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Subject:   Function Machine
Author: Sione
Date: Feb 10 2004
Iulia ,

It is a very interesting applet.

The function machine can be used perhaps to demonstrate number patterns. I
believe, that this is the first introduction at high school level for digital
computing. This is exactly how computers work. When kids reached University
level particularly doing courses in Electronic & Electrical Engineering and
Physics they are taught to design LOGIC CIRCUITS. This is a huge subject and
complex but basically the principles is the same. You have input digital signal
goes into a logic gate(circuits) , does some functional calculation and the
results pops out at the output. Logic Gates are interconnected, that is one
output from one gate can be an input to the next gate and so on...

Input Signal       Logic Gate        Output Signal  
------------       ----------        ----------

     x   --------> |  f(x)   | ------>    y

The function f(x) can be a simple polynomial such as  f(x) = 3*x+1 , so if x=1 ,
then the output of this function machine is y=4 , if x=3 , then y=10 , and so
forth. This type of function machine is called MEMORYLESS systems in the field
of DSP (Digital Signal Processing) - Physics & Electronic Engineering.
Memoryless function machines depend ONLY on the current INPUT.

In specialized branches of Physics (also Electronic Engineering) called DSP
(Digital Signal Processing) and Robust Digital Control (Control Systems Design),
techniques of number patterns are heavily used but instead of using simple
function machine they use "differential Equations" for continuous case
(analogue) and "difference Equations" for discrete case (digital).

The other type of function machines is called MEMORY systems. Memory systems can
depends on current input as well as past inputs , eg, the function machine

     f(n) = 3*x(n) + 1 - 2*x(n-1)

Let    n  = 0, 1, 2, 3, 4,...
Let  x(n) = 3, 4, 5, 6, 7,...

Now figure out the output y(n)=f(n) of the function machine above. The initial
conditions for the difference (discrete) function machine  f(n) , at input  x(n)
is that  x(n)=0 whenever n-1<0 :

So the output y(n) is:

     y(n) = 10, 7, 8, 9, 10,...

Here is a calculation summary of a MEMORY function machine such as:

    f(n) = 3*x(n) + 1 - 2*x(n-1) :

    with initial condition , x(-1) = 0;

    n      x(n)  3*x(n)   x(n-1)  -2*x(n-1)  y(n)=f(n)
    --     ---   -----    -----    --------  ------
    0       3      9        0         0         10
    1       4     12        3        -6          7
    2       5     15        4        -8          8
    3       6     18        5       -10          9
    4       7     21        6       -12         10
    5       8     24        7       -14         11
    6       9     27        8       -16         12
    7      10     30        9       -18         13
    .       .      .        .         .          .
    .       .      .        .         .          .
    .       .      .        .         .          .

 This is how number patterns is taught in Physics and Electronic Engineering at
University level. Perhaps that high school kids can be mentioned how number
patterns is applied in everyday life. The MEMORY function machine or (digital
filter as called in Physics) is in routine use for designing modern electronics
gadget, such as computers, mobile phone, and so on. Radar detection systems is a
MEMORY systems.The signal (discrete sequence of numbers) is emitted by the radar
and when the reflected signal (current input signal that bounces off an object)
it is then compared with the original signal (past input) a process called
signal cross-correlation. It then determine then if there is an object that
has been detected. Radar systems is more complex that I have described, but same


On Feb 10, 2004, Iulia wrote:

 I found this tool kind of easy (I don't know in what class can be used), but
nice.The idea is to put into machine as many of the numbers 1,2,3,4 as desired,
to observe the outputs and then to a pattern ( the function used to get the
outputs). Students should be made aware of the fact that some functions need
more than 4 entries to be identified.
 I think that using this applet to guess patterns can be useful for students (in
the future), especially when they would learn the method of mathematical
induction, since in many problems they will need to find the function and then
to apply the method.

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