1. CMPUT 229 - Computer Organization and Architecture I
2. Number Systems
Decimal
Binary
Addition
Headecimal
Two’s complement
ASCII characters
CMPUT Computer Organization and Architecture I

2. Positional Number System
329
923 3 2 9 9 2 3 9 2 3 9 2 3
329
923
CMPUT Computer Organization and Architecture I

3. CMPUT 229 - Computer Organization and Architecture I
Decimal numbers
329
923
3  100
9  100
3 10 + 91
9  1
329
923
CMPUT Computer Organization and Architecture I

4. Positional Number System
The same positional system works with different basis:
32913
92313
3  130
9  130
3 13 + 91
9  1
54210
155010
CMPUT Computer Organization and Architecture I

5. Binary System 1102 92316 122 + 121 + 020 9162 + 2161 + 3160
In computers we are mostly interested on bases 2, 8, and 16.
1102
92316
122 + 121 + 020
9  160
14 + 12 + 01
9  1
610
233910
CMPUT Computer Organization and Architecture I

6. CMPUT 229 - Computer Organization and Architecture I
Signed Integers
Problem: given 2k distinct patterns of bits, each pattern with k bits, assign integers to the patterns in such a way that:
The numbers are spread in an interval around zero without gaps.
Roughly half of the patterns represent positive numbers, and half represent negative numbers.
When using standard binary addition, given an integer n, the following property should hold:
pattern(n+1) = pattern(n) + pattern(1)
CMPUT Computer Organization and Architecture I

7. Signed Magnitude In a signed magnitude representation we
use the first bit of the pattern to indicate
if it is a positive or a negative number.

8. Signed Magnitude
What do we do with the pattern 1000?

9. Signed Magnitude Having two patterns to represent 0 is wasteful.
The signed magnitude representation
has the advantage that it is easy to
read the value from the pattern.
But does it have the binary
arithmetic property?
For instance, what is the result
of pattern(-1) + pattern(1)?
pattern(-1)
pattern(1) ??
PattPatel pp. 20

10. Signed Magnitude Having two patterns to represent 0 is wasteful.
The signed magnitude representation
has the advantage that it is easy to
read the value from the pattern.
But does it have the arithmetic
property?
For instance, what is the result
of pattern(-1) + pattern(1)?
pattern(-1)
pattern(1)
1010 = ??
PattPatel pp. 20

11. Signed Magnitude Having two patterns to represent 0 is wasteful.
The signed magnitude representation
has the advantage that it is easy to
read the value from the pattern.
But does it have the arithmetic
property?
For instance, what is the result
of pattern(-1) + pattern(1)?
pattern(-1)
pattern(1)
1010 = pattern(-2)
PattPatel pp. 20

12. 1’s Complement A negative number is represented
by “flipping” all the bits of a positive
number.
We still have two patterns for 0.
It is still easy to read a value from
a given pattern.
How about the arithmetic property?
Suggestion: try the folllowing
= ??
= ??
0 + 1 = ??
PattPatel pp. 20

13. CMPUT 229 - Computer Organization and Architecture I
2’s Complement
a representation for negative numbers
The leftmost bit is used to indicate +/-
Positive number starts with 0, negative 1
A negative number is obtained by
Convert the corresponding positive decimal number to a binary
toggle all bits ( all 10 and all 0 1)
Add 1 to the the binary obtained in the previous step
CMPUT Computer Organization and Architecture I

14. 2’s Complement A single pattern for 0. 1111 pattern(-1)
It holds the arithmetic property.
But the reading of a negative pattern is not trivial.
PattPatel pp. 20

15. Binary to Decimal Conversion
Problem: Given an 8-bit 2’s complement
binary number:
a7 a6 a5 a4 a3 a2 a1 a0
find its corresponding decimal value.
Because the binary representation has
8 bits, the decimal value must be in the
[-27; +(27-1)] =[-128;+127] interval.
CMPUT Computer Organization and Architecture I
PattPatel pp. 23

16. Binary to Decimal Conversion
a7 a6 a5 a4 a3 a2 a1 a0
Solution:
negative  false
if (a7 = 1)
then negative  true
flip all bits;
compute magnitude using:
if (negative = true)
then
CMPUT Computer Organization and Architecture I
PattPatel pp. 24

17. Binary to Decimal Conversion (Examples)
Convert the 2’s complement integer to its
decimal integer value.
1. a7 is 1, thus we make a note that this is a negative
number and invert all the bits, obtaining:
2. We compute the magnitude:
3. Now we remember that it was a negative number, thus:
CMPUT Computer Organization and Architecture I
PattPatel pp. 24

18. Decimal to Binary Convertion
We will start with an example. What is the binary
representation of 10510?
Our problem is to find the values of each ai
Because 105 is odd, we know that a0 = 1
Thus we can subtract 1 from both sides to obtain:
CMPUT Computer Organization and Architecture I
PattPatel pp. 24

19. Decimal to Binary Convertion (cont.)
Now we can divide both sides by 2
Because 52 is even, we know that a1 = 0
a2 = 0
a3 = 1
CMPUT Computer Organization and Architecture I
PattPatel pp. 24

20. Decimal to Binary Convertion (cont.)
Thus we got:
a1 = 0
a4 = 0
a5 = 1
a6 = 1
a2 = 0
a3 = 1
a0 = 1
10510 =
CMPUT Computer Organization and Architecture I
PattPatel pp. 25

21. Decimal to Binary Conversion (Another Method)
We can also use repeated long division:
105/2 = 52 remainder 1
52/2 = 26 remainder 0
26/2 = 13 remainder 0
13/2 = 6 remainder 1
6/2 = 3 remainder 0
3/2 = 1 remainder 1
1/2 = 0 remainder 1
CMPUT Computer Organization and Architecture I

