1. Computer Representation of Information
Time: 90 min.

2. Outline Representation of Characters
Mathematical operations on numbers (addition)
Signed integer representation
Sign-magnitude notation
Two’s complement notation
Binary Notations Exercises
Real Numbers Representation
We will cover ….
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3. Representation of Characters
A unique binary pattern/value is used to represent each of the printable characters on your keyboard (e.g. A, a, 4, *, [,’, etc. …), as well as the special control/unprintable characters (e.g. carriage return, line feed, tab, space, etc. …).
A byte (8 bits) gives you the opportunity to have 28 (256) unique representations/patterns/values.
There are standard character coding schemes to ease interchange of information, e.g. ASCII.
The code is designed in such a way not only to preserve uniqueness, but also to keep the order which is often needed in manipulating characters/text, e.g. sorting.
The value/pattern used to represent “A” is less by one than the value used to represent “B”, … etc.
ASCII (American Standard Code for Information Interchange).
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4. Right-hand Left-hand digit digit 3 4 5 6 7 1 2 8 9P p 1 A Q a q 2 B R b r C S c s D T d t E U e u F V f v G W g w 8 H X h x 9 I Y i y J Z j z K k L l M m N n O o
The slide gives some ASCII character representations in hexadecimal.
Appendix A of your book gives the complete ASCII table in decimal.
Each base 16 digit corresponds to 4 binary digits.
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5. Mathematical Operations on Numbers (Addition)
Most of the mathematical operations on numbers from different number systems are conceptually identical to the decimal arithmetic you are used to.
To add two numbers you do the following:
Start at the rightmost digit.
While there are more digits:
Add the current digit of each operand.
If the sum is less than the base/radix
then record that sum,
otherwise record the difference between the sum
and the base/radix,
and add one to the next digit of operand 1.
The slide gives an algorithm for the process of addition.
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6. Mathematical Operations on Numbers (Addition) (cont’d)
Decimal numbers:
1610 +
1510
------
3110
Start by adding The sum (11) is not less than the base (10), so (11-10) record 1, and carry 1 by adding it to the first digit of 16, giving 2, then add the next digit from each operand (2+1) to give 3.
Octal numbers:
168 +
158
338
Start by adding The sum (1110) is not less than the base (8), so (11-8) record 3, and carry 1 by adding it to the first digit of 16, giving 2, then add the next digit from each operand (2+1) to give 3.
The above discussions can be generalized to any base.
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7. Signed Integers
So far, we were representing unsigned integers in binary.
We need to have a way for representing the sign of the number (positive or negative).
We need to have a sign bit, or figure out some other way, so as to be able to represent negative values as well as positive ones.
We will study two notations for representing signed integers:
sign-magnitude notation, and
two’s complement notation.
How can we represent the negative integers in binary?
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8. Sign-Magnitude Notation
It leaves the left most bit for the sign (sign bit), and uses the rest of the bits (m - 1) to represent the integer.
0 in the sign bit is used to represent positive values, and 1 is used to represent negative values.
Therefore, the range of integer values i which can be represented with m bits is:
-(2m-1 – 1) <= i <= +(2m-1 – 1)
Using the sign–magnitude notation (in a byte):
+2910 is represented as
-2910 is represented as
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9. Problems with Sign-Magnitude Notation
You might find this technique simple, natural, and straight forward, as it closely resembles the way you are used to writing numbers.
However this notation creates mainly two problems for the computers:
0 representation ( is not ).
Addition of mixed sign numbers ( (–510) and doesn’t result in –4).
Thus the sign-magnitude notation is not used in computers.
You can easily understand that 0 and -0 are the same, but the computer would need additional circuitry to figure this out.
The straightforward addition algorithm of mixed sign numbers yields an incorrect result.
Addition of a positive and a negative number must be treated as a subtraction problem, and this would need additional logic for the computer to check the sign bits before performing arithmetic.
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10. Two’s Complement Notation
Two’s complement of an m-bit number N =
(Bitwise/one’s complement of N) + 1
Examples of 8-bit numbers and their 2’s complement representation:
+1 =
–1 = =
+29 =
–29 = =
(2’s complement of = = (29))
It is the notation used in computers to represent negative numbers.
Notice that we still have the sign bit to indicate whether the number is a positive or negative one.
You can never read a negative number represented using 2’s complement unless you get its 2’s complement to be able to transform it to decimal unsigned integer.
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11. Two’s Complement Notation (cont’d)
For an m-bit signed integer in two’s complement notation, the range of integer values is from –2m–1 to 2m–1 – 1.
When m = 8, the range is from –128 to 127.
When m = 16, the range is from – to
Two’s complement notation overcomes the problems of sign-magnitude notation.
We have only one representation for the 0.
Addition of mixed sign integers give correct results (with no need for extra logic).
If you add -1 to 1 in two’s complement notation (in 8 bits) you will get 0:
The rightmost 1 is an overflow, as we use only 8 bits to represent our number, and it is lost.
Addition of mixed sign integers are performed in exactly the same way without the need of extra special computer hardware.
Adding -5 to +1:
(-4 in two’s complement)
There are other representations for integers, e.g. BCD (binary coded decimal) system, that we will not study in this course.
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12. Binary Notations Exercises
Representations of integers in a byte (8 bits) using different binary notations:
Integer Unsigned Sign-magnitude 2's-complement  
Not Possible
Not Possible Not Possible
Not Possible Not Possible Not Possible
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13. Exercises (cont’d)
Add (-37) to (25) using 8-bit two’s complement representation.
Ans:
2510
-3710
------
-1210
 Carry + +
Remember that you can always check that your answer is correct.
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14. Exercises (cont’d)
Subtract (39) from (-17) using 8-bit two’s complement representation.
Ans:
-1710
3910
------
-5610
 Carry
Final carry overflows 8 bits. - +
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15. Real Numbers Representation
Real or Floating-Point Numbers, e.g
In binary
101
100
10-1
10-2
10-3 2 . 6 5
decimal point 24 23 22 21 20
2-1
2-2
2-3 1 .
binary point
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16. Real Numbers Representation (cont’d)
exponent
Scientific notation: x 101
Binary scientific notation: x 24
Hence it consists of a sign bit, a mantissa field, and an exponent field.
mantissa
exponent
mantissa
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17. Next lecture will be about Problem Solving Methods
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