- Encoding is a
mapping of
one set of symbols/values into another
set of symbols/values
Comment: encoding is like a kind of "secret word" game that kids like to play where they replace one letter by another. For example, one common code used by kids is "+1 psoition", so that B means A, C means B, and so on (A means Z). So a secret massage "Ifmmp" means "Hello".
- Example encoding: Morse code
encode letters in English alphabet using short and long pulses
Example: . . . is S - - - is O
Try this Morse code applet: click here - Remember that numeric value is something intrinsic
and does not depends on the system of representation:
click here.
- There are 2 types of
codes:
- "Agreement" code
Example:
- Morse code and ASCII code
The meaning of each code word in such code is agreed upon by a group of people
- Mathematical/arithmetic code
Example:
- The binary number system
The meaning of a code word in a Mathematical code "computed" using some formula
Example: the meaing of 00000101(2) is computed using the formula 0×27 + 0×26 + 0×25 + 0×24 + 0×23 + 1×22 + 0×21 + 1×20
The 10s complement code is an Mathematical code:
- The 10s complement code uses
the properities of
the
modulo arithmetic operations
to represent negative values
(Don't worry if you did not have this before, I will not use modulo arithmetic to explain the 10s complement code.
Instead, I will use a car's odometer)
- "Agreement" code
-
The odometer of a
car counts the
number of miles driven:
- If the odometer reading was
999999
- And you drive 1 more mile....
- Then the odometer reading will
become 000000 !!!
- Based on the odometer reading of 000000, you have driven the car 0 miles !!! (i.e.: a new car !!!)
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The odometer in this figure has 6 digits.
Notice that:
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OK, you may wonder why I talked about an odometer when we were discussing modulo arithmetic.... Just keep the odometer in mind for a while, let's explain modulo arithmetic first....
- Modulo arithmetic is an
arithmetic technique where:
- You first perform the arithmetic operation
- Then reduce the result by subtracting a fixed value until the result is between 0 and that fixed value
- Example: module 9 arithmetic
3 * 4 (modulo 9) = 12 (modulo 9) = 3 // 12 - 9 = 3 7 * 4 (modulo 9) = 21 (modulo 9) = 3 // 21 - 9 - 9 = 3In modulo 9 arithmetic, the fixed value used is 9.
After computing the result of some computation, we must reduce the result to a value between 0 .. 8 by subtracting 9 repeatedly until we get a value between 0 .. 8
- The odometer is using
(a very simple)
modulo arithmetic:
The odometer in the above figure is using the modulo 1000000 arithmetic:
- When a result is
greater than
1000000, the
odometer displays
only the
last 6 digits
This has the same effect as subtracting 1000000 repeated from the result !!!
Due to the modulo 1000000 arithmetic, the value 1000000 is equal to 000000 !!!
- When a result is
greater than
1000000, the
odometer displays
only the
last 6 digits
- The 10s complement code is
usually introduced by using
odometer numbers
- The 10s complement code is
the odometer code that I will
explain next.
Sometimes, I call this code "odometer code" because it's related to how the odometer works.
The official name of this code is the 10s complement code....
I will use an odometer with 3 (decimal) digits to keep the values small:
Note:
- There are many different
10s complement (or odometer) codes
depending on
the number of digits in the
odometer number !!!!
Each one of these code operate the same way.
That's why I am using a 3 digit code to teach this material
- The 10s complement code is
the odometer code that I will
explain next.
- Suppose we have a
very special odometer:
- When the car goes forward, its odometer will increase by 1 after driving 1 mile
- When the car goes backwards (i.e.: drive in reverse), its odometer will decrease by 1 after driving 1 mile
Question:
- Starting with odometer reading 001:
we drive in reverse 1 mile.
What's the new odometer reading ?
Answer:
Another question:
- Starting with odometer reading 000:
we drive in reverse 1 mile.
What's the new odometer reading ?
Answer:
- We define the following 3 digit 10s complement code
(= a mapping)
between a 3 digit number
and the
signed values [−500:499]:
3 digit code: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: ... **** *** ** * * ** *** ...
Although the above table is the accurate representation of what the 3 digit code really means, it is easier to used western style notation to represent the values.
So we will write the mapping of the 3 digit 10s complement code as follows:
3 digit 10s compl code (= Odometer reading): 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499Comments:
- Negative values
are represented in the
3 digit code
- Values greater than 499 and less than −500 are not represented (they are "out of range")
- Negative values
are represented in the
3 digit code
- In order to effectively use the
odometer code, you need to be able to:
- Encode any valid value using the odometer code
- Decode an odometer code (= translate the odometer code to its corresponding decimal value)
Let's learn to do these 2 things first....
