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# Binary Numbers

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by

Tweet## andrea nino

on 7 January 2013#### 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!!!

If not, check your work

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-

11100 2 Answer-

1 1 1 0 0

2 2 2 2 2 16+8+4+0+0=28 Additional Examples 1) 101110 3)100 2)1101 4) 10110 Answer Key 1) 1 0 1 1 1 0

2 2 2 2 2 2

32+0+8+4+2+0= 46 2) 1 1 0 1

2 2 2 2

8+4+0+1=13 3) 1 0 0

2 2 2

4+0+0=4 4) 1 0 1 1 0

2 2 2 2 2

16+0+4+2+0=22 Answer Key 2 2 2 2 0 1 2 3 4 5 4 3 2 1 0 3 2 1 0 2 1 0 4 3 2 1 0 Let's connect what we

learned to real life Binary The use of the decimal system (base 10) may be related to the fact that most humans have 10 fingers, and used them for counting at some moment. But mathematically, the number 10 has no merit over other numbers; any base (like 6, 12, or 17) would work as well. In fact, the number base 6, or a multiple like 12, would make it somewhat easier to work with fractions - being based on the smallest two prime numbers. computer electronics use voltage levels to indicate their present state. For example, a transistor with five volts would be considered "on", while a transistor with no voltage would be considered "off." Not all computer hardware uses voltage, however. CD-ROM's, for example, use microscopic dark spots on the surface of the disk to indicate "off," while the ordinary shiny surface is considered "on." Hard disks use magnetism, while computer memory uses electric charges stored in tiny capacitors to indicate "on" or "off." The origin of the computers and the way to get information in them was initially a two stage process. As logic as the decimal system is to humans using hands and fingers, the binary system with only two digits is necessary for systems with two values . Computer systems are constructed of digital electronics. That means that their electronic circuits can exist in only one of two states: on or off. Most Videos for further explanation

Full transcriptSystems 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!!!

If not, check your work

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-

11100 2 Answer-

1 1 1 0 0

2 2 2 2 2 16+8+4+0+0=28 Additional Examples 1) 101110 3)100 2)1101 4) 10110 Answer Key 1) 1 0 1 1 1 0

2 2 2 2 2 2

32+0+8+4+2+0= 46 2) 1 1 0 1

2 2 2 2

8+4+0+1=13 3) 1 0 0

2 2 2

4+0+0=4 4) 1 0 1 1 0

2 2 2 2 2

16+0+4+2+0=22 Answer Key 2 2 2 2 0 1 2 3 4 5 4 3 2 1 0 3 2 1 0 2 1 0 4 3 2 1 0 Let's connect what we

learned to real life Binary The use of the decimal system (base 10) may be related to the fact that most humans have 10 fingers, and used them for counting at some moment. But mathematically, the number 10 has no merit over other numbers; any base (like 6, 12, or 17) would work as well. In fact, the number base 6, or a multiple like 12, would make it somewhat easier to work with fractions - being based on the smallest two prime numbers. computer electronics use voltage levels to indicate their present state. For example, a transistor with five volts would be considered "on", while a transistor with no voltage would be considered "off." Not all computer hardware uses voltage, however. CD-ROM's, for example, use microscopic dark spots on the surface of the disk to indicate "off," while the ordinary shiny surface is considered "on." Hard disks use magnetism, while computer memory uses electric charges stored in tiny capacitors to indicate "on" or "off." The origin of the computers and the way to get information in them was initially a two stage process. As logic as the decimal system is to humans using hands and fingers, the binary system with only two digits is necessary for systems with two values . Computer systems are constructed of digital electronics. That means that their electronic circuits can exist in only one of two states: on or off. Most Videos for further explanation