IKI10201 02Data Types

1
IKI1020102-Data Types Representations

• Bobby Nazief
• Semester-I 2005 - 2006

The materials on these slides are adopted from
those in CS231s Lecture Notes at UIUC, which is
derived from Howard Huangs work and developed by
Jeff Carlyle.
2
Road Map
Logic Gates Flip-flops
3
Boolean Algebra
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6
Finite-StateMachines
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4
Sequential DesignTechniques
Logic DesignTechniques
CombinatorialComponents
StorageComponents
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Binary Systems Data Represent.
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5
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Register-TransferDesign
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Generalized FSM
ProcessorComponents
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3
Number Systems

• To get started, well discuss one of the
  fundamental concepts underlying digital computer
  design
• Deep down inside, computers work with just 1s and
  0s.
• Computers use voltages to represent information.
  In modern CPUs the voltage is usually limited to
  1.6-1.8V to minimize power consumption.
• Its convenient for us to translate these
  analogvoltages into the discrete, or digital,
  values 1 and 0.
• But how can two lousy digits be useful for
  anything?
• First, well see how to represent numbers with
• just 1s and 0s.
• Then well introduce special operations
• for computing with 1s and 0s, by treating them
  as
• the logical values true and false.

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Positional number systems decimal

• Numbers consist of a bunch of digits, each with a
  weight
• The weights are all powers of the base, which is
  10. We can rewrite the weights like this
• To find the decimal value of a number, multiply
  each digit by its weight and sum the products.

(1 x 102) (6 x 101) (2 x 100) (3 x 10-1)
(7 x 10-2) (5 x 10-3) 162.375
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Positional number systems binary

• We can use the same trick for binary, or base 2,
  numbers. The only difference is that the weights
  are powers of 2.
• For example, here is 1101.01 in binary
• The decimal value is

(1 x 23) (1 x 22) (0 x 21) (1 x 20) (0 x
2-1) (1 x 2-2) 8 4 0
1 0 0.25 13.25
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Base 8 - Octal

• The octal system uses 8 digits
• 0 1 2 3 4 5 6 7
• Earlier in this history of computing, octal was
  used as a shorthand for binary numbers. (Now
  hexadecimal is more common.)
• Since 8 23, one octal digit is equivalent to 3
  binary digits.
• Numbers like 67 are easier to work with than
  110111.
• Still shows up in some places. For instance, file
  access permissions in Unix file systems.

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Base 16 is useful too

• The hexadecimal system uses 16 digits
• 0 1 2 3 4 5 6 7 8 9 A B C D E F
• For our purposes, base 16 is most useful as a
  shorthand notation for binary numbers.
• Since 16 24, one hexadecimal digit is
  equivalent to 4 binary digits.
• Its often easier to work with a number like B4
  instead of 10110100.
• Hex is frequently used to specify things like
  IPv6 addresses and 24-bit colors.

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Converting decimal to binary

• To convert a decimal integer into binary, keep
  dividing by 2 until the quotient is 0. Collect
  the remainders in reverse order.
• To convert a fraction, keep multiplying the
  fractional part by 2 until it becomes 0. Collect
  the integer parts in forward order.
• Example 162.375
• So, 162.37510 10100010.0112

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Why does this work?

• This works for converting from decimal to any
  base
• Why? Think about converting 162.375 from decimal
  to decimal.
• Each division strips off the rightmost digit (the
  remainder). The quotient represents the remaining
  digits in the number.
• Similarly, to convert fractions, each
  multiplication strips off the leftmost digit (the
  integer part). The fraction represents the
  remaining digits.

162 / 10 16 rem 2 16 / 10 1 rem 6 1 /
10 0 rem 1
0.375 x 10 3.750 0.750 x 10 7.500 0.500 x 10
5.000
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Binary and hexadecimal conversions

• Converting from hexadecimal to binary is easy
  just replace each hex digit with its equivalent
  4-bit binary sequence.
• To convert from binary to hex, make groups of 4
  bits, starting from the binary point. Add 0s to
  the ends of the number if needed. Then, just
  convert each bit group to its corresponding hex
  digit.

261.3516 2 6 1 . 3 516
0010 0110 0001 . 0011 01012
10110100.0010112 1011 0100 . 0010 11002 B
4 . 2 C16
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Practice

• Convert 32103.510 to hexadecimal.
• Convert CAFE16 to decimal.
• Convert 7038 to base-12.

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Convert 32103.510 to hexadecimal.

• 32103 / 16 2006 rem 7
• 2006 / 16 125 rem 6
• 125 / 16 7 rem 13
• 7 / 16 0 rem 7
• .5 16 8.0
• So 7D67.816.

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Convert CAFE16 to decimal.

• C 12 163 49152
• A 10 162 2560
• F 15 161 240
• E 14 16- 14
• 4915225602401451966

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Convert 7038 to base-12.

