1. Binary Fractions

2. 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.

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

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

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

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

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

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

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

10. 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.

11. 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.

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

13. 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

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

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

16. 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.

17. 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.

18. 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

19. 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.

20. 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

21. 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

22. 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

23. 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

24. 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.

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

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

27. 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

28. 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

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

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

31. 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 …

32. 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 …

33. 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

34. An example in 8 bits + .1001 * 22 9/16 * 4 = 36/16 9/16 * 4 = 9/4
As we’ve seen:
* 22
the mantissa can be understood as 9, in the sixteenths column.
Of course, 22 is 4, so we need to solve 9/16 * 4.
9/16 * 4 = 36/16
9/16 * 4 = 9/4
= 2 4/16
= 2 1/4

35. Alternatively
* 22
Multiplying a binary number by 2 shifts the bits left (moves the radix to the right) one position.
So the exponent +2 tells us to shift the radix 2 positions right.
= 2¼

36. Normal Form
The rule of Normal Form guarantees a unique mapping of patterns onto values.

37. Normal Form Consider the following 8-bit pattern in Normal Form:
.1000 1
.0100 2
.0010 3
.0001

38. Normal Form
The exponent represents 0 so the radix is moved 0 positions.
.1000 1
.0100 2
.0010 3
.0001

39. Normal Form
The exponent represents 0 so the radix is moved 0 positions.
.1000 1
.0100 2
.0010 3
.0001

40. Normal Form Now consider this pattern, which is NOT in Normal Form.
.1000 1
.0100 2
.0010 3
.0001

41. Normal Form
The exponent represents +1 so the radix is moved 1 position to the right.
.1000 1
.0100 2
.0010 3
.0001

42. Normal Form
The exponent represents +1 so the radix is moved 1 position to the right.
.1000 1
.0100 2
.0010 3
.0001

43. Normal Form But this is the same result as the first example!
.1000 1
.0100 2
.0010 3
.0001

44. Normal Form
In fact there are MANY possible ways to represent the same value:
.1000 1
.0100 2
.0010 3
.0001

45. 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