1. Representing Numbers: Integers
Humans use Decimal Number System
Computers use Binary Number System
Important to understand Decimal system before looking at binary system
Decimal Numbers - Base 10
10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Positional number system: the position of a digit in a number determines its value
Take the number 1649
The 1 is worth 1000
The 9 is worth 9 units
Formally, the digits in a decimal number are weighted by increasing powers of 10 i.e. they use the base 10. We can write 1649 in the following form:
1* * * *100
Comp 1001: IT & Architecture - Joe Carthy

2. Representing Numbers: Integers
weighting:
Digits
1649 = 1* * * *100
Least Significant Digit: rightmost one - 9 above
Lowest power of 10 weighting
Digits on the right hand side are called the low-order digits (lower powers of 10).
Most Significant Digit: leftmost one - 1 above
Highest power of 10 weighting
The digits on the left hand side are called the high-order digits (higher powers of 10)
Comp 1001: IT & Architecture - Joe Carthy

3. Representing Numbers: Decimal Numbers
Largest n-digit number ?
Made up of n consecutive 9’s (= 10n -1 )
Largest 4-digit number if 9999
9999 is
Distinguishing Decimal from other number systems such as Binary, Hexadecimal (base 16) and Octal (base 8)
How do we know whether the number 111 is decimal or binary
One convention is to use subscripts
Decimal: Binary:1112 Hex: Octal: 1118
Difficult to write use keyboard
Another convention is to append a letter (D, B, H, O)
Decimal: 111D Binary:111B Hex: 111H Octal: 111O
Comp 1001: IT & Architecture - Joe Carthy

4. Representing Numbers: Binary Numbers
Binary numbers are Base 2 numbers
Only 2 digits: 0 and 1
Formally, the digits in a binary number are weighted by increasing powers of 2
They operate as decimal numbers do in all other respects
Consider the binary number
Weight
bits
= 0*27 + 1*26 + 0*25 + 1*24 + 1*23 + 1*22 + 0*21 + 0*20 =
= 9210
Comp 1001: IT & Architecture - Joe Carthy

5. Representing Numbers: Binary Numbers
Leftmost bit is the most significant bit (MSB).
The leftmost bits in a binary number are referred to as the high-order bits.
Rightmost bit is the least significant bit (LSB).
The rightmost bits in a binary number are referred to as the low-order bits.
Largest n-bit binary number ?
Made up of n consecutive 1’s (= 2n -1)
e.g. largest 4-bit number: 1111 = = 15
Comp 1001: IT & Architecture - Joe Carthy

6. Representing Numbers: Binary Numbers
Exercises
Convert the following binary numbers to decimal:
(i) (ii) (iii)
(iv) (v) (vi)
Joe Carthy Formatting Convention
In these notes we insert a space after every 4 bits to make the numbers easier to read
Comp 1001: IT & Architecture - Joe Carthy

7. Representing Numbers: Converting Decimal to Binary
To convert from one number base to another:
you repeatedly divide the number to be converted by the new base
the remainder of the division at each stage becomes a digit in the new base
until the result of the division is 0.
Example: To convert decimal 35 to binary we do the following:
Remainder
35 / 2 1
17 / 2 1
8 / 2 0
4 / 2 0
2 / 2 0
1 / 2 1
The result is read upwards giving 3510 =
Comp 1001: IT & Architecture - Joe Carthy

8. Representing Numbers: Converting Decimal to Binary
Exercise: Convert the following decimal numbers to binary
(1) 64 (2) 65 (3) 32 (4) 16 (5) 48
Shortcuts
To convert any decimal number which is a power of 2, to binary, simply write 1 followed by the number of zeros given by the power of 2.
For example, 32 is 25, so we write it as 1 followed by 5 zeros, i.e ; 128 is 27 so we write it as 1 followed by 7 zeros, i.e
Remember that the largest binary number that can be stored in a given number of bits is made up of n 1’s.
An easy way to convert this to decimal, is to note that this is the same as 2n - 1.
For example, if we are using 4-bit numbers, the largest value we can represent is 1111 which is 24-1, i.e. 15
Comp 1001: IT & Architecture - Joe Carthy

9. Representing Numbers: Converting Decimal to Binary
Binary Numbers that you should remember because they occur so frequently
Binary
Decimal
111 7
1111 15
127
255
Comp 1001: IT & Architecture - Joe Carthy

10. Comp 1001: IT & Architecture - Joe Carthy
Review
Decimal Number System - Base 10
Significant Digits
Binary Number System - Base 2
Notation (B,D,H,O)
Binary to Decimal
Decimal to Binary
Shortcuts and Common Binary Numbers
Review Questions
What is a positional number system ?
What is the MSB and the LSB. Give an example of each one.
Show how the weights of the bits in an
8-bit binary number
16-bit binary numbe
What is the weight of the MSB in (a) 8-bit number (b) 16-bit number (c) 32-bit number
Convert 48D, 65D, 31D, 15D to binary
Convert B, B and B to decimal
Comp 1001: IT & Architecture - Joe Carthy

