Binary mathematics

Binary Mathematics


Counting system
 There are three kinds of people in the world: those who can count, and those
who can not.

Wisdom

- Unknown

 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
 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

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