Number Systems and Binary Math

1
Number Systems and Binary Math
2
Why do computers use switches?

• Why did engineers select a binary-state (two
  state) switch as the fundamental computer
  component building block.
• It was simple and
• reliable and
• it supported built-in error checking.

3
Adding Switches Doubles Capacity

• 1 switch 2 numbers
• 2 switches 4 numbers
• 3 switches 8 numbers
• 4 switches 16 numbers

4
Units of Computer Measurement

• The smallest unit of computer measurement is a
  binary digit (or more commonly, the bit).
• The bit represents a single binary digit, either
  a 1 or a 0.

5
Larger Units of Measurement

• Bit A single binary digit
• Nibble 4 Bits
• Byte 2 Nibbles 8 Bits
• Word 2 Bytes 16 Bits
• Double Word 2 Words 32 Bits
• Kilobyte 1024 Bytes
• Megabyte 1024 Kilobytes
• Gigabyte 1024 Megabytes

6
Starting with Numbers

• Each switch can either have sufficient voltage
  (ON) or have insufficient voltage (OFF).
• Another way of stating voltage 1 or 0
• This is a common approach in science to use
  abstract, notational systems to describe things.

1
0
7
Numbering Systems

• To figure the answer to this question, lets step
  back and look at numbering systems in general.
• Originally, human being counted with tally math
  systems, with a mark for each sheep they were
  counting
• If you had 10 sheep, you made 10 marks

8
Numbering Systems

• Later, there were advancements in tally math,
  using special symbols for special numbers.
• Instead of 10 marks for 10 sheep, lets make a
  special mark for 10 sheep
• But this notational system was cumbersome you
  had to learn a lot of symbols, and arithmetic was
  difficult

9
Positional Numbering Systems

• Wonderful advancement over tally math.
• Lets keep a few, core numbers (digits).
• We can use these digits to represent REALLY big
  numbers if we say that the value of a number is
  now made up of 2 parts
• A count value
• A place value

10
Place Values
3 7

• Consider the number 37
• The 7 stands for 7 (its count value) ones (its
  positional value). Together, 7 times 1 is 7

11
Place Values
3 7

• Consider the number 37
• The 3 stands for 3 (its count value) tens (its
  positional value). Together, 3 times 10 is thirty

12
Place Values
3 7

• Consider the number 37
• Total value 7 (7 1, from the ones position)
  and thirty (3 10, from the 10 position) is 37!

13
Place Values
3 7
Ones
Tens
7 1
3 10
14
Base-10 (Decimal)

• A base refers to the number of digits (counting
  numbers) available in a numbering scheme.
• The highest single digit in base is one less than
  the base number.
• In Base-10, the digits are
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

15
Base-10 (Decimal)

• What happens after 9?
• The counter in the right column resets itself to
  0 and 1 is added to the value of the next
  left-most column

0 9
1 0
16
Base-10 (Decimal)

• Interpreting a multiple digit number simply means
  to expand its notation, and write it out by each
  positional value

7 4 2
Hundreds
Ones
Tens
2 1
4 10
7 100
17
Base-10 (Decimal)

• What about using some math to describe the
  number

7 4 2
100
101
102
2 100
4 101
7 102
18
Using Expanded Notation

• We can use our expanded notation to figure the
  value of a number

19
A Note on Superscripts

• Weve seen superscripts in expressions before
  today. In math, they are used to indicate an
  exponent in an expression

xy
20
A Note on Subscripts

• Lets introduce one more notation the
  subscript. It is used to indicate the base of a
  number (the number of digits available in a
  numbering system)

xy
21
A Note on Subscripts

• Remember our previous example?
• It had a subscript of 10, meaning that the
  number we use the Base-10 (Decimal) numbering
  system to decode this number.

74210
22
Thinking in Binary (Base-2)

• Instead of having 10 digits, the binary (Base-2)
  numbering systems has 2 base digits (0 and 1).
• Counting in Base-2 0, 1, 10, 11, 100, 101, 110,
  111, 1000, 1001, 1010
• In binary, after the right-most placeholder
  reaches one, it resets to 0 and advances the next
  left-most placeholder.
• Our base number in binary is 2, therefore we can
  use exponents of 2 for placeholder notation

23
Binary Placeholders

• Lets deconstruct 10112

24
Binary Placeholders

• Now, lets use exponents

1 0 1 1
20
21
22
23
25
Converting Base-2 to Base-10
1 0 1 1

• Lets convert 10112.
• STEP 1 Find all switches that are turned OFF.
  Cross them out and bring down the zero.

