Fundamentals

1
Fundamentals

• Terms for magnitudes
• logarithms and logarithmic graphs
• Digital representations
• Binary numbers
• Text
• Analog information
• Boolean algebra
• Logical expressions and circuits

2
Information Technology Magnitude Terms

• Large
• kilo 1,000
• mega 1,000,000
• giga 1,000,000,000
• tera 1,000,000,000,000
• peta 1,000,000,000,000,000
• Small
• milli 1/1,000
• micro 1/1,000,000
• nano 1/1,000,000,000
• pico 1/1,000,000,000,000
• femto 1/1,000,000,000,000,000

3
Logarithms

• Because of these great differences in magnitudes,
  often use logarithms to represent values
• a logarithm is the power to which some base must
  be raised to get a particular value

For example, the base 10 logarithm of 1000
(written log10 1000) is 3, since 103 1000
4
Logarithm Scale Graphs

• Graphs often use a logarithmic scale on one axis
  so that the data fit on a reasonable size graph

5
Logarithm Scale Graphs (continued)

• The problem with this is that such graphs lose
  the impact of how rapidly the magnitudes change

6
Binary Numbers

• Digital systems operate using the binary number
  system
• only two digits, 0 and 1
• can be represented in computer several ways
• voltage high or low
• magnetized one direction or another
• each digit is a binary digit, or bit
• referred to as being in base 2
• Magnitudes of binary numbers determined using
  positional notation, just like decimal
• 269110 1100 9101 6102 2103
• 1001012 120 021 122 023 024 125

7
Converting Between Number Systems

• To convert binary to decimal, simply perform
  arithmetic in base 10
• 1001012 120 021 122 023 024 125
  1 4 32 37
• To convert decimal to binary, divide the decimal
  value by 2
• remainder is rightmost digit of binary number
• repeat on quotient

37/2 18 remainder 1 18/2 9 remainder 0 9/2
4 remainder 1 4/2 2 remainder 0 2/2
1 remainder 0 1/2 0 remainder 1
binary number is 100101
8
Converting Between Number Systems (continued)

• Alternatively, build a table of powers of 2,
  write 1 by largest magnitude less than value to
  convert, then subtract that from the number and
  repeat until get to 0
• produces number most significant digit first

9
Binary Arithmetic

• What happens if you add two digits in base 10 and
  get a result greater than 9?
• generate a carry
• Same thing happens if you add two binary digits
  and get a result greater than 1

5 5 10
0 0 1 1 0 1 0 1 0
1 1 10
10
Binary Arithmetic (continued)

• To do addition, we need just one more piece of
  information
• Then, we can add two binary numbers by using the
  four cases on the previous slide and the identity
  above

10 1 11
carry 1 1 1 1 0 0 0 addend 1 0 1 1 0 1
0 addend 1 1 1 1 0 0 result 1
0 0 1 0 1 1 0
11
Binary Arithmetic (continued)

• Subtraction uses a similar idea, that of a borrow
  from the next column left when were trying to
  subtract a larger digit from a smaller
• With binary digits, the same thing holds

1 6 - 9 7
0 1 1 10 - 0 - 0 - 1 - 1 0
1 0 1
12
Binary Arithmetic (continued)

• Consider a couple examples

1 0 0 0 - 1 1 1 1
1 0 0 1 0 1 0 - 1 1 1 0 1 1 1 1 0
0
13
Binary Arithmetic (continued)

• Multiplication is simple
• 0 times anything is 0
• 1 times anything is that thing again
• For example

1 0 1 1 0 0 1 x 1
0 0 1 0 1 1 0 1 1 0 0 1
0 0 0 0 0 0 0 1 0 1 1 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1
0 0 1 1 0 1 1 1 0 1
14
Binary Arithmetic (continued)

• For division, the divisor is either
• less than or equal to what its dividing into, so
  the quotient is 1
• greater than what its dividing into, so the
  quotient is 0
• For example

1 0 1 1 1 0 quotient 1 0 1 0
1 1 1 0 1 1 0 0 1 1
1 0 1 1
1 0 1 0
1 0 0 1 remainder
15
Octal and Hexadecimal

• Reading and writing binary numbers can be
  confusing, so we often use octal (base 8) or
  hexadecimal (base 16) numbers
• group binary number into sets of 3 (octal) or 4
  (hex) bits, then replace each group by its
  corresponding digit from the tables
• to convert back to binary, just replace each
  octal or hex digit with its binary equivalent

16
Real Numbers

• Previous numeric values were all integers
• We commonly use real numbers (with decimal point
  and fractional part) as well
• 24.125 2x101 4x100 1x10-1 2x10-2 5x10-3
• Same idea holds for binary numbers
• 11000.001 1x24 1x23 0x22 0x21 0x20
  0x2-1 0x2-2 1x2-3
• Can also write these in scientific notation
• 0.24125 x 102
• 0.11000001 x 25
• Referred to as floating point numbers in
  computer speak

