Binary Numbers 723 Binary Numbers 723 = 7x100 + 2x10 + 3x1 Binary - - PowerPoint PPT Presentation

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Binary Numbers 723 Binary Numbers 723 = 7x100 + 2x10 + 3x1 Binary - - PowerPoint PPT Presentation

Oct 02, 2022 470 likes •1.23k views

Binary Numbers 723 Binary Numbers 723 = 7x100 + 2x10 + 3x1 Binary Numbers 723 = 7x100 + 2x10 + 3x1 = 7x10 2 + 2x10 1 + 3x10 0 Binary Numbers 5349 = 5x10 3 + 3x10 2 + 4x10 1 + 9x10 0 Binary Numbers Why base 10? Binary Numbers 257 (base 8)

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Binary Numbers

723

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Binary Numbers

723 = 7x100 + 2x10 + 3x1

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Binary Numbers

723 = 7x100 + 2x10 + 3x1 = 7x102 + 2x101 + 3x100

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5349 = 5x103 + 3x102 + 4x101 + 9x100

Binary Numbers

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Why base 10? Binary Numbers

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257 (base 8) 2x82 + 5x81 + 7x80 2x64 + 5x8 + 7x1 175 (base 10)

Binary Numbers

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0110 (base 2) 0x23 + 1x22 + 1x21 + 0x20 0x8 + 1x4 + 1x2 + 0x1 6 (base 10)

Binary Numbers

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2-bit binary number

00 0 01 1 10 2 11 3

2120

max value = 22 -1

Binary Numbers

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3-bit binary number

000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7 max value = 23 -1

Binary Numbers

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4-bit binary number

0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 10 1011 11 1100 12 1101 13 1110 14 1111 15

max value = 24 -1

Binary Numbers

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reliability! Binary Numbers (why?)

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9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1

4 6 7 Binary Numbers (why?)

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9 8 7 6 5 4 3 2 1 9 8 7 6 5 4 3 2 1

1 Binary Numbers (why?)

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Binary Numbers How do we encode negative numbers?

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use left-most bit to represent sign 0 = “+” 1 = “-” Binary Numbers

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3-bit signed binary number

000 0 001 1 010 2 011 3 100 101 110 111

sign 2120

Binary Numbers

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3-bit signed binary number

000 0 001 1 010 2 011 3 100 -0 101 -1 110 -2 111 -3

sign 2120

??? Binary Numbers

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Binary Numbers (two’s complement)

  • 1. start with an unsigned 4-bit binary number where left-

most bit is 0

  • 0110 = 6
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  • 1. start with an unsigned 4-bit binary number where left-

most bit is 0

  • 0110 = 6
  • 2. complement your binary number (flip bits)
  • 1001

Binary Numbers (two’s complement)

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  • 1. start with an unsigned 4-bit binary number where left-

most bit is 0

  • 0110 = 6
  • 2. complement your binary number (flip bits)
  • 1001
  • 3. add one to your binary number
  • 1010 = -6

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 0 1 001 -1 2 010 -2 3 011 -3

positive negative complement +1

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 0 1 001 110 111 -1 2 010 -2 3 011 -3

positive negative complement +1

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 -3

positive negative complement +1

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3

positive negative complement +1

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 111 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3

positive negative complement +1

Binary Numbers (two’s complement)

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3-bit signed binary number

0 000 111 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3

positive negative

!!!

complement +1

Binary Numbers (two’s complement)

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we lost a number?

0 000 111 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3

positive negative complement +1

Binary Numbers (two’s complement)

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we lost a number?

