CONVERTING BETWEEN VALUES
You may be asked, for example, how many Mi are in a Ti. You don’t need to learn all
of the different numbers, just remember the patterns and the rule for dividing
numbers with powers:

​Decimal prefixes
These work in the same way. However, for a decimal number, these increase by a power
of 3 each time.

The decimal prefixes are the ones most commonly known and used in computing, e.g. a kilobyte (kB) is 1000 bytes, whereas a kibibyte (KiB) is 1024 bytes.

​Binary prefixes
Different quantities of bits can be described with specific names and prefixes. You should
know quite a few of these terms, but you also need to know what they represent.
■ At the very basic level you have bits, nibbles and bytes.
■ A bit is just 1 binary digit, e.g. 1 or 0.
■ A nibble is 4 bits, e.g. 0110 or 1101.
■ A byte is 8 bits (or 2 nibbles), e.g. 10101101 or 01011011.After a byte you start recording the number of bits in powers of 10.

​These are called prefixes because they will all come before what they represent, which is
the number of bytes:

Converting a base 10 to a base 2 or base 16 number
Using division, you can convert a base 10 (decimal) to both base 2 (binary) and base
16 (hexadecimal).

​■ Divide the number by the base you want it in until you get 0.
■ You want base 2 so you divide by 2.
Step 1: divide the number by 2.
29 ÷ 2 = 14 r 1
Step 2: take the quotient (the result) and divide it by 2.
14 ÷ 2 = 7 r 0
Step 3: repeat this process until you have 0 — quotient divided by 2.
7 ÷ 2 = 3 r 1
Still not 0? Repeat — quotient divided by 2.
3 ÷ 2 = 1 r 1
Still not 0? Repeat — quotient divided by 2.
1 ÷ 2 = 0 r 1
​Make sure you write your remainder, as this will be used to make the answer.
Step 4: now you have 0, write all the remainders from the last to the first; 11101.

Step 1: divide 100 by 2.
100 ÷ 2 = 50 r 0
Step 2: divide the quotient by 2.
50 ÷ 2 = 25 r 0
Step 3: repeat until you have 0.
25 ÷ 2 = 12 r 1
12 ÷ 2 = 6 r 0
6 ÷ 2 = 3 r 0
3 ÷ 2 = 1 r 1
1 ÷ 2 = 0 r 1
Continue dividing by 2 until you have 0.
Step 4: write all the remainders, from the last to the first, 1100100.

Step 1: divide the number by 16.
29 ÷ 16 = 1 r 13
Step 2: divide the quotient by 16.
1 ÷ 16 = 0 r 1
​Step 3: repeat until you have 0.
Step 4: write the remainders, from the last to first; 1 13.
Step 5: check for numbers over 9. You can’t use the
number 13 in hexadecimal. You need to replace it with
the appropriate letter, D
1 13 becomes 1D

​Worked examples
a Convert the binary number 0101 to decimal.
Step 1: put the binary number into the table format as shown in Table

Step 2: binary is base 2, so replace the word ‘base’ with 2.

Step 3 calculate each column 

​Step 4: add the columns together.
0 + 4 + 0 + 1 = 5

Convert the hexadecimal number 1D9 to decimal.
Step 1: put the hexadecimal number into the table format as shown in Table

​Step 2: hexadecimal is base 16, so replace the word ‘base’ with 16.

​Step 3: calculate each column, replacing any letters with their numerical equivalent. (D = 13)

​Step 4: add the columns together.
256 + 208 + 9 = 473

​‘Number base’ means the number of different symbols that are used. So if you have base
10 (known as decimal, or denary), there are 10 different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
These symbols are then combined to represent any number, e.g. 12, 99, 386.
Binary is base 2 and there are two different symbols: 0 and 1.
Hexadecimal is base 16 and there are 16 different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,
C, D, E, F. Letters are used in hexadecimal instead of two-digit numbers.
A = 10
B = 11
C = 12
D = 13
E = 14
F = 15
The base of a number can be identified by its subscript:
■ If the subscript is 10, e.g. 910
, this means it is a base 10 number (decimal).
■ If the subscript is 2, e.g. 10112, this means it is a base 2 number (binary).
■ If the subscript is 16, e.g. 28116, this means it is a base 16 number (hexadecimal).

​Worked examples
a) Find A, where A = {x | x ∈ ℤ ∧ x > –3}.
Step 1: understand the set comprehension statement. This is asking for all of the values of x, such
that x is a member of the set of integers AND x is larger than –3.
Step 2: integers are negative and positive whole numbers. Therefore integers larger than –3 are
–2, –1, 0, 1, 2, 3, 4 and so on. This set continues infinitely upwards.
Step 3: as the set shows a regular sequence, use the … notation to show that the values continue.
Therefore, A = {–2, –1, 0, 1, 2, 3, …}

​b)If A = {7, 8, 9, 10} and B = {5, 6, 7, 8}, find:
i A ∪ B
The union of two sets includes the elements that are members of either set (or both). Be careful
not to repeat any elements. It may be helpful to think of this as joining the sets together and
removing any duplicates. A ∪ B = {5, 6, 7, 8, 9, 10}
ii A ∩ B
The intersection of two sets includes the elements that are members of both sets. A ∩ B = {7, 8}
iii A \ B
The difference of two sets is simply the elements that are in the first which are not in the second.
A \ B = {9, 10}

​Set operations
The main operations that can be carried out on sets are union, intersection, difference
and Cartesian product.
■ The union of sets A and B is the set of all elements that are a member of either or
both sets. The symbol ∪ is used to show union. If A = {1, 3, 7, 9} and B = {1, 2, 4, 7}
then A ∪ B = {1, 2, 3, 4, 7, 9}.
■ The intersection of sets A and B is the set of all elements that are members of both
sets. The symbol ∩ is used to show intersection. Using the same sets for A and B
above then A ∩ B = {1, 7}.
■ The difference of sets A and B is the set of all elements that are members of A but not
members of B. The symbol \ is used to show difference. Again using the same sets as
above, A \ B = {3, 9}.
■ The Cartesian product of two sets is the set of all possible ordered pairs between the
two sets. The symbol × (the same symbol that is used for multiplication) is used to
denote a cartesian product. For example, if A = {1, 7} and B = {True, False} then
A × B = {(1, True), (1, False), (7, True), (7, False)}.

​Subsets
A subset is defined as a set that is wholly contained within another set. For example,
if A = {1, 2, 3, 4} and B = {2, 3} then B is a subset of A. The symbol ⊆ is used
to mean subset, so B ⊆ A reads ‘B is a subset of A’. The line in this symbol is
reminiscent of an equals sign because a subset can technically have exactly the same
elements as the superset. That is, if D = {1, 2, 3, 4} and E = {1, 2, 3, 4}, D = E and
D ⊆ E are both true.
A proper subset uses the symbol ⊂ and is defined as a subset where at least one element
is in the superset that is not in the subset. For example, {1, 2, 3} is a proper subset of ℕ
because there is at least one element in ℕ that is not in {1, 2, 3}.