17
Holding Binary Numbers in a Computer

• Computer memory is organized into chunks of 8
  bits, called bytes
• The range of values that an integer can hold
  depends on how many bytes of memory are used
• 1 byte 0 - 255
• 2 bytes -32,768 - 32,767
• 4 bytes -2,147,483,648 - 2,147,483,647
• Floating point numbers usually have 4 or 8 byte
  representations
• separate exponent and magnitude

0
1
7
8
31
magnitude
exp
sign
63
magnitude
exp
18
Representing Text

• Text is an example of discrete information
• like integers - there are only certain values
  that are allowed
• Representing text in a computer is simply a
  matter of defining a correspondence between each
  character and a unique binary number
• called a code
• need different numbers for upper and lower case
  representation of same letter
• need representation for digits 0 - 9 as
  characters
• want A to be less than B so its possible to
  alphabetize character information

19
Representing Text (continued)

• American Standard Code for Information
  Interchange (ASCII) code is standard for most
  computers
• 7-bit code (128 possible characters)
• stored in memory as single byte
• Wont represent non-Roman characters easily
• new 16-bit UniCode will

20
Representing Analog Information

• If the data we want to represent in a computer is
  not discrete but continuous, need to turn it into
  a sequence of numerical values by sampling
• examples are sound, pressures, images, video
• sequence of samples approximates original signal

21
Representing Analog Information (continued)

• Values used for the samples determine precision
  of measurement
• too coarse a division of the range of possible
  input values yields a poor approximation
• too fine a division wastes storage space (since
  more bits needed for each sample)
• 8 bits, 256 levels 16 bits, 65,536 levels

22
Representing Analog Information (continued)

• Number of samples in given time period is called
  the frequency or sample rate
• defined by number of measurements per second (Hz)
• sample rate needed depends on how rapidly the
  input signal changes

23
Representing Analog Information (continued)

• Need to trade off sampling rate and precision to
  achieve acceptable approximation without letting
  resulting digital data get too large
• Audio CD
• 44.1 KHz sampling rate
• 16 bit precision
• 1 minute of CD-quality stereo is almost 10.6
  Mbytes
• For images
• resolution (number of samples in horizontal and
  vertical direction) takes role of sampling rate
• Precision is measured by number of bits per
  sample (samples are called pixels)

per channel
24
Original 1600 x 800 Pixels, 24 Bits per Pixel
25
Lower Resolution 300 x 150 Pixels, 24 Bit
26
Lower Resolution 150 x 75 Pixels, 24 Bit
27
Lower Resolution 50 x 25 Pixels, 24 Bit
28
Lower Resolution 25 x 12 Pixels, 24 Bit
29
Base Image 300 x 150, 24 Bit
30
Lower Precision 300 x 200 Pixels, 8 Bit
31
Lower Precision 300 x 200 Pixels, 4 Bits
32
Lower Precision 300 x 200 Pixels, 1 Bit
33
Boolean Algebra

• Developed in 1854 by English mathematician George
  Boole
• logical algebra in which all quantities are
  either true or false
• fits well with binary representations (1 true,
  0 false)
• Foundation of all computer hardware design
• Three fundamental logical operations

example of a truth table
34
Boolean Algebra (continued)

• Its important that the possible values for A and
  B are assigned so they cover all the possible
  combinations
• assign methodically as shown on preceding slide

35
Boolean Algebra (continued)

• Two other logical operations (combinations of the
  fundamental ones) are important
• not or (nor)
• not and (nand)
• any logic function that can be expressed using
  and, or, not can also be expressed using just one
  of nand, nor

think of as or followed by not, or and followed
by not
36
Logical Expressions

• Can combine these logical operations just as we
  combine arithmetic expressions, to produce
  logical expressions
• order of operations is not first, then and, then
  or
• do equal precedence operations left to right
• change order with parentheses

37
Implementing Logical Expressions

• To convert the logical expression to a circuit
  that calculates the equivalent logical value,
  simply provide a circuit for each of the terms of
  the logical expression

38
Implementing Logical Expressions (continued)

• Of course, its not really as simple as this
• there may be many possible logical expressions
  that produce the same output of 0s and 1s
• the hardware designer must choose the optimal one
  based on one or more criteria
• minimum number of logic functions
• fewest different types of logic functions
• fewest levels of logic functions between inputs
  and outputs

39
Remembering the Past

• The previous logic circuit is an example of a
  combinational circuit
• the output at any given time depends solely on
  the current values of the input
• Another kind of logic circuit is a sequential
  circuit
• the output at any given time depends on the
  current values of the input and the current value
  of the output