1. Binary Fractions

2. Fractions A radix separates the integer part from the fraction part of a number. 101.101 Columns to the right of the radix have negative powers of 2.

3. Fractions 2 2121 2020.2 -1 2 -2 2 -3

4. Fractions 2 2121 2020.2 -1 2 -2 2 -3 421.½¼⅛

5. Fractions 2 2121 2020.2 -1 2 -2 2 -3 421.½¼⅛ 101.101

6. Fractions 2 2121 2020.2 -1 2 -2 2 -3 421.½¼⅛ 101.101 4+1+½+⅛

7. Fractions 2 2121 2020.2 -1 2 -2 2 -3 421.½¼⅛ 101.101 4+1+½+⅛ 5⅝5⅝

8. Fractions Another way to assess the fraction: 4 2 1. ½ ¼ ⅛ 1 0 1. 1 0 1 Just as this is 5

9. Fractions Another way to assess the fraction: 4 2 1. ½ ¼ ⅛ 1 0 1. 1 0 1 Just as this is 5 so is this!

10. Fractions Another way to assess the fraction: 4 2 1. ½ ¼ ⅛ 1 0 1. 1 0 1 Just as this is 5 so is this Except it’s not 5 ones. It’s 5 eighths.

11. Fractions Another way to assess the fraction: 4 2 1. ½ ¼ ⅛ 1 0 1. 1 0 1 Just as this is 5 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 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 × 10 b 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 0.00000000000000000000000000000091093826 kg. In scientific notation, this is written: 9.1093826×10 −31 kg.

15. Scientific Notation - examples An electron's mass is about 0.00000000000000000000000000000091093826 kg. In scientific notation, this is written: 9.1093826×10 −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: 5.9736×10 24 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: 5.9736×10 24 kg.

18. E Notation To allow values like this to be expressed on calculators and early terminals ‘× 10 b’ was replaced by ‘Eb’ So 9.1093826×10 −31 becomes 9.1093826E−31 And 5.9736×10 24 becomes 5.9736E+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? 01101001

26. An example in 8 bits If the following pattern stores a floating point value, what is it? 01101001 Separate it into its components:

27. An example in 8 bits If the following pattern stores a floating point value, what is it? 01101001 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? 0 110 1001 Separate it into its components: sign exponent mantissa

29. An example in 8 bits 0 110 1001 A sign bit of 0 means the number is…?

30. An example in 8 bits 0 110 1001 A sign bit of 0 means the number is positive. 110 in Excess Notation converts to …?

31. An example in 8 bits 0 110 1001 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 0 110 1001 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 0 110 1001 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 … +.1001 * 2 2

34. An example in 8 bits +.1001 * 2 2 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. + 10.01 = 2¼

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

36. Normal Form Consider the following 8-bit pattern in Normal Form: 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

37. Normal Form The exponent represents 0 so the radix is moved 0 positions. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

38. Normal Form The exponent represents 0 so the radix is moved 0 positions. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

39. Normal Form Now consider this pattern, which is NOT in Normal Form. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

40. Normal Form The exponent represents +1 so the radix is moved 1 position to the right. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

41. Normal Form The exponent represents +1 so the radix is moved 1 position to the right. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

42. Normal Form But this is the same result as the first example! 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

43. Normal Form In fact there are MANY possible ways to represent the same value: 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000

44. Normal Form The first bit of the mantissa must be 1 to prevent multiple representations of the same value. 0 100 1000 0.1000 0 101 0100 1.0100.1000 0 110 0010 2.0010.1000 0 111 0001 3.0001.1000