• 7 7 82 448
• 0 0 81 0
• 3 3 80 3
• 44803 451
• 451 / 12 37 rem 7
• 37 / 12 3 rem 1
• 3 / 12 0 rem 3
• So 31712.

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Number Systems Summary

• Computers are binary devices.
• Were forced to think in terms of base 2.
• Today we learned how to convert numbers between
  binary, decimal and hexadecimal.
• Weve already seen some of the recurring themes
  of architecture
• We use 0 and 1 as abstractions for analog
  voltages.
• We showed how to represent numbers using just
  these two signals.

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Additional information

• Assistants
• Panca Novianto panca.novianto_at_mhs.cs.ui.ac.id
• Evan Jonathan Winata evan.winata_at_mhs.cs.ui.ac.id

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Review - Positional number systems binary

• We can use the same trick for binary, or base 2,
  numbers. The only difference is that the weights
  are powers of 2.
• For example, here is 1101.01 in binary
• The decimal value is

(1 x 23) (1 x 22) (0 x 21) (1 x 20) (0 x
2-1) (1 x 2-2) 8 4 0
1 0 0.25 13.25
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Review - Converting decimal to binary

• To convert a decimal integer into binary, keep
  dividing by 2 until the quotient is 0. Collect
  the remainders in reverse order.
• To convert a fraction, keep multiplying the
  fractional part by 2 until it becomes 0. Collect
  the integer parts in forward order.
• Example 162.375
• So, 162.37510 10100010.0112

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Addition and Subtraction of Binary Numbers

• Arithmetic is the most basic thing you can do
  with a computer, but its not as easy as you
  might expect!
• These next few lectures focus on addition and
  subtraction.
• Computers were designed to compute, so arithmetic
  is at the heart of a CPU.

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Binary addition by hand

• You can add two binary numbers one column at a
  time starting from the right, just as you add two
  decimal numbers.
• But remember that its binary. For example, 1 1
  10 and you have to carry!

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Subtraction

• Computers do subtraction by adding the minuend
  with the negative representation of the
  subtrahend
• The main problem is representing negative numbers
  in binary. We introduce three methods, and show
  why one of them is the best.
• With negative numbers, well be able to do
  subtraction using the adders we made last time,
  because A - B A (-B).

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Negative Numbers
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Signed magnitude representation

• Humans use a signed-magnitude system we add or
  - in front of a magnitude to indicate the sign.
• We could do this in binary as well, by adding an
  extra sign bit to the front of our numbers. By
  convention
• A 0 sign bit represents a positive number.
• A 1 sign bit represents a negative number.
• Examples

11012 1310 (a 4-bit unsigned number) 0 1101
1310 (a positive number in 5-bit signed
magnitude) 1 1101 -1310 (a negative number in
5-bit signed magnitude)
01002 410 (a 4-bit unsigned number) 0 0100
410 (a positive number in 5-bit signed
magnitude) 1 0100 -410 (a negative number in
5-bit signed magnitude)
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Signed magnitude operations

• Negating a signed-magnitude number is trivial
  just change the sign bit from 0 to 1, or vice
  versa.
• Adding numbers is difficult, though. Signed
  magnitude is basically what people use, so think
  about the grade-school approach to addition. Its
  based on comparing the signs of the augend and
  addend
• If they have the same sign, add the magnitudes
  and keep that sign.
• If they have different signs, then subtract the
  smaller magnitude from the larger one. The sign
  of the number with the larger magnitude is the
  sign of the result.
• This method of subtraction would lead to a rather
  complex circuit.

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Ones complement representation

• A different approach, ones complement, negates
  numbers by complementing each bit of the number.
• We keep the sign bits 0 for positive numbers,
  and 1 for negative. The sign bit is complemented
  along with the rest of the bits.
• Examples

11012 1310 (a 4-bit unsigned number) 0 1101
1310 (a positive number in 5-bit ones
complement) 1 0010 -1310 (a negative number in
5-bit ones complement)
01002 410 (a 4-bit unsigned number) 0 0100
410 (a positive number in 5-bit ones
complement) 1 1011 -410 (a negative number in
5-bit ones complement)
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Why is it called ones complement?

• Complementing a single bit is equivalent to
  subtracting it from 1.
• 0 1, and 1 - 0 1 1 0, and 1 - 1 0
• Similarly, complementing each bit of an n-bit
  number is equivalent to subtracting that number
  from 2n-1.
• For example, we can negate the 5-bit number
  01101.
• Here n5, and 2n-1 3110 111112.
• Subtracting 01101 from 11111 yields 10010

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Ones complement addition

• To add ones complement numbers
• First do unsigned addition on the numbers,
  including the sign bits.
• Then take the carry out and add it to the sum.
• Two examples
• This is simpler and more uniform than signed
  magnitude addition.