0 000 111 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3 100

positive negative complement +1

Binary Numbers (two’s complement)

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100

complement

  • 1

Binary Numbers (two’s complement)

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011 100

complement

  • 1

010 011 100 101 110

  • 1

+1 +2

  • 2

Binary Numbers (two’s complement)

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100 011 100

complement

  • 1

Binary Numbers (two’s complement)

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4 100 011 100

complement

  • 1

Binary Numbers (two’s complement)

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0 000 111 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3 100 -4

positive negative complement +1

n-bit unsigned binary numbers: 0...2n-1 Binary Numbers (two’s complement)

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0 000 111 000 0 1 001 110 111 -1 2 010 101 110 -2 3 011 100 101 -3 100 -4

positive negative complement +1

n-bit signed binary numbers: -2n-1... 2n-1-1 Binary Numbers (two’s complement)

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0010 0010 + ---- 0100

summing unsigned binary numbers is easy

2 2 + - 4

Binary Numbers (two’s complement)

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summing signed binary numbers

0010 1010 + ---- 1100 2

  • 2

+ - ?

Binary Numbers (two’s complement)

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summing signed binary numbers

0011 1011 + ---- 1110 3

  • 3

+ - ?

Binary Numbers (two’s complement)

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summing signed (2‘s complement) binary numbers

0010 + ---- 2

  • 2

+ -

0010 -> 1101 -> 1110

Binary Numbers (two’s complement)

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summing signed (2‘s complement) binary numbers

0010 1110 + ---- 0000 2

  • 2

+ -

0010 -> 1101 -> 1110

Binary Numbers (two’s complement)

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0011 + ---- 3

  • 3

+ -

0011 -> 1100 -> 1101

summing signed (2‘s complement) binary numbers Binary Numbers (two’s complement)

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0011 1101 + ---- 0000 3

  • 3

+ -

0011 -> 1100 -> 1101

summing signed (2‘s complement) binary numbers Binary Numbers (two’s complement)

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Binary Numbers (decoding two’s complement) 4-bit signed (two’s complement) binary number

0111 = ?

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4-bit signed (two’s complement) binary number

0111 = 7

Binary Numbers (decoding two’s complement)

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4-bit signed (two’s complement) binary number

1011 = ?

Binary Numbers (decoding two’s complement)

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4-bit signed (two’s complement) binary number

subtract 1

1011 1010

Binary Numbers (decoding two’s complement)

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4-bit signed (two’s complement) binary number

subtract 1

1011 1010 0101

complement

Binary Numbers (decoding two’s complement)

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4-bit signed (two’s complement) binary number

subtract 1

1011 1010 0101 5

complement

Binary Numbers (decoding two’s complement)

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4-bit signed (two’s complement) binary number

1011 = -5

Binary Numbers (decoding two’s complement)

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Binary Numbers How do we encode fractional numbers?

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Binary Numbers ± mantissa x base ± exponent

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Boolean Logic (variables) 1 = True 0 = False

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a and b Boolean Logic (truth tables)

a b a and b

1 1 1 1 1

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a or b

a b a or b

1 1 1 1 1 1 1

Boolean Logic (truth tables)

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not a

a not a

1 1

Boolean Logic (truth tables)

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  • utput

(boolean variable)

a and b a or b not a input

(boolean variable)

a, b Boolean Logic (truth tables)

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a and b

Gates

a b a and b

1 1 1 1 1

⋅

a b

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a b a or b

1 1 1 1 1 1 1

a or b a b

+

Gates

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not a a not a

1 1

a

Gates

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Building Gates (transistors)

input power input power

1

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power

Building Gates (transistors)

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Building Gates (transistors)

power

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1 power

Building Gates (transistors)

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Building Gates (transistors)

1 power

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1 power

Building Gates (transistors)

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Building Gates (transistors)

1 power

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1 1 power

Building Gates (transistors)

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power 1

Building Gates (transistors)

1 1

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power 1 power 1 power 1 1 power 1

Building Gates (transistors)

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AND gate Building Gates (transistors)

power 1 power 1 power 1 1 power 1

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OR gate

1

Building Gates (transistors)

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OR gate

1 power 1

Building Gates (transistors)

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OR gate

1 1 power 1

Building Gates (transistors)

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OR gate

power

Building Gates (transistors)

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NOT gate

power

1

junk

resistor

Building Gates (transistors)

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NOT gate

1 power junk

resistor

Building Gates (transistors)