11. Hexadecimal Number System
Base-16 number System
16 digits: 0, 1, 2, .., 9, A, B, C, D, E, F.
We use the letters A to F to represent numbers 10 to 15 using a single symbol
A = 10; B = 11; C = 12; D = 13; E = 14; F = 15;
Use H at right hand side to indicate Hexadecimal
Used because
binary numbers are very long and so are error prone
Easy to convert between hexadecimal and binary than between decimal and binary
Example: Convert 2FAH to decimal
weighting:
digits F A
2FA = 2 * F * A * 160
= 2 * * * 160 =
= 50610
Comp 1001: IT & Architecture - Joe Carthy

12. Hexadecimal Number System
Comp 1001: IT & Architecture - Joe Carthy

13. Hexadecimal Number System
Example 2: Convert FFFH to decimal
weighting:
digits F F F
2FA = F * F * F * 160
= 15 * * * 160 = =
Example 3: Convert 65D to Hexadecimal
65 / 16 = 4 Remainder 1
4 / 16 = 0 Remainder 4
65D = 41H
Exercise: Convert 97D, 48D, 255D to Hexadecimal
Comp 1001: IT & Architecture - Joe Carthy

14. Hexadecimal Number System
Hexadecimal to Binary
Convert each Hex digit to a 4-bit binary number
Example: 7FAH
7 F A
7FA = B
Exercise: Convert FFH, FEH, BBH to binary
Comp 1001: IT & Architecture - Joe Carthy

15. Hexadecimal Number System
Binary to Hexadecimal
Break binary numbers into groups of 4-bits from right hand side
Pad with 0’s on left if necessary
Convert each group of 4-bits to its equivalent Hex digit
Example 1: B
Decimal
Hex F B
B = F3BH
Exercise: Convert B; B and B
to Hexadecimal
Comp 1001: IT & Architecture - Joe Carthy

16. Comp 1001: IT & Architecture - Joe Carthy
Signed Numbers
How do we represent negative numbers ?
Humans use a symbol to indicate number sign: “-” or “+”
In computer we only have binary: 1’s and 0’s .
Two common methods for representing signed numbers
Signed Magnitude
Two’s Complement (2’s Complement)
In the following assume we are working with 8-bit numbers
We designate the leftmost bit i.e the MSB as a sign bit
The sign bit indicates whether a number is positive or negative
0 sign bit => positive number
1 sign bit => negative number
The remaining bits give the magnitude of the number
Comp 1001: IT & Architecture - Joe Carthy

17. Comp 1001: IT & Architecture - Joe Carthy
Signed Numbers
Example +15 and -15 as 8-bit numbers
+15 => B MSB = 0 => +
-15 => B MSB = 1 => -
Note the magnitude is comprised of 7 bits
Largest positive number is
=> +127
Largest negative number is
=> -127
Two representations of zero ! and
Comp 1001: IT & Architecture - Joe Carthy

18. Signed Numbers: 2’s Complement
In a complementary number system each number has a unique representation.
Two’s complement is a complementary number system used in computers
It is the most commonly used method for representing signed numbers
Uses only one representation of zero:
Uses a sign bit as for signed magnitude
0 sign bit => positive number
1 sign bit => negative number
In the case of positive numbers, the representation is identical to that of signed magnitude,
the sign bit is 0 and the remaining bits represent the positive number.
In the case of negative numbers, the sign bit is 1 but the bits to the right of the sign bit do not directly indicate the magnitude of the number.
Comp 1001: IT & Architecture - Joe Carthy

19. Signed Numbers: 2’s Complement
In negative numbers the sign bit carries a negative weight while all other bits
carry a positive weight e.g. the 2’s complement number B is
weighted as follows
Bits
Weights
Value = *1 + 1*1 = = -125D
So B = -125D
Exercise: Convert the 2’s complement numbers B and B to decimal.
Comp 1001: IT & Architecture - Joe Carthy

20. Signed Numbers: 2’s Complement
Example 2: The 2’s complement number B
Bits
Weights
Value = _+1
= = -1D
In 2’s complement each number has a unique representation i.e. the negative representation of a number uses a completely different bit pattern than its positive counterpart.
In any number system: +x - x = 0
Example: = 0
In 2’s complement: B B B
Comp 1001: IT & Architecture - Joe Carthy