20
21
22
23
0
26
Converting Base-2 to Base-10
1 0 1 1

• STEP 2 For each placeholder turned ON,
  calculate its exponential expression.

20
21
22
23
0
1
2
8
27
Converting Base-2 to Base-10
1 0 1 1

• STEP 3 Add the results of the exponents. The sum
  is the Base-10 equivalent.

20
21
22
23
0
1
2
8




1110
28
Converting from Base-10 to Base-2

• Consider 2310. How can we translate this to its
  Base-2 equivalent?
• Well use successive division. Essentially, that
  means well divide by 2 over and over and keep
  track of our remainders.

29
Converting Base-10 to Base-2

• STEP ONE Draw a table with three columns. Label
  the last two columns quotient and remainder,
  respectively.

30
Converting Base-10 to Base-2

• STEP TWO In the first available left-most
  column, put the expression 23/2. Calculate the
  division and put the quotient and remainder in
  their respective columns.

31
Converting Base-10 to Base-2

• STEP THREE Take the quotient of the previous
  expression and divide it by two. Repeat until
  have 0 for a quotient.

32
Converting Base-10 to Base-2

• STEP FOUR Read the REMAINDER column from the
  bottom to the top. The 1s and 0s in the remainder
  column represent the binary number.

33
Converting Base-10 to Base-2

• 2310 101112

34
Checking the Results

• So.. 2310 101112!
• Lets check our answer by reversing the process
• 101112 (1 24) (0 23 ) ( 1 22 ) (1
  21) (1 20)
• 16 0 4 2 1 2310

35
Groups of Switches

• We have already seen that a single switch cant
  encode much information. However, if we group
  switches together, we can expand our encoding
  ability.
• Computers often encode many types of data,
  graphics and large numbers for example, not in
  Base-2, but in Base-8 or Base-16.

36
Introducing Octal

• Computer scientists are often looking for
  shortcuts to do things. One of the ways in which
  we can represent Base-2 numbers is to use their
  octal (Base-8) equivalents.
• This is especially helpful when we have to do
  fairly complicated tasks using large numbers.

37
Introducing Octal

• The octal numbering system includes eight base
  digits, 0-7.
• After 7, the next placeholder to the right begins
  with a 10, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13
  ...

38
Octal Digits
Click to Return to the Last Slide Viewed
39
Octal Placeholders
Ones
Placeholder Name
Eights
Sixty-Fours
642
84
11
Value
Exponential Expression
822
814
801
40
Converting Base-2 to Base-8
100011001010012
STEP ONE Take the binary number and from right
to left, group all placeholders in triplets. Add
leading zeros, if necessary
010
001
101
001
100
Click to See Octal Digits Table
41
Converting Base-2 to Base-8
100011001010012
214518

STEP TWO Convert each triplet to its
single-digit octal equivalent. (HINT For each
triplet, the octal conversion is the same as
converting to a decimal number)
010
001
101
100
001
2
1
1
5
4
42
Converting Base-8 to Base-2
435208
1000111010100002

Take each octal digit and convert each digit to a
binary triplet. Insert leading zeros, if needed
4
3
0
2
5
100
000
010
011
101
Click to See Octal Digits Table
43
Converting Base-8 to Base-10
23748 N10
Multiply each octal digit by the exponential
expression that represents its placeholder and
then add the values
283 2512 1024 8 382 364 192 8 781
78 56 8 480 41 4 8 127610
44
Converting Base-10 to Base-8

• To convert Base-10 to Base-8, well use
  successive division
• Our new problem483210 ?8

45
Converting Base-10 to Base-8

• STEP ONE Draw a table with three columns. Label
  the last two columns quotient and remainder,
  respectively.

46
Converting Base-10 to Base-8

• STEP TWO In the first available left-most
  column, put the expression 4832/8. Calculate the
  division and put the quotient and remainder in
  their respective columns.

4832/8
604
0
47
Converting Base-10 to Base-8

• STEP THREE Take the quotient of the previous
  expression and divide it by eight. Repeat until
  youve have 0 for a quotient.

4832/8
604
0
48
Converting Base-10 to Base-8

• STEP FOUR Read the REMAINDER column from the
  bottom to the top. The numbers in the remainder
  column represent the octal number.

49
Converting Base-10 to Base-8

• 483210 113408

50
Base-16

• Whats tricky about Base-16?
• This time, we are converting to a base that has
  more base digits than Base-10 (Base-2 and Base-8
  have fewer digits than Base-10)
• So what?

