Data Representation

Question 1

What is a bit?

Answer 1

A binary digit (0 or 1) (no voltage/voltage) (off/on)

Question 2

What is a byte

Answer 2

8 bits

Question 3

What is word size

Answer 3

The number of bits the processor can deal with in a single operation

E.g. if you have a 64 bit PC, it has a word size of 64 bits

Question 4

How do you convert binary to decimal

Answer 4

Write column headings over each binary digit

128 64 32 16 8 4 2 1
1 0 1 0 1 0 1 1

128 + 32 + 8 + 2 + 1 = 171

Question 5

How do you convert decimal to binary

Answer 5

Start with your decimal number (e.g. 83)

Write the column headings up to the first number bigger than 83

128 64 32 16 8 4 2 1
0 1 0 1 1 0 1 1

83 - 64 = 29
Put a 1 under the first number smaller or the same size as 29

29 - 16 = 13
Same again

16 - 13 = 3
Same again

3 - 2 = 1
Same again

Question 6

How do you do binary addition

Answer 6

Like column addition

Each bit is added together

If the result is 2 or 3 then the first bit is carried over to the next column

Question 7

Why do we use hexadecimal?

Answer 7

Because it keeps the numbers much shorter and easier to read

Question 8

What are the 16 symbols used in hexadecimal

Answer 8

0 - 9 and A - F

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Question 9

How do you convert binary to hexadecimal

Answer 9

Split into groups of 4 bits and write the column headings 8, 4, 2, 1 over each set
8 4 2 1 8 4 2 1
E.g. 11101011 is therefore 1 1 1 0 | 1 0 1 1

1st set of 4 = 8+4+2 = 14, in hexadecimal that is E

2nd set of 4 = 8+4+2 = 14 in hexadecimal that is E

Therefore EE is the answer

Question 10

Convert hexadecimal to binary

Answer 10

E.g. 30F

8 4 2 1 |8 4 2 1|8 4 2 1
0 0 1 1 |0 0 0 0|1 1 1 1

Proof:
First set of 4 = 2+1 = 3, in hexadecimal that is 3
Second set of 4 = 0, in hexadecimal that is 0
Third set of 4 = 8+4+2+1 = 15, in hexadecimal is F

Answer is 30F

Question 11

How many characters are in the ASCII character set

Answer 11

127 characters

Question 12

What does the ascii character set do

Answer 12

The ascii character set allocates each character with a number between 0 and 127

Question 13

What is the recommend method for representing characters in modern day?

Answer 13

Unicode

Question 14

How are integers usually represented

Answer 14

They are usually represented in Twos complement

Question 15

How are real numbers represented (numbers that include fractions/values after the decimal point)

Answer 15

Normalised floating point

Question 16

What is a method for recording signed integers

Answer 16

Sign and magnitude

The left-most bit is used as the sign bit

0 = +
1 = -

Question 17

What is the range of numbers that can be stored in 8 bits (sign and magnitude)

Answer 17

-127 to +127

Question 18

What is the range of numbers that can be stored in 8 bits unsigned

Answer 18

0 to +255

Question 19

How do you make a binary number twos complement

Answer 19

To make 44 into -44 we must do a process called taking the twos complement

I.e. start at the right hand side, copy each bit up to and including the first 1 and then reverse the rest

Question 20

What is the range of numbers that can be held in 8 bits twos complement

Answer 20

-128 to +127

Question 21

How do you do binary arithmetic subtraction

Answer 21

25 - 10 will be 25 + (-10)

25: 00011001
10: 00001010

Taking the 2s complement to make +10 into - 10

-10: 11110110

Then do column method

= 15

Question 22

What is a logical shift

Answer 22

Logical shift left

Before: 1 0 0 1 1 0 1 1

After: 0 0 1 1 0 1 1 0

All bits move left, bit falls off the end and a 0 appears on the right

Logical shift right

Before: 1 0 0 1 1 0 1 1

After: 0 1 0 0 1 1 0 1

All bits move right, bit falls off the end and a 0 appears on the left

Question 23

What is a arithmetic shift

Answer 23

• arithmetic shift left is identical to logical shift left I.e. 0 always appears. 1 shift multiples the number by 2 (shift twice to multiply by 4, three times to multiply by 8 etc)
• arithmetic shift right looks at the sign bit. If it is a 1 a 1 appears as the new digit. If it is a 0 a 0 appears as the new digit. Shift right divides the number by 2 (shift twice to divide by 4, three times to divide by 8 etc.)

