1. Data Representation
Number Systems

2. Binary
The binary number system is a means of representing quantities using only 2 digits:
0 and 1.
Like other number systems it’s based on
Positional Notation.

3.  Positional Notation
The Decimal system is based on the number of digits we have. 
Positional Notation allows us to count past 10 by organizing numeric digits in columns.
Each column of a number represents a power of the base.
The base is 10.
The exponent is the order of magnitude for the column.

4. Positional Notation
103
102
101
100
10001
1001 11
The exponent is the order of magnitude for the column.
The Least Significant digit is in the right-most column.
The Most Significant digit is in the left-most column.

5. Positional Notation
103
102
101
100
10001
1001 11
The base is 10.

6. Positional Notation
103
102
101
100
10001
1001 11
The magnitude of the column is base exponent

7. Positional Notation
=27916
Consider a number like the one above.
How many does it represent?

8. Positional Notation
=27916
The size of a number is determined by multiplying the magnitude of the column by the digit in the column and summing the products.

9. Positional Notation
=27916
The columns are labelled with their exponents.

10. Positional Notation
=27916
The base of the system is 10.

11. Positional Notation
=27916
The magnitude of the column is base exponent

12. Positional Notation
*2 *7 *9 *1 *6
=27916
Multiply the magnitude of the column by the digit in the column.

13. Positional Notation
*2 *7 *9 *1 *6
27 thousand, 9 hundred, sixteen
Sum the products.

14. Positional Notation
In Binary, the columns have the expected exponents 23 22 21 20 81 41 11

15. Positional Notation
In Binary, the columns have the expected exponents
but the base of the system is 2. 23 22 21 20 81 41 11

16. Positional Notation
In Binary, the columns have the expected exponents
but the base of the system is 2.
So the column magnitudes are powers of 2. 23 22 21 20 81 41 11

17. Binary
Rather than referring to each of the numbers as a binary digit, we shorten the term to bit.
To facilitate addressing, binary values are typically stored in units of 8 bits, which is called a byte.
Large values occupy multiple bytes.

18. Natural Numbers in Binary
Consider the pattern:
To calculate the Decimal equivalent of this number, multiply each digit by the value of the column and sum the products.

19. Natural Numbers in Binary

20. Natural Numbers in Binary

21. Natural Numbers in Binary

22. Natural Numbers in Binary

23. Natural Numbers in Binary
=149

24. Natural Numbers in Binary
=255
This is the largest unsigned value that can be stored in 8 bits.
How many different patterns are there?

25. Natural Numbers in Binary
Conversion from Decimal to Binary uses the same technique, in reverse.
Consider the value 73.
This is 7 units of 10, plus 3 units of 1.

26. Natural Numbers in Binary
We need to express the value in terms of powers of 2. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

27. Natural Numbers in Binary
What is the largest power of 2 that is included in 73? 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

28. Natural Numbers in Binary
64 is the largest power of 2 that is included in 73, so a 1 is needed in that position 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

29. Natural Numbers in Binary
Subtracting 64 from 73 leaves 9, which cannot include 32, or 16, but does include 8. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

30. Natural Numbers in Binary
Subtracting 8 from 9 leaves 1, which cannot include 4, or 2, but does include 1. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

31. Natural Numbers in Binary
So the 8 bit binary representation of 73 is:

32. Arithmetic in Binary

33. Binary Addition
A “Rule of Addition” is a statement of the form:
3 + 5 = 8
How many such rules are there in Decimal?

34. Binary Addition “Rules of Addition”
10 digits can appear as the first addend,
and 10 can appear as the second,
so Decimal has 10 * 10, or 100
“Rules of Addition”

35. 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 Binary Addition
In Binary there are 4 “Rules of Addition”:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10

36. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

37. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

38. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

39. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

40. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

41. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

42. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

43. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

44. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

45. Binary Addition
To add any 2 binary numbers, simply apply the rules on each column, starting at the right – just like Decimal.

46. Binary Multiplication
Any multiplication can be re-expressed as a series of additions.
For example, 7 * 3 (seven times three) is simply the sum of 3 sevens.
7 * 3 =
Since we already know how to perform addition, binary multiplication becomes easy.