- If you look at the
positive values
in this table:
Odometer reading: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499
you can see that:
- The odometer code for a positive value is obtained by adding preceeding 0's to the value
- If you look at the
negative values
in this table:
Odometer reading: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499 Odometer reading value represented by odometer reading 500 = 1000 - abs(-500) 996 = 1000 - abs(-4) 997 = 1000 - abs(-3) 998 = 1000 - abs(-2) 999 = 1000 - abs(-1)
you can see that:
- Odometer code = 1000 − abs(negative value)
- Algorithm
to convert
a value v
to its 3 digit odometer
(= 10s complement) code:
- If value v ≥ 0:
- just write the number out in 3 digits with leading 0
- If value v < 0:
- Compute 1000 − abs(v)
- Write the result out in 3 digits with leading 0 digits
Example:
v = 3 3 digit 10's complement representation: 003 v = 103 3 digit 10's complement representation: 103 v = -3 3 digit 10's complement representation: 1000 - 3 = 997 v = -103 3 digit 10's complement representation: 1000 - 103 = 897
- If value v ≥ 0:
- We first note that in the
3 digit 10s complement code:
Odometer reading: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499
that:
- Code words that begins with the digits 0, 1, 2, 3, 4 represent positive values
- Code words that begins with the digits 5, 6, 7, 8, 9 represent negative values
- The decoding algorithm is the
reverse of the
previous algorithm
To convert a 3 digit 10's complement code c to its corresponding value
- If the 10s complement code
c begins with
0, 1, 2, 3 or 4
(i.e.: c represents
a positive value):
- The value represented by c is the "face value" (without the leading 0's)
- If the 10s complement code
c begins with
5, 6, 7, 8 or 9
(i.e., c
represents a negative
value):
- The value represented by c is equal: − (1000 - c)
Example:
code c = 413 -> it is a positive number the value = 413 code c = 613 -> it is a negative number... the value = - (1000 - 613) = -387
- If the 10s complement code
c begins with
0, 1, 2, 3 or 4
(i.e.: c represents
a positive value):
- Recall that
the
mapping between
a 3 digit 10s complement code
and the signed values [−500:499]
is:
3 digit code: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: ... **** *** ** * * ** *** ...
We use the western representation for the signed values for simplicity:
3 digit 10s compl code (= Odometer reading): 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499To fully understand the discussion below, try do the following:
- Imagine that
you have never learned the
western notation for
signed values !!!
- Replace every western notation of a signed value by its cultural independent representation (= colored dots !!!)
- Imagine that
you have never learned the
western notation for
signed values !!!
- Properties of
10s complement code:
- The is
only
one representation (000)
for value ZERO:
3 digit code: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: ... **** *** ** * * ** *** ...
- The meaining of arithmetic operations do not depend on the values of their operands - see examples below
- The is
only
one representation (000)
for value ZERO:
- Examples that
show that
the meaining of
arithmetic operations do
not depend on
the
values of
their operands
- Adding 10's complement numbers:
Values 3 digit 10's compl repr Adding 2 positive 5 005 values + 9 + 009 ----- ------ 14 014 -> represents 14 Adding positive + 5 005 negative + -9 + 991 ----- ------ -4 996 -> represents -4 Adding negative + -5 995 positive + 9 + 009 ----- ------ 4 004 -> represents 4 Adding 2 negative -5 995 values + -9 + 991 ----- ------ -14 986 -> represents -14 - Subtracting 10's complement numbers:
Values 3 digit 10's compl repr Subtract 2 positive 5 005 values - 9 - 009 ----- ------ -4 996 -> represents -4 Subtract positive - 5 005 negative - -9 - 991 ----- ------ 14 014 -> represents 14 Subtract negative - -5 995 positive - 9 - 009 ----- ------ -14 986 -> represents -14 Subtract 2 negative -5 995 values - -9 - 991 ----- ------ 4 004 -> represents 4
- Adding 10's complement numbers:
- Modulo arithmetic can be
represented using a
circle.
The 3 (decimal) digit 10s complement code uses the modulo 1000 arithmetic.
We can represent the modulo 1000 arithmetic on a circle as follows:
- Additing is the same
an moving a number of
steps in the
clock wise direction:
- Subtraction is the same
an moving a number of
steps in the
anti-clock wise direction:
Notice that the value −1 is represented by the code 999.
The reason is:
- When we add
the value −1, we
must walk 1 position in the
anti-clock wise direction
Example: 2 + (−1) = 2 − 1 = 1
Using the 10s complement code representation, we can obtain the correct representation for the result by walking 999 positions in the clock wise direction:
Therefore:
- If we represent the value −1 using the code 999, we can add without needing to use any gimmicks !!!
So the 10s complements code uses the modulo arithmetic to obtain the correct representation for arithmetic operations !!!
(Since multiply is repeated addition, it also works !!!)
- When we add
the value −1, we
must walk 1 position in the
anti-clock wise direction
- Fact:
- Every
Mathematical code has
the overflow problem:
- An overflow has occured is the result of a computation exceeds the range of the represented values in a code
- Every
Mathematical code has
the overflow problem:
- Example:
the 3 (decimal) digit 10s complement code
can represent the
value
−500..499:
Odometer reading: 500 ... 996 997 998 999 000 001 002 003 ... 499 ----------------- +--------+---+---+---+---+---+---+---+---------+ Value represented: -500 ... -4 -3 -2 -1 0 1 2 3 ... 499
Suppose we compute:
Add the values: 499 + 2 In 10s complement code: 499 // repr the value 499 + 002 // repr the value 2 ----- 501 ---> represents the value -499 !
- What to do when there is overflow ?
- You will need to use a longer (= more digit) code when overflow occurs.
Longer codes means more data to store and you need more time to perform the computation
That's why the computer provide different length codes to represent integers
But instead of using 10s complement codes, the computer (because it uses binary numbers) uses 2s complement codes
We will study the 2s complement codes next.