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# Binary Numbers

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## andrea nino

on 7 January 2013

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#### Transcript of Binary Numbers

By Andrea Nino Binary Numbers Decimal System Why we use decimal and Binary
Systems Positional system Binary Numbers Base two system Decimal In a position base-b numeral system, b represents the number of symbols used and the power the symbols are multiplied by Base 2 to base 10 To write from base2 to base10 simply find the powers of base2 used and add them Base ten system Decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) each digit representing a quantity To understand binary numbers first, let's take a look at the number
system we are used to It's called base ten because the position of a digit is used to signify the power of ten that the digit is to be multiplied with For example- The number 451 = (1* 10 )

(1*1)

1 power of ten Position each power
represents (5* 10 )

(5*10)

50 (4* 10 )

(4*100)

400 Important points to remember digit power of ten Base ten has 10 values (digits) H | T | O 4 5 1 = First let's take a look at how in a number system symbols can represent a quantity Also known as place-value notation, was developed in India on the 5th century + +

+ +

+ + 10 | 10 | 10 position * * * Each digit is multiplied by a power of ten Each power represents a position From right to left, each position represent a power
of the base, starting with the base to the zero power, then first power, etc. the symbol represents a quantity and are placed right to left The symbol is multiplied by the power depending
on its position digit power of ten Now lets take a look at binary numbers The values added represents the desired
quantity Compared to base ten system, binary system has two symbols (digits) 0 and 1. Binary numbers are base two system because depending on their position, the symbols are multiplied by a power of two. In a binary system, multiplying by one and zero allow simplicity, because either a power of two is used(multiplied by one) or not (multiplied by zero) To write a binary number lets write down the powers of two 2

2

2

2

2
2

2

2

2
0

1 2 3 4 = = = = 5 6 7 8 9 = = = = = 2 1 2 4 8 16 32 64 128 256 512 = Now try-

write 687 in base 10 answer-

6*10 + 8*10 + 7*10
0 1 2 0 1 2 2 1 0 Now for example, if we want to write the number 26 from base ten to base two 2nd-Subtract the power by the number 2 =32
32-26= 5 3rd-Repeat first and second step until equal to zero 2=4 2 =2
-4= -2=0 2 1 1st-Find the highest value power of two in which 26 fits To write the binary number, start by placing a one if a power of two is used and a zero if not used starting with two to the zero power and ending with the highest value power of two 2 2 2 2 2 2 1 2 3 4 5 0 1 0 0 1 1 0 2 The final answer is

100110 subscript to make it clear its a
binary number Now write 150 in base 2 6 6 2 2 If your answer is 10010110 you're right!!!

2 =128 150-128=22
2 =16 22-16=6 2 2 2 2 2 2 2 2
2 =4 6-4=2 1 0 0 1 0 1 1 0
2 =2 2-2=0 2 7 4 2 1 7 6 5 4 3 2 1 0 2 Additional
Examples Change from base ten to base two

1) 175

2)96

3)69

4)55

5)84 6)300
2 =1 2 =32
2 =2 2 =64
2 =4 2 =128
2 =8 2 =256
2 =16 2 =512 0 1 2 3 4 5 6 7 8 9 10101111

2 =128 175-128=47
2 =32 47-32=15 2 2 2 2 2 2 2 2
2 =8 15-8=7 1 0 1 0 1 1 1 1
2 =4 7-4=3
2 =2 3-2=1
2 =1 1-1=0 7 5 3 2 1 0 7 6 5 4 3 2 1 0 2 2 1) 2) 1100000

2 =64 96-64=32 2 2 2 2 2 2 2
2 =32 32-32=0 1 1 0 0 0 0 0
3) 1000101

2 =64 69-54=5 2 2 2 2 2 2 2
2 =4 5-4=1 1 0 0 0 1 0 0
2 =1 1-1=0
6 5 4 3 2 1 0 2 6 5 2 6 5 4 3 2 1 0 2 2 6 2 0 4) 110111

2 =32 55-32=23 2 2 2 2 2 2
2 =16 23-16=7 1 1 0 1 1 1
2 =4 7-4=3
2 =2 3-2=1
2 =1 1-1=0 2 5 4 2 1 0 5 4 3 2 1 0 2 5) 1010100

2 =64 84-64=20 2 2 2 2 2 2 2
2 =16 20-16=4 1 0 1 0 1 0 0
2 =4 4-4=0 2 6 4 2 6 5 4 3 2 1 0 2 6) 100101100

2 =256 300-256=44
2 =32 44-32=12 2 2 2 2 2 2 2 2 2
2 =8 12-8=4 1 0 0 1 0 1 1 0 0
2 =4 4-4=0 2 8 5 3 2 8 7 6 5 4 3 2 1 0 2 Example- 101010 2 1 0 1 0 1 0 2 2 2 2 2 2 5 4 3 2 1 0 32 + 0 + 8 + 0 + 2 + 0 = 42 Now try-