51
Hexadecimal Numbering

• There are new symbols for the Base-16 equivalents
  of the Base-10 numbers 10, 11, 12, 14 and 15.
  Examine

52
Converting Base-10 to Base-16

• To convert Base-10 to Base-16, well again use
  successive division
• Our new problem21410 ?16

53
Converting Base-10 to Base-16

• STEP ONE Draw a table with three columns. Label
  the last two columns quotient and remainder,
  respectively.

54
Converting Base-10 to Base-16

• STEP TWO In the first column, write the
  expression 214/16. Calculate the expression,
  taking care not to divide past the decimal point.
  Write the quotient and remainder in their
  respective columns.

55
Converting Base-10 to Base-16

• STEP THREE Keep bringing down the quotient of
  the previous expression and dividing it by 16
  until you have a quotient of zero.

56
Converting Base-10 to Base-16

• STEP FOUR Convert all remainders to their hex
  equivalents.

57
Converting Base-10 to Base-16

• STEP FIVE Read the numbers in the REMAINDER
  column from the bottom up to get your hex
  equivalent.

21410 D616
58
Binary Arithmetic

• Computer processing requires a little more
  capability in binary math, notably arithmetic.
• Lets consider addition and subtraction

59
Binary Addition

• Just a few combinations are possible
• 02 02 02
• 02 12 12
• 12 02 12
• What about 12 12?

60
Binary Addition

• 12 12 102
• Remember what that 10 means it means a zero in
  the right column, and a one in the left column
• When you carry in base two, you are bringing a
  power of two to the left

61
Binary Addition
1
1
11102 10112
1410
1110
1
0
0
2510
2
62
Binary Subtraction

• Subtraction in base two is similar to addition
• 112 112 002
• 112 002 112
• 102 002 102
• What about 102 - 12?

63
Binary Addition

• 102 - 12 12
• Think about borrowing in decimal you are really
  borrowing a power of your base
• In Base-10, we borrow 10 from the left column and
  give it to the next column to the right. So
• In Base-2, we borrow 2 from the left column and
  give it to the next column to the right.

64
Binary Addition
1
1
1
001100112 -000101102
0
0
5110
2210
1
0
1
1
1
0
0
0
2910
2
65
Binary Borrowing A Little Trick

• Many students find it convenient to just mentally
  convert the binary number to decimal, complete
  the subtraction, and then convert the answer back
  to binary.
• Consider 102 12 ?
• A one in binary is still a one in decimal
• When you borrow for the zero, you bring over a
  power of two, which is equal to two
• The decimal conversion for the borrowing is 2-1,
  which is 1
• And 110 12
• Be sure to add your answer back up as a check

66
Negative Numbers

• We have mastered binary math for positive
  integers
• But what about negative numbers?
• Handling the negative sign has proven somewhat
  problematic, as we will see

67
Negative Integers Sign/Magnitude Notation

• The first solution for encoding negative numbers
  in binary form was to dedicate a switch (which we
  now know is a memory location) to communicate
  whether a number was positive or negative
• The only thing required was to agree, in the bank
  of switches representing the number, which switch
  referred to the sign
• In one of the more popular encoding notations
  for negative numbers, the first switch is the
  sign.

68
Negative Integers Sign/Magnitude Notation

• A 0 setting means the number is positive, a 1
  setting means the number is negative.
• This encoding scheme (for signed integers) is
  called sign/magnitude notation.
• The first switch is the sign, the remaining
  switches represent the binary magnitude of the
  number.
• Which highlights one of our key concepts about
  binary encoding the leftmost digit could be a
  negative sign, or a one in magnitude
• The computer must be told which
  interpretation/decoding scheme to use

69
Sign/Magnitude The Problem

• What was wrong with this scheme?
• Great for all numbers except 0
• This encoding scheme allows you to come up with 2
  distinct encoding schemes for 0, one with a
  positive sign, and one with the negative sign
• The one with the negative sign has no
  mathematical meaning, and the ambiguity of two
  patterns representing the same number is
  problematic.
• So, most computers use a different scheme than
  sign/magnitude for representing negative integers
  

70
Twos Complement

• Twos Complement solves the ambiguity problem.
• The scheme works because we are in binary math,
  which has only 2 digits.
• Offsetting one moves you to the only other
  possible digit, etc.

71
Twos Complement

• In Twos Complement, you encode positive integers
  normally.
• To use Twos Complement for negatives
• Complement every digit in the number (change 1s
  to 0s and 0s to 1s)
• Add 1 to the complemented number
• Example101 010 1 011

72
Twos Complement Math

• Twos Complement allows us to add the
  representation and processing of negative
  integers to our growing computing capability.
• To add a positive and negative number, first
  perform the twos complement trick on the negative
  number, and then add normally.
• To add two negative numbers, first encode both of
  them in 2s complement, and then add normally.

