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. 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

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 2Base 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

Base 10 to Binary

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

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

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