Base Numbering Systems

1
Base Numbering Systems

• Stewart Blakeway
• FML 208
• blakews_at_hope.ac.uk

2
Session Aims

• To explain what a base numbering system is
• How to work out representations of numbers
• How to convert from Base 10 to Base 2 and back
  again
• Performing arithmetic with Base 2
• Representing Negative Numbers

3
Base 10 - Decimal

• 4,367
• Four Thousands
• Three Hundreds
• Six Tens
• Seven Units
• Each position of a digit represents a different
  weight

4
Position of Digits

• 9735 (9 103) (7 102) (3 101) (5
  100).

Calculator
5
Question

• Why is it called the Base 10 numbering system?

Because there are 10 digits. 1 2 3 5 5 6 7 8 9 0
6
Question

• What is 11 11
• 23
• 6
• 12
• 110

The answer depends on the base numbering system
being used. In a base 10 system the answer would
be 22. In a base 2 system the answer would be
110.
7
Computers use Binary

• Data is represented using the binary system.
• Data is stored on main memory using electrical
  pulses, a strong electrical pulse through a
  wire/circuit will mean a "1" state and a weaker
  or no electrical pulse would mean a "0" state.

8
Base 2 - Binary

• How many digits does a base 2 numbering system
  use?

TWO.

• What are they?

1 and 0
9
Binary

• Computers use the base 2 number system
• The digits are known as BINARY DIGITS
• This is where the term BIT comes from
• BINARY DIGITS

10
More BITs

• Obviously a BIT is useless by itself. A
  collection of bits can represent data.
• If 2 bits were put together, we would have the
  following possible 4 states
• 1st state 0 02nd state 0 13rd state 1
  04th state 1 1

11
Collections of BITs

• Question A collection of 8 BITS is known as
  what?
• It takes 8 BITS to store a single character!
• How many different combinations can you have with
  an 8 Bit number?

A Byte
256, or 28
12
Position of the Numbers

• In Base 10 (Decimal) we know that 12 is one 10
  and 2 units.
• We know this because of the position of the
  digits.
• The same applies in Base 2 (Binary).
• Base 2 uses the same weighting system but with
  different weights!

13
Question

• Why does Binary not use the same weights as
  Decimal?

Answer There are only 2 digits in Binary.
Therefore the weighing of the adjacent digit is a
multiple of 2.
Still confused?
14
Weighting

• Remember we all read numbers from right to left

One Unit
Question How many units does 1 multiple signify
in Decimal?
A multiple of Units
15
Weighting Base 10
We know the Number to read Thirty Two Thousand
One Hundred and Forty Two because of the position
of the numbers
16
Weighting

• Remember we read numbers from right to left

One Unit
Question How many units does 1 multiple signify
in Binary?
A multiple of Units
17
Weighting Base 2
We know the Number to read 1 instance of 16 0
instance of 8 1 instance of 4 0 instance of 2 1
instance of 1
Remember Base 2 Only has 2 digits. 1 or 0 On
or Off True of False
18
Converting

• You will be expected to convert Binary to
  Decimal!
• And perform arithmetic on Binary numbers

19
Base 2 - Position of Digits

• 1101 is equivalent to
• 1101 (1 23) (1 22) (0 21) (1 20)

Calculator
20
Weighting Base 2
We know the Number to read 1 instance of 8 1
instance of 4 0 instance of 2 1 instance of 1
8 4 1 13 is the Decimal. Easy!
21
Binary Arithmetic

• Binary Arithmetic is easy once you understand the
  rules. Much easier than decimal.
• The key fact to remember is that it is a base 2
  numbering system.
• Remember that you have to carry across every 2
  and you cant go wrong. Well, theoretically!

22
Question

• What is 1010
• 1111
• ___________
• 11001
• Remember start from the right!
• 0 1 1
• 1 1 2 (no such number, so it must be a
  multiple)
• 1 1 2 (no such number, so it must be a
  multiple)
• 1 1 1 3 (2 is the multiple so it leaves 1)
• 0 1 1

23
Does it work with 3 numbers?

• What is 0110011
• 1100111
• 1011001
• ___________

24
Does it work with 4 numbers?

• What is 0110011
• 1100111
• 1110110
• 1011001
• ___________

25
Subtraction?

• What is 1111
• 1010 -
• ___________

Remember start from the right! 1 - 0 1 1 - 1
0 1 - 0 1 1 - 1 0 Answer 0101
26
What if we have to borrow?

• The last example was deliberately easy!
• What if we have to borrow from the previous
  column?
• Lets take a look at base 10 to remind ourselves

27
Base 10 Subtraction

• What is 364
• 291 -
• ___________

Remember start from the right! 4 - 1 3 6 - 9
Cant do! We have to borrow. It is base 10 so
we borrow 10. 16 - 9 7 2 - 2 0 (Its 2
because we borrowed 1) Answer 073
28
Base 2 Subtraction with borrow

• Same rules apply across all numbering systems.
• We however borrow 2 when dealing with base 2.

29
Subtraction?

• What is 1101
• 0111 -
• ___________

Remember start from the right! 1 - 1 0 0 - 1
Cant do! We have to borrow. Its base 2 so
borrow 2. 2 - 1 1 0 - 1 Cant do! We have to
borrow. Its base 2 so borrow 2. 2 - 1 1 0 - 0
0 Answer 0110
30
Computers do not subtract!

• The last example is how we subtract as humans and
  probably how we were all taught in school.
• Computers do not subtract, they add!

31
How it works

• This is achieved by using a mathematical rule
  that always works
• First the number that is to be subtracted is
  turned into its negative representation
• The numbers are then added together
• In base 2 it is called twos compliment
• It works in all base numbering systems

32
Base 10

• 1000 -
• 345
• --------
• 655

• 1000
• -345
• -------
• 655

33
Negative Numbers in Binary!

• If in binary 101 represents the decimal 5. What
  is this as a negative?
• Decimal is easy. We just stick a at the
  beginning to signify that it is minus 5.
• Binary is easy too, however we do not have the
  luxury of a minus sign!

34
Converting Positive Binary to Negative

• Converting positive to negative is relatively
  straight forward. For the purpose of this
  lecture we are assuming that we are using 4 bit
  addresses.
• 0110 becomes 1001 (we just flip each digit)
• Then we add 1. 1001 becomes 1010

35
Question

• If 0110 is the decimal for 6. 1010 must be the
  negative representation.
• Isnt 1010 the decimal number 10? How does the
  computer know it is a negative number?

36
Subtraction by Addition

• Write the sum as normal.
• Flip the smaller number.
• Perform an addition.
• Disregard the leftmost number.
• Add 1.

37
Subtraction?

• What is 1101
• 0111 -
• ___________

38
Any Questions?

• You will be given exercises in your seminar.
• You will be tested on this.
• Hint One of the questions may ask you how
  computers know if its a negative or positive
  number!