3.5.4.4 Numbers With a Fractional Part.

Question 1

Outline the us of fixed point representation.

Answer 1

Used if we want to represent fractions.

We can fix the decimal point by defining our byte to represent fractions (but not actually storing the point).

Question 2

How do we represent decimals and what does this involve?

Answer 2

Ways we can represent decimals:
325.5 can be represented as 0.3255 x 10^3

This way of representing numbers in different forms involves moving (floating) the decimal point to a new position.
We need a consistent method of representing decimals.

Question 3

Outline the process of floating point notation.

Answer 3

In floating point notation, real numbers are represented in the following way:

• A sign ( 0 indicates a positive, 1 a negative number).
• some significant digits expressed as a number with a fractional part (mantissa).
• and an integer power of 2 (exponent).

0●1011000 0011
mantissa exponent.

Question 4

When is the implied binary point?

Answer 4

The implied binary point is always after the first digit.

Question 5

What is the exam question structure for Floating Point Numbers.

Answer 5

Exam questions will use 12 bit numbers:
8 bits for the mantissa
4 bits for the exponent.

Question 6

How to calculate floating point numbers.

Answer 6

Find the sign of the mantissa - if the mantissa is negative perform two’s compliment.

• Find the value of the exponent (positive or negative).
• Move the decimal point the distance the exponent asks for (add up the positive values in the exponent) (right for a positive exponent, left for a negative exponent).
• Starting at the decimal point, work out the value of the mantissa.

Question 7

Define a mantissa.

Answer 7

The number you want to store.

Question 8

Define an exponent.

Answer 8

The position of the binary point in the number.

Question 9

What must we remember about the exponent?

Answer 9

The exponent does not form part of the end number.

Question 10

Outline how to float the binary point when working with a negative exponent.

Answer 10

Shift the entire binary number however many values specified by the exponent in the binary waiting line, add 0’s.

Question 11

Outline the need for normalisation.

Answer 11

Normalisation overcomes the loss of a less significant bit, which will then lead to an unnecessary error, when there are leading zeros before the least significant bit.

Question 12

Outline normalised floating point numbers.

Answer 12

With a fixed number of bits, a normalised representation of a number will display the number to the greatest accuracy possible.

Question 13

Summarise what normalised numbers do.

Answer 13

In summary normalised numbers:
Give only one representation of a number.
Save space.
Give the most accurate representation of a number in a given number of bits.

Question 14

Outline how to normalise.

Answer 14

The first two digits must be different.

• 0.1 (positive)
• 1.0 (negative)

Question 15

Examples of normalisation:

Answer 15

(after floating point notation)

1101.1010 is normalised as 1.0110100 0011.

00101 normalised as 0.101000 1010

Question 16

Outline precision in both the mantissa and exponent.

Answer 16

There needs to be a decision between the range of a number and the precision.
- If you want a very precise number, use more digits for the mantissa and less for the exponent as this will allow for more decimal places.
If you want a large range of numbers, use more digits for the exponent and less for the mantissa.

Question 17

Outline what is meant by a rounding error in floating point notation.

Answer 17

When we try to represent some numbers but cant within the space we are given.
E.g. 1/3 = 0.33333333….etc.

Question 18

What will occur if we cannot get perfect precision?

Answer 18

This can lead to errors, namely, a rounding error.

Question 19

Define an absolute error.

Answer 19

The difference between the target number and the closest number achieved.

Question 20

Define a relative error.

Answer 20

The percentage difference between the target number and the rounded value.

Question 21

Example of an absolute error:

Answer 21

If I want to represent 23.27 in binary, and the closest I can get is 23.25, then the absolute error is (23.27 - 23.25) 0.02.

Question 22

Example of a relative error:

Answer 22

If I want to represent 23.27 in binary, and the closest I can get is 23.25, then the relative error is (23.27 - 23.25) /23.27 - 0.09%.

Question 23

Outline an underflow error.

Answer 23

Occurs when the exponent is too small to be displayed accurately in the number of bits available.

Question 24

Outline an overflow error.

Answer 24

Occurs when the exponent is too large to be displayed in the number of bits available.
Essentially, the opposite of and underflow error.

Question 25

Outline a cancellation error.

Answer 25

This type of error occurs during the subtraction of two very similar values where most of the significant digits are lost.