47. Binary Multiplication7
0111
* 3
+ 0111
1110
10101

48. Binary Multiplication7
0111
* 3
+ 0111
1110
10101

49. Binary Multiplication7
0111
* 3
+ 0111
1110
10101

50. Binary Multiplication7
0111
* 3
+ 0111
1110
10101

51. Binary Multiplication7
0111
* 3
+ 0111
1110
10101

52. Binary Multiplication
The multiplication of large values will take many such additions, but the machine doesn’t mind.
Even so, we look for faster more efficient techniques, and there is one available.
It’s based on the Algebraic property called Distribution.

53. Binary Multiplication
The Distribution Property can be expressed this way:
a * (x + y) = a * x + a * y
Any number can be expressed as the sum of other numbers:
106 * 37
= 106 * (30 + 7)
= (106 * 30) + (106 * 7)

54. Binary Multiplication
The other insight necessary to optimize multiplication regards multiplying a number by its base.
In Decimal, multiplying anything by 10 is easy -
simply shift the digits to the left,
and pad with a zero.
975 * 10 = 9750

55. Binary Multiplication
This is true in any number system.
In Binary, for example:
001 represents 1
010 represents 2
100 represents 4

56. Binary Multiplication
This is true in any number system.
In Binary, for example:
001 represents 1
010 represents 2
100 represents 4
Each shift/pad doubles the value.

57. Binary Multiplication
Now reconsider this problem:
106 * 37 = ?
Expressed as this distribution:
106 * ( )
= 106 * * * 1

58. Binary Multiplication
Now reconsider this problem:
106 * 37 = ?
Expressed as this distribution:
106 * ( )
= 106 * * * 1

59. Binary Multiplication
Now reconsider this problem:
106 * 37 = ?
Expressed as this distribution:
106 * ( )
= 106 * * * 1

60. Binary Multiplication
? = 106 * * * 1
32 = (25)
4 = (22)
1 = (20)
by substitution…
? = 106 * * * 20

61. Binary Multiplication
? = 106 * * * 1
32 = (25)
4 = (22)
1 = (20)
by substitution…
? = 106 * * * 20

62. Binary Multiplication
? = 106 * * * 1
32 = (25)
4 = (22)
1 = (20)
by substitution…
? = 106 * * * 20

63. Binary Multiplication
? = 106 * * * 1
32 = (25)
4 = (22)
1 = (20)
by substitution…
? = 106 * * * 20

64. Binary Multiplication
? = 106 * * * 1
32 = (25)
4 = (22)
1 = (20)
by substitution…
? = 106 * * * 20

65. Binary Multiplication
? = 106 * * * 20
This term can be simplified.

66. Binary Multiplication
? = 106 * * * 20
106

67. Binary Multiplication
? = 106 * * * 20
106
converts to

68. Binary Multiplication
? = 106 * * * 20
106
106 * 22

69. Binary Multiplication
? = 106 * * * 20
106
106 * 22
shift 2

70. Binary Multiplication
? = 106 * * * 20
106
106 * 22
shift 2

71. Binary Multiplication
? = 106 * * * 20
106
106 * 25

72. Binary Multiplication
? = 106 * * * 20
106
106 * 25
shift 5

73. Binary Multiplication
? = 106 * * * 20
106
106 * 25
shift 5

74. Binary Multiplication
? = 106 * * * 20
106
106 * 22
106 * 25
====================================

75. Binary Multiplication
? = 106 * * * 20
106
106 * 22
106 * 25
====================================
Sum

76. Integers

77. Integer Storage
To store integers, the Most Significant Bit is used to represent the sign.
Thus, half the combinations can be used to represent negative values.
Which value of the sign bit (0 or 1) will represent a negative number?

78. 2’s Complement Notation (examples in 8 bits to save space)
Fixed length notation system.
Uses 1 to represent negative values.
The largest non-negative value:
The smallest non-negative value:
The largest negative value is:
The smallest negative value is:

79. 2’s Complement Notation
Note that the representations of non-negative integers in 2’s Complement look the same as they do for Natural numbers.
However, negative values look VERY different than we might expect.

