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 mantissa 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

• ## ensures that the floating point number is stored in a consistent format, maximizing precision

Question 11

floating vs fixed point : range

Answer 11

fixed point - Limited to a specific range based on the fixed number of digits for the integer part.

floating point - has a large range due to the scaling by the exponent

Question 12

floating point vs fixed point : speed

Answer 12

fixed point - Faster in simple hardware (e.g., embedded systems) since the calculations are straightforward.

floating point -
Slower because floating point operations involve more complex calculations (e.g., adjusting the exponent).

Question 13

uses of fixed vs floating point

Answer 13

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 14

what is an absolute error

Answer 14

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

Question 15

what is a relative error

Answer 15

The absolute error divided by the exact value.

Question 16

compare absolute and relative errors

Answer 16

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 17

what is a rounding error

Answer 17

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.