73
What about fractional values?

• Our goal is to learn how to encode all numbers in
  a way that they can be represented using switch
  settings
• We can encode positive integers
• We can encode negative integers
• But what about decimal numbers (numbers with
  fractional values)?
• To store decimal numbers, most computers utilize
  scientific notation

74
Binary Math and Scientific Notation

• As a civilization, we now work with numbers so
  big and so small that mathematicians have already
  had to wrestle with creating a manageable
  notation.
• Scientists developed scientific notation as a
  standard way to represent numbers.

75
Binary Math and Scientific Notation

• Consider the following example/-5.334
  10/-23
• The 5.334 part is the mantissa.
• The 10 part is the base.
• The 23 part is the exponent.

76
Scientific Notation

• Lets move up a level of abstraction to the more
  generalized case of scientific notation
• /- M B/-E
• M is the mantissa.
• B is the base.
• E is the exponent.
• /- means the number can be positive or negative.

77
Switch Math the Holes

• If we can figure out how to use scientific
  notation to encode base ten decimal numbers in a
  binary form of scientific notation, we can
  represent any finite number in our computer and
  our switch math will be complete

78
Interpreting Scientific Notation

• It turns out that handling scientific notation,
  or any other encoding scheme, is amazingly simple
• We design the computer to know which
  interpretation scheme to use!
• We will just figure out how to program in some
  logic that says if a number is tagged as being
  in scientific notation, just use a look up chart
  to interpret the number.

79
Interpreting Scientific Notation

• Given an encoded number, to decode it such that
  it is in scientific notation, you would need to
  know several things
• How big is the overall memory allocation (how
  many switches were used)?
• How many switches are dedicated to the mantissa,
  and which are they?
• How many switches are dedicated to the exponent,
  and which are they?

80
One Last Thing!

• Okay, so we know we will always need to know the
  encoding scheme to interpret scientific notation.
• But, once we know the encoding scheme for any
  given computer, we need to be able to encode in
  this scientific notation form

81
Fractional Value Numbers

• Consider the decimal (base 10) number 5.75
• How can we encode this is switches?
• We already know how to do half of the encoding
  the integer part (the numbers to the left of the
  decimal point).
• Lets break the problem into two parts, the left
  hand side (of the decimal point) and the right
  hand side

82
Left Side of a Decimal Number

• Okay, our number is 5.75
• To convert the 510 to ?2
• Use successive division

510 1012
83
Right Side of a Decimal Number

• On the left hand side, we divided successively by
  2 to convert
• On the right hand side, we multiply successively
  by 2 to convert
• We are multiplying numbers by 2 to promote them
  to the left hand side of the decimal place.
• Once promoted, we record them as a digit in our
  answer.
• This process is called extraction.

84
Extraction

• STEP ONE Draw a table with two columns. Label
  the columns for the expression and the
  extraction, respectively.

85
Extraction

• STEP TWO Write down your original number and
  multiply it by 2. Put the whole number part in
  the Extraction column.

86
Extraction

• STEP THREE Bring down the fractional part (the
  number to the right of the decimal point) and
  multiply it by two. Extract the whole number, as
  before.Repeat until you extract 0.

87
Extraction

• STEP FOUR Read the extraction column, from top
  to bottom. This number represents your binary
  equivalent.

0.7510 1102
88
Both Sides Now

• 5.7510 101.1102 This isnt in scientific
  notation yet, but at least we have figured out
  how to convert it to binary
• Recall the steps so far
• Do the problem in halves
• For the left side of the decimal, use successive
  division by 2 and read the remainders from bottom
  to top
• For the right hand side of the decimal, use
  successive multiplication by 2 and read the
  extractions from top to bottom

89
Lets Try One More

• Lets try another problem. This time, with a
  number that doesnt convert to binary as easily
• Lets try this number
• 2.5410 ?2

90
Converting

• Remember our algorithm left side by successive
  division, right side by successive multiplication

91
Left Side of 2.54

• 2 by successive division

Reading the remainders from bottom to top, 210
102
92
Right Side of 2.54

• Reading the extraction column, top to bottom
  
• .5410 1000101002
• What would have if we kept extracting, past .24
  2?
• Lets convert our number back to Base-10 to see
  what happens

93
Lets Go Back the Other Way

• Putting 10.1000101002 back to base 10
• Left hand side first, expanded notation
• 102 (0 20) (1 21) 0 2 210
• Now, to convert the right hand side

94
Right Side of a Decimal Number

• Remember, the left hand side represents powers of
  two positive powers of two
• Moving left from the decimal point, we saw the
  unit values were 20, 21, 22, 23, 24, etc.
• Doesnt it seem reasonable that the units to the
  right of the decimal represent negative powers of
  two?
• Moving from the right of the decimal point, we
  will see unit values for 2-1, 2-2, 2-3, 2-4, 2-5,
  etc.