21. Signed Numbers: 2’s Complement
2’s Complement Arithmetic
In 2’s Complement, we do not need subtraction to compute x - y
We simply add -y to x to get the result.
This makes it easier to design the hardware to implement 2’s arithmetic
It much more complicated with signed magnitude e.g to compute we must always subtract the smaller number from the larger one and then take the sign of the larger number.
Quick Conversion to/from 2’s Complement
Use the rule: Flip the bits and Add 1
Comp 1001: IT & Architecture - Joe Carthy

22. Signed Numbers: 2’s Complement
Quick Conversion to/from 2’s Complement
Use the rule: Flip the bits and Add 1
Example 1 Convert 2’s complement number B to decimal
Step 1: Flip the bits: Change 1’s to 0’s and change all 0’s to 1’s (complement of 1 is 0; complement of 0 is 1)
B => B
Step 2: Add 1
B + 1B => B => 1D Remember the sign bit was 1 => negative
So B => -1D
Comp 1001: IT & Architecture - Joe Carthy

23. Signed Numbers: 2’s Complement
Example 2: Convert -1D to 2’s complement
First convert 1D to binary => B
Step 1: Flip the bits
B => B
Step 2: Add 1
B + 1B => B => -1D => B
Exercise: Convert the following
2’s complement numbers to decimal: ; ; ;
Decimal to 2’s complement: -128 ; -65; -2
Comp 1001: IT & Architecture - Joe Carthy

24. Comp 1001: IT & Architecture - Joe Carthy
B (i.e. -128) is the largest negative 8-bit 2’s complement number
(127) is the largest positive 8-bit 2’s complement number
256 numbers can be represented using 8-bit two’s complement numbers from -128 to 127.
There is only one representation for zero.
The table below lists the decimal equivalents of some 8-bit 2’s complement
and unsigned binary numbers.
Comp 1001: IT & Architecture - Joe Carthy

25. Number Range and Overflow
The range of numbers (called the number range) that can be stored in a given number of bits is important.
Given an 8-bit number, we can represent
unsigned numbers in the range 0 to 255 (0 to 28-1) and
two’s complement numbers in the range -128 to +127 (-27 to 27).
Given a 16-bit number, we can represent
unsigned numbers in the range 0 to 65,535 (0 to ) and
two’s complement numbers in the range to (-215 to 215-1).
In general given an n-bit number, we can represent
unsigned numbers in the range 0 to 2n -1 and
two’s complement numbers in the range -2n-1 to 2n-1 -1
Exercise: What is the number range of 4-bit, 10-bit, 20-bit, 30-bit and 32-bit numbers ?
Comp 1001: IT & Architecture - Joe Carthy

26. Number Range and Overflow
The magnitude of an unsigned number doubles for every bit added
10 bits can represent 1024 numbers (1K)
11 bits => 2048 numbers (2K)
12 bits => 4096 numbers (4K) ….
16 bits => 64K numbers ..
20 bits => 220 => 1 Mb
21 bits => 2 Mb
24 bit => 16Mb
30 bits => 230 => 1 Gb
31 bits => 2Gb
32 bits => 4 Gb
Comp 1001: IT & Architecture - Joe Carthy

27. Number of bits in Memory Address
The maximum amount of memory that a processor can access is determined by the number of bits that the processor uses to represent a memory address.
This determines the maximum memory address that can be accessed i.e. is a limit on the maximum amount of RAM a computer can use
For example, a processor that uses 16-bit addresses will only be able to access
up to 65,536 memory locations (64Kb), with addresses from 0 to 65,535.
A 20-bit address allows up to 220 (1Mb) memory locations to be accessed
A 24-bit address allows up to 16Mb (224 bytes) of RAM to be accessed
A 30-bit address allows up to 1Gb (230 bytes) of RAM to be accessed
A 32-bit address allows up to 1Gb (230 bytes) of RAM to be accessed
Most PCs now use 32-bit addresses. Original PC (1981) used 20-bit addresses.
Early Macintoshs used 24-bit addresses.
Comp 1001: IT & Architecture - Joe Carthy

28. Comp 1001: IT & Architecture - Joe Carthy
Review
Hexadecimal: 16 digits: 0 to 9 and A to F. Easy to convert to binary
Signed Numbers: Signed Magnitude and 2’s Complement
Sign bit: MSB 1 => negative
2 Complement: Flip the bits and add 1.
Number range is important
Given an n-bit number, we can represent
unsigned numbers in the range 0 to 2n -1 and
two’s complement numbers in the range -2n-1 to 2n-1 -1.
A 20-bit address allows up to 220 (1Mb) memory locations to be accessed
A 30-bit address allows up to 1Gb (230 bytes) of RAM to be accessed
A 32-bit address allows up to 1Gb (230 bytes) of RAM to be accessed
Comp 1001: IT & Architecture - Joe Carthy