80. 2’s Complement Notation
Complementary numbers sum to 0.
Decimal is a Signed Magnitude system so complements have the same magnitude but different signs: 5 and -5, for example.
2’s Complement is a Fixed Length system. There are no signs, so to find a number’s complement, another technique is needed.

81. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.

82. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:

83. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length

84. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length

85. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length
“flip the bits” (1 → 0, 0 → 1)

86. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length
“flip the bits” (1 → 0, 0 → 1)

87. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length
“flip the bits” (1 → 0, 0 → 1)
add 1 to the new pattern +1

88. 2’s Complement Notation
One such technique is to simply change each bit to its opposite, and then add 1.
To find the 2’s complement notation for -5:
Represent +5 in fixed length
“flip the bits” (1 → 0, 0 → 1)
add 1 to the new pattern +1
to produce -5

89. 2’s Complement Notation
Complementary numbers
sum to 0.
So if to +5
we add -5
we should get
discard the carry bit

90. Arithmetic in 2’s Complement (remember it’s a fixed length system)
= 00
= 01
= 01
= 10 -1
in 2’s complement
+ 1 +
discard the carry bit 1

91. Arithmetic in 2’s Complement
The ability to represent negative values in Binary means that the Addition operation can be used to effect subtraction.
The expression 7 – 3 can be alternatively represented as (+7) + (-3).
With the 4 “Rules of Binary Addition” and 2’s Complement Notation, addition becomes subtraction.

92. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
- (+3)

93. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
- (+3)

94. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
- (+3)

95. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
- (+3)

96. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
- (+3) +1

97. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7) +1
+ (-3)

98. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
+ (-3)
+1 discard the carry bit

99. Arithmetic in 2’s Complement (remember it’s a fixed length system)
(+7)
+ (-3)
+1 discard the carry bit

100. Division

101. Division
Just as multiplication can be expressed as a series of additions, division can be seen as a series of subtractions.
21 ÷ 7 asks how many times 7 can be subtracted from 21.

102. Division In Binary, subtraction is done by adding a negative value…
So division is the repeated addition of a negative value.

103. Division
21 ÷ 7 = ?

104. Division
21 ÷ 7 = ?

105. Division
21 ÷ 7 = ?

106. Division
21 ÷ 7 = ?

107. Division
21 ÷ 7 = ?

108. Division
21 ÷ 7 = ?

109. Division
21 ÷ 7 = ?
keep the carry bits

110. Division
21 ÷ 7 = ?

111. Division
21 ÷ 7 = ?

112. Division
21 ÷ 7 = ?

113. Division
21 ÷ 7 = ?

114. Division
21 ÷ 7 = ?
quotient
remainder

115. Division This is integer division. It produces a quotient, and
a remainder.
Of course, the remainder won’t always be 0!

116. Excess Notation
The other way to represent Integers.

117. Excess Notation (examples are in 8 bits to save space)
Fixed length notation system.
Uses 0 to represent negative values.
The largest non-negative value:
The smallest non-negative value:
The largest negative value is:
The smallest negative value is:

118. Excess Notation (examples are in 8 bits to save space)
To better understand how binary patterns unpack under the 3 integer notations, let’s look at an example.
Consider the pattern
Show the value it represent if it’s:
an unsigned integer
signed integer, 2’s Complement Notation
signed integer, Excess Notation

119. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:

120. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:
the MSB

121. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:
the MSB
the rest

122. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:
the MSB
the rest
Let’s look at “the rest”:

123. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:
the MSB
the rest
Let’s look at “the rest”:
As a Natural number, this is ,
or 57.

124. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) is, therefore, 57 greater than – regardless of the meaning of the MSB.

125. Excess Notation (examples are in 8 bits to save space)
As an unsigned integer, is 128

126. Excess Notation (examples are in 8 bits to save space)
As an unsigned integer, is 128
In 2’s Complement, is the smallest, negative value…

127. Excess Notation (examples are in 8 bits to save space)
As an unsigned integer, is 128
In 2’s Complement, is the smallest, negative value…
In Excess Notation, is the smallest, non-negative value…

128. Excess Notation (examples are in 8 bits to save space)
So the pattern is 57 greater than:
128 if it’s natural (57+128=185)
-128 if it’s 2’s Complement (57-128= -71)
0 if it’s Excess (57+0= 57)

129. Binary Fractions

130. Fractions
A radix separates the integer part from the fraction part of a number.
Columns to the right of the radix have negative powers of 2.

