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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 • 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 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