Binary, Octal and Hexidecimal numbers

1
Binary, Octal and Hexidecimal numbers

• Mr. Simon
• Lowell High School
• San Francisco, CA

2
Numeral Systems

• The common system of numerals is base 10 called
  decimal
• You can have other systems with a different base
  (also called radix)
• In computers, the common systems are
• Base 2 called binary
• Base 8 called octal
• Base 16 called hexidecimal

3
Place value and expanded notation

• In Elementary school, students are asked to write
  numbers in expanded notation
• 1,235 1000 200 30 5
• 1 (103) 2 (102) 3 (101) 5 (100)
• In a decimal number each digit has a place value
  that is a power of 10

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Binary place values are powers of 2

• In Binary, each digit has a place value that is a
  power of 2
• 1011

5
Converting Binary to Decimal

• To convert binary to decimal, just write the
  number out in expanded notation
• 1011 1 (23) 0 (22) 1 (21) 1 (20)

6
Converting Binary to Decimal

• To convert binary to decimal, just write the
  number out in expanded notation
• 1011 1 (23) 0 (22) 1 (21) 1 (20)
• 18 04 12 11

7
Converting Binary to Decimal

• To convert binary to decimal, just write the
  number out in expanded notation
• 1011 1 (23) 0 (22) 1 (21) 1 (20)
• 18 04 12 11
• 8 0 2 1

8
Converting Binary to Decimal

• To convert binary to decimal, just write the
  number out in expanded notation
• 1011 1 (23) 0 (22) 1 (21) 1 (20)
• 18 04 12 11
• 8 0 2 1
• 11

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Powers of 8

• 20 1
• 21 2
• 22 4
• 23 8
• 24 16
• 25 32
• 26 64
• 27 128

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Binary numbers often comein groups of eight
called octets

• If all 8 bits are 0s 00000000 then the value of
  the octet is 0.
• If all 8 bits are 1s, 11111111 then the value of
  the octet is 255 (1286432168421).
• If the 8 bits are mixed, such as 10101000, add
  the place values of the 1 bits 128 32 8 168

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Binary Game!
12
Octal Numbers

• Octal is base 8
• Binary numbers are long and unwieldy
• Octal makes it easier to represent binary numbers
  than decimal 3 binary digits map to one Octal
  digit
• So the same number in binary has three times as
  many digits as it would in Octal
• Octal is used in Linux, for instance to set user
  permissions using the chmod command

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Place Value

• In base 8, each place value is a power of 8
• 1,245 (base 8) 1x83 2x82 4x81 5x80

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Converting to base 10

• To convert a number from another base to base 10
• Just write out the number in expanded notation,
  and evaluate it
• 1,245 (base 8) 1x83 2x82 4x81 5x80

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Converting to base 10

• Just write out the number in expanded notation,
  and evaluate it
• 1,245 (base 8) 1x83 2x82 4x81 5x80
• 512 128 32 40

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1,245 (base 8) 712 (base 10)

• Just write out the number in expanded notation,
  and evaluate it
• 1,245 (base 8) 1x83 2x82 4x81 5x80
• 512 128 32 40
• 712 (base 10)

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Converting from base 10

• Converting from base 10 to another base is the
  opposite
• First, figure out what the highest place value is
• Then, work from the highest place value to the
  lowest

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197 (base 10) ?? (base 8)
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197 (base 10) ?? (base 8)

• What's the largest power of 8 that will go into
  197?
• 80 1
• 81 8
• 82 64
• 83 512

20
19710 ? x 82 ? x 81 ? x 80

• Since 64 is the largest power of 8 that will fit,
  we know that the number will have these three
  place values
• 80 1
• 81 8
• 82 64
• 83 512

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19710 ? x 82 ? x 81 ? x 80

• How many times will 64 go into 197?
• 64 x 1 64
• 64 x 2 128
• 64 x 3 192

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19710 3 x 82 ? x 81 ? x 80

• Now how much room do we have for the next power
  of 8?
• 197 192 5
• 81 x ?

