=10

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

Wisdom said

- There are 10 kinds of people in the world, those who use binary counting system, and those who don’t.

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
- 10001101+11011100=?

Answer

- 10100101 (Binary) = 165 (Decimal)
- 176 (Decimal)= 10110000 (Binary)
- BADDEF=1011,1010,1101,1101,1110,1111
- The result of summation
- 101101001

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