1. Learning Objectives
3.3.1f - Describe the nature and uses of floating point form
3.3.1h - Convert a real number to floating point form
Learn how to normalise a floating point binary numbers
Learn why we want to normalise floating point

2. Floating Point Denary 1.637 x 105 Recall Mantissa and Exponent
Start with 1.637
Shift point 5 places to the right.
What base will we use for binary?

3. Shift point 5 places to the right.
Floating Point Binary
Recall Mantissa and Exponent
x 25
Start with
Shift point 5 places to the right.

4. Converting Binary Floating Point to Decimal
20 and 7/8

5. Floating Point Representation
2’s complement is used for negatives
Recall Mantissa and Exponent • 1
Markout Exponent, Mantissa, and Decimal
Sign Bit is Positive on exponent and mantissa
Convert Exponent
Positive exponent means shift decimal in mantissa +4 right
Convert to decimal
Positive Numbers +4
/8 = d

6. Negative Exponent 1 • 2’s complement is used
Markout Exponent, Mantissa, and Decimal
Sign Bit is Negative on exponent and mantissa means 2’s complement numbers
Convert Exponent From 2’s Complement But Remember It’s Negative
Convert Mantissa From 2’s Complement But Remember It’s Negative
Shift decimal -4 places means left
Convert to decimal, Remember it’s negative
Negative Numbers
= -4
-1/64 + 1/ /512 =

7. 1100111000001000 (6-bit exp) Mark Out Determine Signs|
Determine Signs
Exponent is positive; Mantissa is negative
Convert Exp to decimal; Remember sign
Convert 2s complement +8
If mantissa is negative, convert it; Remember sign
Convert 2s complement 
Shift mantissa 
Convert mantissa; Remember sign
 -100

8. Your Turn
8-bit mantissa, 4-bit exponent
2s complement, Normalised
0.125
-14.5

9. Normalisation
Move
That
Decimal

10. Big Problem Normalisation is the way around this. 0.100000000 000010
x 22 2
x 23 2
x 24 2
If we let any number of leading 0s in and adjust the exponent we get multiple ways to represent the same number. 
If we have more than 1 way to represent a number it reduces the "pool" of possible numbers we can represent.
Normalisation is the way around this.

11. Another Big Problem Normalisation is the way around this.
x 22 2
Let’s add in some number of leading 0s and adjust the exponent.  
We also have to drop the 101 off the end because we don't have enough bits to store it.   
We've reduced the precision of our number.
Normalisation is the way around this.

12. Normalisation Ensures only 1 way to represent each number
Ensures maximum range available
Normalised numbers always begin with two bits that are different
You may have to move the decimal yourself and adjust the exponent accordingly.
Positive Numbers
0.1
Negative Numbers
1.0

13. Normalised Good (Normalised) Bad (Unnormalised) 0.100110011 110001
Bad (Unnormalised)
10 bit mantissa, 6 bit exponent
Normalised begin with different patterns
Remember, may need to change your exponent to make it normalised

14. Your Turn
Exponent is negative, convert it back to decimal
= 25 (negative)
Move decimal to normalised location
To get back to original location, I have to move decimal -2 places
= -27 is the new exponent
 2s comp
Float =

15. Your Turn
Exponent is positive, convert it back to decimal
= 19 (positive)
Move decimal to normalised location
To get back to original location, I have to move decimal -2 places
= 17 is the new decimal
100010
Float =

16. Converting Fractions to Binary

17. Convert Fractions to Binary
This is a simple algorithm for converting fractional parts of numbers to binary
Put the fraction into decimal format
Multiply by 2
If the whole number part of result is 0, then binary digit is 0; if the whole number part of result is 1, then binary digit is 1
Mod the result (take the fractional part only)
Keep going until you get 0 as a result or fill all of your binary digit slots

18. .011110 Fill in with following 0s to number of bits you need
Example
15/32 =
.01111
Multiply
Result
Binary
x 2
x 2
1.8750 1
.8750 x 2
1.75
.75 x 2
1.5
.5 x 2
1.0
Fill in with following 0s to number of bits you need

19. Converting Decimal to Binary Floating Point (The Whole Thing Together)

20. Given a Decimal Number
BinInteger = Convert integer part of number to binary
BinFraction = Convert fractional part of number to binary
If decimal number is negative
BinInteger = 2s complement of BinInteger
BinFraction = 2s complement of BinFraction
NonNormalFloat = write BinInteger + “.” + BinFraction
NormalFloat = Normalise NonNormalFloat
Exponent = Count decimal movement & determine sign
Convert Exponent to binary
If Exponent is negative
Exponent = 2s complement of Exponent
If NormalFloat has empty slots
Fill with 0s on right
If Exponent has empty slots
Fill with 0s if positive or 1s if negative on left
Answer = NormalFloat join Exponent

21. Full Example: -27.625 10-bit Mantissa 6-bit Exponent 0000011011
BinInteger = Convert integer part of number to binary
BinFraction = Convert fractional part of number to binary
If decimal number is negative
BinInteger = 2s complement of BinInteger
BinFraction = 2s complement of BinFraction
NonNormalFloat = write BinInteger + “.” + BinFraction
NormalFloat = Normalise NonNormalFloat
Exponent = Count decimal movement & determine sign
Convert Exponent to binary
If Exponent is negative
Exponent = 2s complement of Exponent
If NormalFloat has empty slots
Fill with 0s on right
If Exponent has empty slots
Fill with 0s if positive or 1s if negative on left
Answer = NormalFloat join Exponent
.101
Yes
.011 +5
101
101000

22. Homework
Assume 6-bit exponent, 10-bit mantissa, 2s complement, normalised form
Convert to decimal

23. Homework Assume 10-bit mantissa, 6-bit exponent, 2s complement.
Convert to normalised floating point
123
0.1875
-15/32