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19710 3 x 82 0 x 81 ? x 80

• Now how much room do we have for the next power
  of 8?
• 197 192 5
• 81 x 0 0
• 81 x 1 8

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19710 3 x 82 0 ? x 80

• Now how much room do we have for the next power
  of 8?
• 197 192 - 0 5
• 80 x 4 4
• 80 x 5 5
• 80 x 6 6

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19710 3 x 82 0 x 81 5 x 80

• So, 197(base 10) 305(base 8)

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Luckily, you dont usually convert back and forth
from decimal to octal

• Octal converts easily back and forth from binary
• One Octal digit maps to three Binary Digits
• So, 101001 is

27
Luckily, you dont usually convert back and forth
from decimal to octal

• Octal converts easily back and forth from binary
• One Octal digit maps to three Binary Digits
• So, 101 001 is

28
Luckily, you dont usually convert back and forth
from decimal to octal

• Octal converts easily back and forth from binary
• One Octal digit maps to three Binary Digits
• So, 101 001 is
• 5 1

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Luckily, you dont usually convert back and forth
from decimal to octal

• Octal converts easily back and forth from binary
• One Octal digit maps to three Binary Digits
• So, 1010012 is 518
• Its that easy!

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Convert 4728 to binary

• Again, each Octal digit maps to three Binary
  digits

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Convert 4728 to binary

• Again, each Octal digit maps to three Binary
  digits
• So 472 is

32
Convert 4728 to binary

• Again, each Octal digit maps to three Binary
  digits
• So 4 7 2 is

33
Convert 4728 to binary

• Again, each Octal digit maps to three Binary
  digits
• So 4 7 2 is
• 100 111 010

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Convert 4728 to binary

• Again, each Octal digit maps to three Binary
  digits
• So 4728 is 1001110102
• Easy!

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In Class

• Convert each octal value to binary, and each
  binary value to octal
• 63
• 70
• 100100101
• 001101101

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Hexadecimal Numbers

• hex for short
• Hexadecimal is base 16
• It uses sixteen distinct symbols, the symbols 09
  to represent values zero to nine, and A, B, C, D,
  E, F to represent values ten to fifteen

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Mac addresses are usually in hex

• 00-13-D3-A2-8F
• Some IP addresses (IPv6) are also written in hex
• Encryption keys in wireless routers can be in hex
• You can usually spot hex because it has only the
  letters A-F

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Converting Hex to Decimal

• In a hex number, each place value is a power of
  16
• 13FA

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Converting Hex to Decimal

• In a hex number, each place value is a power of
  16
• 13FA 1 (163)3 (162)15 (161)10 (160)

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Converting Hex to Decimal

• In a hex number, each place value is a power of
  16
• 13FA 1 (163)3 (162)15 (161)10 (160)
• 4096 768 240 10

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Converting Hex to Decimal

• In a hex number, each place value is a power of
  16
• 13FA 1 (163)3 (162)15 (161)10 (160)
• 4096 768 240 10
• 5114

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If you use a table, converting back and forth
from hex to binary is easy

• To convert hexadecimal F8 to binary, write down
  the binary for F first, then the binary for 8.
• F 8
• 1111 1000
• So, the answer is 11111000
• Thats all there is to it!

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Converting Binary to Hex

• Converting Binary to Hex is just as easy
• 01011110101101010010
• 0101 1110 1011 0101 0010
• 5 E B 5 2
• 5EB52

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In Class

• Convert each hex value to binary, and each binary
  value to hex
• E1
• 15
• 10010010
• 00110110

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Sometimes binary and hexcome up in Java

• The getRGB() function returns a single int that
  has all three values
• //get RGB values at mouseX and mouseY
• int c image.getRGB(mouseX, mouseY)
• //use bitwise logical operations
• int red (c 0x00ff0000) gtgt 16
• int green (c 0x0000ff00) gtgt 8
• int blue c 0x000000ff
• So let's say we clicked on a magenta area. . .

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Sometimes binary and hexcome up in Java

• getRGB() will return 0x00FF00FF
• which in binary is
• 00000000 11111111 00000000 11111111
• The octets are opacity, red, green and blue

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The bitwise

• int red (c 0x00ff0000) gtgt 16
• The single is the bitwise logical AND
• We're looking for the bits that are 1 in both
  numbers
• 00000000 11111111 00000000 11111111
• 00000000 11111111 00000000 00000000

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The bitwise

• int red (c 0x00ff0000) gtgt 16
• So this expression evaluates to
• 00000000 11111111 00000000 11111111
• 00000000 11111111 00000000 00000000
• 00000000 11111111 00000000 00000000

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gtgt bit-shift operator

• int red (c 0x00ff0000) gtgt 16
• This means shift to the right 16 places
• 00000000 11111111 00000000 00000000

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gtgt bit-shift operator

• int red (c 0x00ff0000) gtgt 16
• This means shift to the right 16 places
• 00000000 11111111 00000000 00000000
• which means that red is assigned
• 00000000 00000000 00000000 11111111
• which is 255 in decimal