Question 24

How do you do binary multiplication

Answer 24

-works similar to long multiplication

0  0  0  1  1  0  1  0
                 0  1  0  1   x
-----------------------------------
    1  0  1
        1  0  1
             1  0  1
-----------------------------------
1  0  0  0  0  0  1  0        = 130

• left shift second number until right most bit is under each 1 in the first
• add up each value

Question 25

How do you work out fractions in binary

Answer 25

E.g. 0 1 1 0 1 . 1 1 0

16 8 4 2 1 . ½ ¼ ⅛
0 1 1 0 1 . 1 1 0

8+4+1+½+¼ = 13¾(13.75)

Question 26

How to convert decimal fraction to binary fraction

Answer 26

E.g. 0.6875

1) multiply by 2:
0. 6875 x 2 = 1.375 the digit before the decimal is a 1 so the first binary digit to right of point is a 1

So far: 0.6875 = 0.1??????

2) discard the whole number part of the previous result (1.375) so it becomes (0.375) now multiply by 2 again
3) 0.375 x 2 = 0.75 the digit before the decimal point is a 0 so the 2nd digit is a 0

So far: 0.6875 = 0.10???????

4) 0.75 x 2 = 1.5 the digit before the decimal is a 1 so the third binary digit to right of point is a 1

So far 0.6875 = 0.101???? Discard 1 and continue

5) 0.5 x 2 = 1.0 the digit before decimal is 1, so that is the next digit

Discarding 1 means we are left with 0 so we are finished

Answer is 0.1010

Question 27

How do we write decimal numbers in standard form

Answer 27

Mantissa x 10^exponent

E.g.

0.000101 can be written as 0.101 x 2^-3

(Which would be 0.101 x 2^101 using 2s complement)

Question 28

What are standard form numbers called when written in binary

Answer 28

Normalised floating point

Can be written in the format (mantissa x 2^exponent)

Question 29

How would 7.25 be recorded in normalised floating point

Answer 29

7.25 = 7¼

16 8 4 2 1 . ½ ¼ ⅛
0 0 1 1 1 . 0 1 0

Therefore 7.25 = 00111.010

Now write this number straight into the mantissa but ignore the leading 0s. The first 2 digits of the mantissa are always different. Fill in remaining digits of mantissa with zeros.

Question 30

What is truncation

Answer 30

Number is approximated to the whole number/tenth/hundredth etc. That is nearer to zero

I.e. number is always rounded down

Question 31

What is truncation used for

Answer 31

For dealing with situations where there are not enough bits to represent all of the number to be stored. The extra bits are just missing off

E.g. 0.0101101 would be stored in 4 bits as 0.010

Question 32

How do you calculate absolute error

Answer 32

Original - truncated = answer

Question 33

How do you calculate the relative error

Answer 33

Absolute error/original = answer%

Question 34

What is rounding

Answer 34

Number is approximated to nearest whole number

If digit after last digit in binary is to be represented is a 1, then the previous digit is increased by 1. Therefore:

Original: 0.0101101
New: 0.011

Question 35

Overflow

Answer 35

Occurs when a number is too large to be stored satisfactorily by the computer

Question 36

When can overflow occur

Answer 36

When ghe results of a computation is too large to be represented in the system

E.g. multiplying a large number by a large number

Question 37

Underflow

Answer 37

Occurs when the result of a computation is close to zero, and is too small to be represented in the system and becomes 0 instead.

E.g. ⅛ x ¼ is 0.001 x 0.010 = 0.00001 I.e. 1/32