131. Fractions 22 21 20 .
2-1
2-2
2-3

132. Fractions 22 21 20 .
2-1
2-2
2-3 4 2 1 ⅛

133. Fractions 22 21 20 .
2-1
2-2
2-3 4 2 1 ⅛

134. Fractions 22 21 20 .
2-1
2-2
2-3 4 2 1 ⅛ +

135. Fractions 22 21 20 .
2-1
2-2
2-3 4 2 1 ⅛ + 5⅝

136. Fractions
Another way to assess the fraction:
½ ¼ ⅛
Just as this is 5

137. Fractions
Another way to assess the fraction:
½ ¼ ⅛
Just as this is so is this!

138. Fractions
Another way to assess the fraction:
½ ¼ ⅛
Just as this is so is this
Except it’s not 5 ones.
It’s 5 eighths.

139. Fractions
Another way to assess the fraction:
½ ¼ ⅛
Just as this is so is this
Except it’s not 5 ones.
It’s 5 eighths.

140. Fractions But there’s no dot (.) in Binary!
We need to explore a method for storing fractions without inventing another symbol.

141. Scientific Notation a × 10b 1 ≤ |a| < 10
Very large and very small numbers are often represented such that their orders of magnitude can be compared.
The basic concept is an exponential notation using powers of 10.
a × 10b
Where b is an integer, and a is a real number such that:
1 ≤ |a| < 10

142. Scientific Notation - examples
An electron's mass is about
 kg.
In scientific notation, this is written:
×10−31 kg.

143. Scientific Notation - examples
The Earth's mass is about
5,973,600,000,000,000,000,000,000 kg.
In scientific notation, this is written:
5.9736×1024 kg.

144. E Notation
To allow values like this to be expressed on calculators and early terminals
× 10b
was replaced by Eb
So ×10−31
becomes E−31
And ×1024
becomes E+24

145. E Notation
The ‘a’ part of the number is called the mantissa or significand.
The ‘Eb’ part is called the exponent.
Since exponents could also be negative they would typically have a sign as well.

146. Floating Point Storage
In floating point notation the bit pattern is divided into 3 components:
Sign – 1 bit (0 for +, 1 for -)
Exponent – stored in Excess notation
Mantissa – must begin with 1

147. Mantissa Assumes a radix point immediately left of the first digit.
The exponent will determine how far and in which direction to move the radix.

148. An example in 8 bits
If the following pattern stores a floating point value, what is it?

149. An example in 8 bits
If the following pattern stores a floating point value, what is it?
Separate it into its components:

150. An example in 8 bits
If the following pattern stores a floating point value, what is it?
Separate it into its components: sign exponent mantissa

151. An example in 8 bits
If the following pattern stores a floating point value, what is it?
Separate it into its components: sign exponent mantissa

152. An example in 8 bits
A sign bit of 0 means the number is…?

153. An example in 8 bits
A sign bit of 0 means the number is positive.
110 in Excess Notation converts to …?

154. An example in 8 bits
A sign bit of 0 means the number is positive.
110 in Excess Notation converts to +2.
Place the radix in the mantissa …

155. An example in 8 bits
A sign bit of 0 means the number is positive.
110 in Excess Notation converts to +2.
Place the radix in the mantissa .1001
Put it all together …

156. An example in 8 bits
A sign bit of 0 means the number is positive.
110 in Excess Notation converts to +2.
Place the radix in the mantissa .1001
Put it all together …
* 22

157. An example in 8 bits + .1001 * 22 + 10.01 = 2¼
Multiplying a binary number by 2 shifts the bits left (moves the radix to the right) one position.
So this exponent tells us to shift the radix 2 positions right.
= 2¼

158. Normal Form
The first bit of the mantissa must be 1 to prevent multiple representations of the same value.
.1000 1
.0100 2
.0010 3
.0001