P2 T1 L8 - Numerical errors

Question 1

Sometimes it is not possible to store an exact binary equivalent of the actual real number. (e.g. 1/3 as a decimal or binary)

This will affect the ________ of the stored numbers and introduce _________.

Answer 1

precision

errors

Question 2

State the 7 error types

Answer 2

1. Precision
2. Absolute
3. Relative
4. Rounding
5. Cancellation
6. Underflow
7. Overflow

Question 3

Define precision error
(1 point)

Increasing significant bits used makes a number more accurate but reduces the exponent so the maximum size (number range) is _______

Answer 3

1. When the number of bits used to represent the mantissa is too small to represent the number accurately.

reduced

e.g.
Implications of the following (Using 16 bits):
12 bit mantissa and 4 bit exponent more accurate but smaller number range
8 bit mantissa and 8 bit exponent less accurate but larger number range

Question 4

Define absolute errors (1 point)

What is the formula for an absolute error?

Answer 4

1. The difference between an actual number and the nearest value that can be represented

Absolute error = (actual value – approximation)

e.g.
A bag of apples weighs 474g. Unfortunately the scales can only measure to 450g or 500g.

The absolute error is the actual difference: 24

Question 5

Define relative error (1 point)

What is the formula for relative error?

Answer 5

1. A way of identifying the scale of an absolute error as a percentage

Relative error = Absolute error / actual value * 100

e.g.
A bag of apples weighs 474g. Unfortunately the scales can only measure to 450g or 500g.
The absolute error is the actual difference: 24

The relative error is
24 / 474 = 0.0506 * 100 = 5.06%

Question 6

Define rounding errors

2 points

Answer 6

1. Loss of value caused by rounding a number to nearest value that may be stored
2. This may introduce an absolute error

E.g. 5.063291139240506 to 2dp is 5.06, in rounding we lost 0.003291139240506

Question 7

When do cancellation errors occur?

1 point

Answer 7

1. These occur during addition or subtraction of numbers of widely different sizes in a restricted number of bits

E.g. adding a very big number to a very small number.
(1.011x 2^8) + (1.1 x2^−5) = 1.011 x 2^8

Question 8

Define underflow (1 point)

When might it happen? (1 point)

Answer 8

1. A number is too small (close to zero) to be represented in available number of bits
2. Might happen when you are dividing a small number by a very large number

E.g. represent .000002123456789 with 8 bits

Question 9

Define overflow (1 point)

When might it happen? (1 point)

Answer 9

1. A number is too large to be represented in available bits
2. Might happen when you are multiplying two large values together

E.g. represent 2123456789 with 8 bits