22. Decimal to Binary Conversion (Another Method)
We can also use repeated long division:
rightmost digit
105/2 = 52 remainder 1
52/2 = 26 remainder 0
26/2 = 13 remainder 0
13/2 = 6 remainder 1
10510 =
6/2 = 3 remainder 0
3/2 = 1 remainder 1
1/2 = 0 remainder 1
CMPUT Computer Organization and Architecture I

23. Decimal to Binary Conversion (Negative Numbers)
What is the binary representation of in 8 bits?
We know from the previous slide that: =
To obtain the binary representation of a negative
number we must flip all the bits of the positive
representation and add 1:
Thus: =
CMPUT Computer Organization and Architecture I
PattPatel pp

24. Hexadecimal Numbers (base 16)
By convention, the characters 0x
are printed in front of an hexadecimal
number to indicate base 16.
If the number 0xFACE represents
a 2’s complement binary number,
what is its decimal value?
First we need to look up the binary
representation of F, which is 1111.
Therefore 0xFACE is a negative number,
and we have to flip all the bits.
CMPUT Computer Organization and Architecture I
PattPatel pp

25. Hexadecimal Numbers (base 16)
It is best to write down the binary
representation of the number first:
0xFACE =
Now we flip all the bits and add 1:
= 0x0532
Then we convert 0x0532 from base 16 to base 10:
0x0532 = 0   160
= 0 + 5 16 + 21 = =
0xFACE =
CMPUT Computer Organization and Architecture I
PattPatel pp

26. CMPUT 229 - Computer Organization and Architecture I
Binary Arithmetic
Decimal 19
+ 3 22
Binary
010011
010110
Binary
001110
000101
Decimal 14
- 9 5
910 =
-910 =
CMPUT Computer Organization and Architecture I
PattPatel pp. 25

27. Overflow
What happens if we try to add +9 with +11 in a 5-bit 2-complement
representation?
Decimal 9
+ 11 20
Binary
01001
10100
= -12 ?
The result is too large to represent in 5 digits,
i.e. it is larger than =
When the result is too large for the representation
we say that the result has OVERFLOWed the
capacity of the representation?
CMPUT Computer Organization and Architecture I
PattPatel pp. 27

28. CMPUT 229 - Computer Organization and Architecture I
Overflow Detection
What happens if we try to add +9 with +11 in a 5-bit 2-complement
representation?
Decimal 9
+ 11 20
Binary
01001
10100
= -12 ?
We can easily detect the overflow by detecting that the
addition of two positive numbers resulted in a negative result.
CMPUT Computer Organization and Architecture I
PattPatel pp. 28

29. Overflow (another example)
Could overflow happen when we add two negative numbers?
Decimal
- 12
+ -6
-18
Binary
10100
01110
= +14 ?
Again we can detect overflow by detecting that we
added two negative numbers and got a positive result.
Could we get overflow when adding a positive and
a negative number?
CMPUT Computer Organization and Architecture I
PattPatel pp. 28

30. Sign-extension What is the 8-bit representation of +510? 0000 0101
What is the 8-bit representation of -510?
What is the 8-bit representation of -510?
CMPUT Computer Organization and Architecture I
PattPatel pp. 27

31. Sign-extension What is the 8-bit representation of +510? 0000 0101
To sign-extend a
number to a larger
representation, all
we have to do is to
replicate the sign bit
until we obtain the
new length.
What is the 16-bit representation of +510?
What is the 8-bit representation of -510?
What is the 8-bit representation of -510?
CMPUT Computer Organization and Architecture I
PattPatel pp. 27

32. CMPUT 229 - Computer Organization and Architecture I
Storing characters
ASCII (American Standard Code for Information Interchange)
One byte
UTF-8 (8-bit Unicode Transformation Format)
One to four bytes
Coincides with ASCII
Be able to represent any languages
CMPUT Computer Organization and Architecture I

33. Endianess
In the previous example, there are two ways to store at
the address 0x and at the address 0x : 0x 0x 0x 0x 0x 0x 0x 0x
0x00
0x13
0xFF
0x97
Address
Value 0x 0x 0x 0x 0x 0x 0x 0x
0x13
0x00
0x97
0xFF
Address
Value
What is the difference?
CMPUT Computer Organization and Architecture I

34. Little-End and Big-End
+1910 =
(binary)
Little end of +1910
Big end of +1910
+1910 = 0x
(hexadecimal) =
+1910 = 0xFFFF FF97
Little end of +1910
Big end of +1910
(binary)
(hexadecimal)
CMPUT Computer Organization and Architecture I

35. Endianess
The question is: which end of the integer do we store first in memory?
Big end of +1910
Little end of +1910 0x 0x 0x 0x 0x 0x 0x 0x
0x00
0x13
0xFF
0x97
Address
Value 0x 0x 0x 0x 0x 0x 0x 0x
0x13
0x00
0x97
0xFF
Address
Value
Big Endian Byte Order
Little Endian Byte Order
DECstations and Intel 80x86 are little-endians.
Sun SPARC and Macintosh are big-endians.
CMPUT Computer Organization and Architecture I

36. CMPUT 229 - Computer Organization and Architecture I
Floating Number
How to represent a very large number?
How about a number of the form
x 10
Floating number
IEEE standard -8
CMPUT Computer Organization and Architecture I

37. CMPUT 229 - Computer Organization and Architecture I
IEEE 754 Floating Number
S exponent fraction (precision)
S exponent – 127
N = x 1. Fraction x 2
CMPUT Computer Organization and Architecture I

38. CMPUT 229 - Computer Organization and Architecture I
Example 5 8 -6
CMPUT Computer Organization and Architecture I