1. Objectives Today: P4 Data Types – Floating Points P4 Variable Quiz P3 Iteration and Selection Practical
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Unit 6 Software Development Assignment 1

2. Quiz?
Folks, discuss in pairs the following questions and write your answers down on the worksheet. (2 mins)
What are integers? (2 points)
What is a bit? (2 points)
How many bits in a byte? (2 points)

3. What do you remember from yesterday?
Do Floating Points include?
1. A Fraction system
2. Use of bytes
Which of these are used to create the floating point?
MRT
Engagement
Mantissa
Expurgator
Mantisigator
Exponent

4. What is a Floating Point?
How many bits are in 2 bytes?
Answer: 16 bits.
We can use many bytes to hold numbers especially very large numbers.
Work out the following using 1 byte?
Convert 136 to binary
128 ¦ 64 ¦ 32¦ 16 ¦ 8 ¦ 4 ¦ 2 ¦ 1
(5 points) 1 1

5. How to represent fractions?
Next problem…
The number is the equivalent to /2 and 1/4
1 / 2 = 0.5 (1/2)
1 / 4 = 0.25 (1/4)
1/8 = (1/8)
1/16 = (1/16)
Convert in to binary
128 ¦ 64 ¦ 32¦ 16 ¦ 8 ¦ 4 ¦ 2 ¦ 1 . 1/2 ¦ 1/4 ¦ 1/8 ¦ 1/16
What is the binary representation of .75? (5 points)

6. This is a very big number! 1,200,000,000,000.00000000000001
Another purpose of a floating point is to represent numbers of different magnitudes (sizes) and help maintain the accuracy of the number.
This involves the use of many bytes.
Physicists will use huge numbers in their calculations where as a microchip designer will need accuracy (1/10 of a millimetre is crucial).

7. Memory
Since computer memory is limited (only so many ‘storage-cells’) when storing data, there is a limitation on the number of digits to represent real numbers. An example:
is the same as 3.45?
Answer: 32 Bits
How many bits in 4 bytes?
1st byte
2nd byte
What is the largest number represented by 32 bits that you can remember - maintain the accuracy of the number (every digit must be correct)?
3rd byte
4th byte

8. Single and Double Precision What is a single and a double?
Single – a number with a decimal part. Double is a much bigger number with a decimal part.
Format
Total bits
Significand bits
Exponent bits
Smallest number
Largest number
Single precision
7 significant digits 32
sign 8
1.2 x 10-38
3.4 x 1038
Double precision
14 significant digits 64
sign 11
5.0 x
1.8 x 10308

9. Floating Points in our Decimal System
In decimal we can display large numbers as a MANTISSA and an EXPONENT
1,200,000,000,000 can be written as
0.12 X 1013
0.12 is called the mantissa and 1013 is the exponent
Holds the digits and the exponent defines where to place the decimal point
The decimal point has been moved 13 places to the left.

10. Floating Point in our Binary System
The same techniques can be used for binary numbers
2 bytes (16 bits) can be divided into 10 bits for the mantissa (1 sign bit and 9 digits) and 6 bits for the exponent.
Mantissa (10 bits)
Exponent (6 bits)
000011
Sign shows whether the number is positive (0) or negative (1)

11. Floating Point in our Binary System
Let’s see what is happening with the Exponent…
We can work out what the exponent is in denary
We can work out the exponent equals the number?
2 + 1 = 3

12. Floating Point in our Binary System
Let’s use the exponent (which we know is 3) on the mantissa
We move the binary point three places to the right;
the number now becomes which is…
30 seconds convert this binary number in to denary
¦ 4 ¦ 2 ¦ ½ ¦ ¼

13. Quiz – what is the correct answer?
Is the number
3.5
6.75
8 ¦ 4 ¦ 2 ¦ ½ ¦ ¼
8.5
6.5 =
12.75
0.5

14. Convert the following binary numbers to denary
Exponent is 2
Answer is = 2.625
Exponent is 4
Answer is = 13.5

15. Summing Up Quiz Why are Floating points used?
Represent fraction system
Represent large numbers
What are floating points made up of?
Mantissa with 1 sign bit
Exponent
 How are these parts used?
Mantissa holds the digits representing the number
Determines where the binary point needs to be placed

16. Use your answers Yippee!!
Use your answers to write up how floating points are used for Assignment 1 Task 4.
Print off this assignment for marking.
Yippee!!