Arithmetic

Question 1

What variable type is the representation of numbers generally based on?

Answer 1

Integers

Question 2

How are numbers represented?

Answer 2

binary notation

Question 3

Why is range and resolution of the representation limited?

Answer 3

Because the number of bits used to represent data is limited

Question 4

What are the three rules for addition in computer arithmetic?

Answer 4

1. 0+0 = 0
2. 1+0 = 1
3. 1+1= 10 (0 carry 1)

Question 5

E.g. can you add 96 and 37 in binary?

Answer 5

1100000
+0100101
10000101 == 133

Question 6

What is the range number (binary) for unsigned integers using one byte?

Answer 6

• 1 byte = 8 bits

- range: 0 (00000000) to 255 (11111111)

Question 7

What method do we use for negative number notation?

Answer 7

Two’s Complement

Question 8

What is the two’s complement rule for changing the sign of a number ?

Answer 8

1. Invert all bits

2. Add 1 to the result?

Question 9

Can you write two’s complement notation for numbers +7 to -8?

Answer 9

Yes / no?

Question 10

What rules are used for adding of two’s complement signed integers?

Answer 10

same rules for adding unsigned integers

Question 11

E.g. can you do the summation: 3 -2 in twos complement notation? What is wrong?

Answer 11

0011
1110
10001
There is an extra 1, called a carry out bit which should be ignored in order to obtain the right answer (1)

Question 12

How can two’s complement theory be summed up in a sentence?

Answer 12

Mathematically, we have defined that if A is an n-bit binary number, then -A is represented as 2^(n)-A. (this turns out to be the same as inverting all the bits and adding one)

Question 13

How is multiplication by the nth power of 2 done? can you do 5 * 2 in binary?

Answer 13

Simply by shifting the data to the left n places.

00101

Question 14

How is division by the nth power of 2 done? Can you do 12/ 2 in binary?

Answer 14

Simply by shifting the data right n places.

1100&raquo_space; 0110 (6)

Question 15

How can we divide and multiply by non powers of 2? Can you do 3 * 10 in binary?

Answer 15

Using a combination of shifting and adding:
310 = 32^3 + 3*2^1
= 0011

Question 16

What is overflow?

Answer 16

When an arithmetic operation results in a value that is outside the range which can be represented by the variable.

Question 17

What happens if overflow occurs when a variable is represented by >1 byte?

Answer 17

The lower byte results in a carry bit being taken to the higher byte:
high byte 00000000 00000001
low byte 11000001 *2 (

Question 18

How can integers represent fractions?

Answer 18

Decimal numbers can be scaled to be represented by integers

Question 19

Example: a thermometer measures temps between -40 and 60 degrees. We are using an 8bit ADC. How can the degrees be mapped to units?

Answer 19

8bit ADC - measurements must fit into one byte.

We map -40 degrees to 0 and 87.5 degrees to 255… (with numbers inbetween e.g. -39.5 maps to 1)

Question 20

What value is 000101.11 in fixed (binary) point notation?

Answer 20

2^2 + 2^0 + 2^-1 + 2^-2 = 5.75

Question 21

What is the disadvantage of fixed point numbers?

Answer 21

The binary point is at a fixed location so the range and resolution are limited. E.g. in q20 notation (assuming unsigned numbers) the range is 0 ….63.75 and the maximal resolution is 2^(-2) , 0.25

Question 22

Why is floating point notation used?

Answer 22

To cope with very large and very small numbers.

Question 23

E.g. how may 11011100.0 be written in floating point notation?

Answer 23

1.1011100 * 2^7 (note first bit of the number is always1)

Question 24

What is the standard IEEE 754 notation?

Answer 24

(-1)^s * (1+F) * 2^E

Question 25

What do s, F and E represent in IEEE 754 notation?

Answer 25

``` s = sign F = fraction E = exponent ```

Question 26

How is the single precision format represented?

Answer 26

- 32 bit - s = 0 (positive) or 1 (negative) [1 bit] - Exponent E = -126 .... 127 [8 bits] - Fraction F = bit 22*2^-1 .... bit0*2^-23 [23 bits]

Question 27

How is zero defined in IEEE 754 floating point notation?

Answer 27

If E = 0 and F = 0

Question 28

How is infinity defined in IEEE 754 floating point notation?

Answer 28

If E=Emax and F=0

Question 29

How is not a number defined in IEEE 754 floating point notation?

Answer 29

If E=Emax and F!=0 (NaN)

Question 30

What is the main reason for using double precision numbers (8bytes) instead of single precision numbers (4bytes)?

Answer 30

Not the increased range but the improved precision

Question 31

What is the precision of a single precision number?

Answer 31

Fractional part originally 1 + 0 Next number 1 + 2^-23 Precision is 2^-23 in 1 i.e. 1 part in 2^23 or 1 part in 10^7

Question 32

What is the precision of a double precision number?

Answer 32

Fractional part originally 1+0 Next number 1 + 2^-52 Precision is 1 part in 2^52 or 1 part in 10^15

Question 33

Name 6 types of integers?

Answer 33

byte, short, int, unsigned int, long, unsigned long

Question 34

how many bits in a short?

Answer 34

16 bits

Question 35

How to work out the range for an unsigned integer?

Answer 35

from 0 to 2^n -1 where n is the number of bits

Question 36

How to work out the range for a signed integer?

Answer 36

lowest -2^(n-1) - 1 and highest 2^(n-1)

Question 37

What are the two conclusions that can be made about the efficiency of types of arithmetic?

Answer 37

1. using integer arithmetic is much more efficient than using floating point: the logic circuits for integer arithmetic are very simple, whereas floats require complex operations 2. Using mcs native data size is most efficient (but might use more memory). Other data sizes require additional memory operations.