Lecture 4 - An introduction to computer arithmetic

Question 1

What are numbers on a computer?

Answer 1

A collection of binary digits (bits)

Question 2

What are the two different ways numbers are represented on a computer?

Answer 2

Fixed point and floating point

Question 3

What is a byte?

Answer 3

Short for binary term.
8 bits
Enough to hold a single character of text

Question 4

What is a word?

Answer 4

Natural data size of a computer.

Depends on the CPU but for most desktop systems it is 32 bits (64 for newer systems)

Question 5

What is a kilobyte?

Answer 5

2**10 bytes

Question 6

What is a megabyte?

Answer 6

2**20 bytes

Question 7

What is a Gigabyte?

Answer 7

2**30 bytes

Question 8

What does the first number of a fixed point number represent?

Answer 8

The sign of the integer
0 - positive
1 - negative

Question 9

What base are bits in?

Answer 9

Base 2

Question 10

How do you multiply a fixed point number by a power of 2?

Answer 10

Shift the bits left

Question 11

How do you multiply a fixed point number by a number which is not a power of 2?

Answer 11

Use the shift and ass algorithm. Express as a sum of powers.

Question 12

How do you divide a fixed point number by a factor of 2?

Answer 12

Shift right. Drop the fractional part

Question 13

What is overflow?

Answer 13

When the fixed point number you are trying to produce cannot be written with the number of bits being used.

Question 14

What is fixed point data called in R?

Answer 14

Integers

Question 15

How many bits are used to store fixed point data in R?

Answer 15

32

Question 16

What is returned when you overflow a fixed point number on R?

Answer 16

N/A

Question 17

Advantages of fixed point numbers in R.

Answer 17

• Results are exact

- Fast

Question 18

Disadvantages of fixed point numbers in R.

Answer 18

• Limited applicability: integers and relaxed

- Limited range.

Question 19

What are the 4 different components to make up a floating point number in R?

Answer 19

Sign
Fraction - unsigned integer
Base
Exponent - signed integer

Question 20

What are the number of bits assigned to each component making up a floating point number on R using an 8 bit system?

Answer 20

1 bit for sign
3 bits for exponent
4 bits for the fraction

Question 21

Why are there limitations with floating point numbers?

Answer 21

As the fraction and exponent are stored in a fixed number of bits.

Question 22

What is the largest possible floating point number in an 8 bit system?

Answer 22

7.5

Question 23

What is the smallest possible floating point number in an 8 bit system?

Answer 23

0.0078125

Question 24

What is the machine epsilon and what does it govern?

Answer 24

The machine epsilon is the smallest positive integer such that epsilon + 1.0 is not equalto 1.0. It governs the relative accuracy of the system.

Question 25

What is the machine epsilon equal to?

Answer 25

2**(1-d) where d is the number of bits in the fraction.

Question 26

What are the consequences from a lack of accuracy of floating point numbers?

Answer 26

Things rarely add up because of rounding errors. Order of summation matters. Maximum error when adding n values to x is < (n*epsilon*x)/2 Mixed signs can cause problems.

Question 27

What are floating point numbers called in R?

Answer 27

Numeric

Question 28

How many bits are used for floating point numbers in R?

Answer 28

64

Question 29

How are the bits distributed to the components making up a floating point number in R?

Answer 29

1 - sign 11 - exponent 53 - fraction

Question 30

What is the largest possible floating point number in R?

Answer 30

8.98 x 10**(307)

Question 31

What is the machine epsilon in R?

Answer 31

2.20x10**(-16)

Question 32

What is returned for overflow of a floating point number in R?

Answer 32

+inf and -inf

Question 33

What does NaN mean in R?

Answer 33

Not a number

Question 34

What are underflows for floating point numbers set to in R and what is the warning message?

Answer 34

0, no warning generated.

Question 35

How can you deal with floating point limitations?

Answer 35

- rewrite expressions to avoid overflow - normalise scalars - change exact tests to good enough tests - be careful about order of operations - use double precision (uses twice the memory) Many of these show a trade between speed and accuracy

Question 36

What are the advantages of floating point numbers in R?

Answer 36

Wider applicability | Wider range than fixed point

Question 37

What are the disadvantages of floating point numbers in R?

Answer 37

Results are not exact | Slow

Question 38

How is the correspondence between stored numbers and displayed symbols determined?

Answer 38

By a character set

Question 39

What are 3 examples of character sets and what are the numbers of bits in the systems?

Answer 39

ASCII - 7 bit system Extended ASCII - 8 bit system UNICODE - 16 bit system