95
Okay, But What Does That Mean?

• Exactly what is a number to a negative power,
  such as 2-1, 2-2, 2-3?
• These are numbers you already know we are just
  used to seeing them written in a different
  format

96
Back to Our Conversion

• Remember, we are translating 10.100010100
• We finished the left hand side 102 210
• Now for the right hand side
• .100010100
• (1 2-1) (0 2-2) (0 2-3) (0 2-4)
  (1 2-5) (0 2-6)
• (1 2-7) (0 2-8) (0 2-9)
• Which is equal to ½ 1/32 1/128 (we can throw
  out the zero terms)
• Which is equal to 69/128 .539
• Combining both sides, 10.1000101002 2.53910
• But we started with 2.54! Why did we get the
  error?

97
The Imprecision of Conversions

• We started with 2.5410, took it to switch units
  in binary, and then back to decimal
• Our result was 2.539
• This is a precision error
• 2.539 / 2.540 .99, so our percentage error is
  1!!!
• As you might imagine, in large, multi-step
  calculations these errors can become significant
  enough for you to see if you are working in a
  package such as Excel.

98
Storing Binary in Scientific Notation

• Remember, we said if we could encode a decimal
  number in switch math, we were good to go.
• Decimal numbers are usually stored in a binary
  version of scientific notation.
• If we can take our translation of 2.54, and store
  it in a binary version of scientific notation, we
  can quit!

99
Last Step

• There are several switch encoding schemes for
  scientific notation.
• The good news, remember, is that you (and the
  computer!) will always have to be told which
  encoding scheme to use.
• One common encoding scheme is 16-bit encoding

100
16 Bit Scientific Notation Encoding

• In 16 bit encoding, 16 bits (or switches) are
  allocated to the numeric part of a number
  expressed in scientific notation.
• There are a few different flavors of 16 bit
  encoding, with a different allocation of switches
  between the mantissa and the exponent.

101
One 16 Bit Encoding in Detail

• Remember the parts of a number in scientific
  notation? the mantissa, base and exponent?
• Each is assigned to specific locations in the 16
  bit switch pattern. In the variation we are
  looking at first
• Switch 1 Sign of the mantissa
• (Decimal point assumed to go next, but isnt
  stored)
• Switches 2-10 next 9 switches are for the
  mantissa
• Switch 11 Sign of the exponent
• Switches 12-16 next 5 switches are for the
  exponent

102
16 Bit More

• What happened to the base? We dont have to store
  that, because it will always be base two, the
  binary base from the world of switches
• Dont need to store the decimal point, because
  the encoding scheme will guarantee it is placed
  correctly when decoded
• Theres only one problem our number must be
  normalized before encoding
• This means that we need to adjust the exponent as
  required to insure that the first significant
  digit is the first digit to the right of the
  decimal point

103
Normalizing

• Take the number 101.112, which is really 101.11
  20
• Before we can store this number in scientific
  notation, we must normalize it.
• We have to adjust the exponent until the first
  significant digit, which is the leftmost 1, is
  the first digit to the right of the decimal
  point.
• In other words, we have to move the decimal place
  in our example 3 places to the left, so that the
  number become .10111.

104
Normalizing

• How can we legally do this, without changing the
  value of the number?
• Every time you move the decimal place to the
  left, you must adjust the exponent by a positive
  increment.
• In our case, we moved 3 decimal points to the
  left, so our new exponent is 23
• In other words, 101.11 20 .10111 23

105
Normalizing Examples

• A couple more, to make sure we have it
• 1101.11 20 .110111 24
• 101.1 20 .1011 23
• .01 20 .1 2-1
• .001 20 .1 2-2

106
Algorithm

• 1. Convert the base ten number to base two, one
  side of the decimal place at a time
• Left hand side, successive division
• Right hand side, successive multiplication
• In our case, 2.5410 10.1000101002

107
Algorithm (cont)

• Put number in normalized form10.100010100 20
  .10100010100 22
• Determine what the encoding scheme is
• For the sake of our problem , assume the
  following 16 bit encoding
• 1 Sign of mantissa
• 2-12 Mantissa
• 13 Sign of exponent
• 14-16 Exponent

108
Algorithm (cont)
Using the described notation system, the decimal
2.5410 would be stored in switch binary math as
the following pattern of zeros and
ones 0101000101000010
109
Questions?