floating point form

Question 1

what is fixed point binary

Answer 1

when the numbers have a predetermined number of bits before and after the point

Question 2

advantages of fixed point

Answer 2

makes numbers easier to process

Question 3

disadvantages of fixed point

Answer 3

they cannot represent the range or accuracy of numbers that may be required

Question 4

what are floating point numbers

Answer 4

when the number is expressed in m x 10^n where m is known as the mantissa and n is the exponent

Question 5

if the number is positive, what should it start with, in floating point form

Answer 5

01

Question 6

if the number is negative, what should it start with in floating point form

Answer 6

10

Question 7

what does it mean for a number to be normalised

Answer 7

it either starts with 10 or 01

Question 8

what does the exponent do

Answer 8

scales the mantissa by a power of two

Question 9

floating point formula

Answer 9

FloatingPoint=Mantissa×2
^Exponent

Question 10

why is normalisation important

Answer 10

-Maximises precision / accuracy for given number of bits;
- Unique representation of each number // simpler to test for equality of numbers;

Question 11

uses of fixed vs floating point

Answer 11

Fixed point: Good for applications where speed is important and the range of values is small (e.g., financial systems).

Floating point: Better for representing a wide range of values but more complex in terms of calculations (e.g., scientific calculations, graphics processing).

Question 12

what is an absolute error

Answer 12

The difference between the exact value and the approximate value stored in the system.

Question 13

what is a relative error

Answer 13

The absolute error divided by the exact value.

Question 14

compare absolute and relative errors

Answer 14

For Large Magnitude Numbers:

Absolute errors tend to be larger, but relative errors may be small since the error is small relative to the large value.
Example: A small absolute error on a large value like
10^6 may result in a very small relative error.

For Small Magnitude Numbers:

Even a small absolute error can cause a large relative error. This can be problematic when dealing with numbers close to zero.
Example: If the exact value is 0.001 and the stored value is 0.0009, the relative error will be much larger compared to a large number.

Question 15

what is a rounding error

Answer 15

occurs when a number can’t be represented exactly within the limits of a system’s precision, and so it must be rounded to the nearest representable value.

Question 16

how to represent more numbers using the same number of bits in floating point

Answer 16

Move a/some bit(s) from the exponent to the mantissa

Question 17

Explain why the relative error is usually considered to be a more important measure of
error than the absolute error.

Answer 17

The impact of an error depends on its size relative to the number that is represented

Question 18

advantages of floating vs fixed point

Answer 18

floating point :
- a floating point system can represent numbers with a
greater range than a fixed point system;
- can represent numbers much closer to zero // can represent much smaller numbers
- can represent much larger numbers

fixed point:

• a fixed point system can represent (some) numbers more
  precisely than a floating point system;
  Calculations can be performed more quickly