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Twos complement

• Our final idea is twos complement. To negate a
  number, complement each bit (just as for ones
  complement) and then add 1.
• Examples

11012 1310 (a 4-bit unsigned number) 0 1101
1310 (a positive number in 5-bit twos
complement) 1 0010 -1310 (a negative number in
5-bit ones complement) 1 0011 -1310 (a
negative number in 5-bit twos complement)
01002 410 (a 4-bit unsigned number) 0 0100
410 (a positive number in 5-bit twos
complement) 1 1011 -410 (a negative number in
5-bit ones complement) 1 1100 -410 (a negative
number in 5-bit twos complement)
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More about twos complement

• Two other equivalent ways to negate twos
  complement numbers
• You can subtract an n-bit twos complement number
  from 2n.
• You can complement all of the bits to the left of
  the rightmost 1.
• 01101 1310 (a positive number in twos
  complement)
• 10011 -1310 (a negative number in twos
  complement)
• 00100 410 (a positive number in twos
  complement)
• 11100 -410 (a negative number in twos
  complement)

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Comparing the signed number systems

• Here are all the 4-bit numbers in the different
  systems.
• Positive numbers are the same in all three
  representations.
• Signed magnitude and ones complement have two
  ways of representing 0. This makes things more
  complicated.
• Twos complement has asymmetric ranges there is
  one more negative number than positive number.
  Here, you can represent -8 but not 8.
• However, twos complement is preferred because it
  has only one 0, and its addition algorithm is the
  simplest.

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Converting signed numbers to decimal

• Convert 110101 to decimal, assuming this is a
  number in
• (a) signed magnitude format
• (b) ones complement
• (c) twos complement

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Example solution

• Convert 110101 to decimal, assuming this is a
  number in
• Since the sign bit is 1, this is a negative
  number. The easiest way to find the magnitude is
  to convert it to a positive number.
• (a) signed magnitude format
• Negating the original number, 110101, gives
  010101, which is 2110 in decimal. So 110101
  must represent -2110.
• (b) ones complement
• Negating 110101 in ones complement yields
  001010 1010, so the original number must have
  been -1010.
• (c) twos complement
• Negating 110101 in twos complement gives
  001011 1110, which means 110101 -1110.
• The most important point here is that a binary
  number has different meanings depending on which
  representation is assumed.

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Addition Subtraction of 2s complement numbers

• Subtraction
• Form 2s complement of the subtrahend
• Add the two numbers as in Addition

• Addition
• Just add the two numbers
• Ignore the Carry-out from MSB
• Result will be correct, provided theres no
  overflow

0 1 0 1 (5)0 0 1 0 (2) 0 1 1 1 (7)
0 1 0 1 (5)1 0 1 0 (-6) 1 1 1 1 (-1)
0 0 1 0 (2) 0 0 1 0?0 1 0 0 (4) 1 1 0
0 (-4) 1 1 1 0 (-2)
1 0 1 1 (-5)1 1 1 0 (-2)11 0 0 1 (-7)
0 1 1 1 (7)1 1 0 1 (-3)10 1 0 0 (4)
1 1 1 0 (-2) 1 1 1 0?1 0 1 1 (-5) 0 1 0
1 (5) 10 0 1 1 (3)
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Overflow

• Examples 7 3 10 but ...
  
• - 4 5 - 9 but ...

1
1
1
0
1
0
1
1
1
1
1
0
0
7
4
3
5
0
0
1
1

1
0
1
1

1
0
1
0
0
1
1
1
6
7
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Binary Multiplication

• Since we always multiply by either 0 or 1, the
  partial products are always either 0000 or the
  multiplicand (1101 in this example).
• There are four partial products which are added
  to form the result.

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Shifting the multiplicand
1 1 0 1 Multiplicand x 0 1 1 0 Multiplier
0 0 0 0 first partial products 0 0 0 0 sh
ifted zeros 0 0 0 0 second partial
products 1 1 0 1 shifted multiplicand 1 1
0 1 0 third partial products 1 1 0 1 shifted
multiplicand 1 0 0 1 1 1 0 fourth partial
products 0 0 0 0 shifted
zeros 1 0 0 1 1 1 0 Product
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Binary Division
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Floating-Point Numbers
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Scientific notation review
6.02 x 1023

• Normalized form no leadings 0s (exactly one
  digit to left of decimal point)
• Alternatives to representing 1/1,000,000,000
• Normalized 1.0 x 10-9
• Not normalized 0.1 x 10-8,10.0 x 10-10

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Scientific notation for binary numbers
1.0two x 2-1

• Computer arithmetic that supports it called
  floating point, because it represents numbers
  where binary point is not fixed, as it is for
  integers
• Declare such variable in C as float

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Floating point representation

• Normal format 1.xxxxxxxxxxtwo2yyyytwo
• Multiple of Word Size (32 bits)

• S represents SignExponent represents
  ysSignificand represents xs
• Represent numbers as small as 2.0 x 10-38 to as
  large as 2.0 x 1038

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BCD
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Character Codes
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Codes for Error Detection Correction

• Please read yourself.

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Hamming Codes

• Please read yourself.