Thursday, August 21, 2014

Even and odd functions

From  Khan Academy and Wikipedia

A function is even if f(x)=f(x) for all values of x.
A function is odd if f(x)=f(x) for all values of x.
 The graph of an even function is symmetric with respect to the y-axis.
 The graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin   


Advantages and Disadvantages of Digital Circuits

From http://www.asic-world.com/digital/intro1.html#Numerical_Presentation
     
http://nptel.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/digital_circuit/frame/index.html


Advantages of Digital Circuits
  • Easier to design. Exact values of voltage or current are not important, only the range (HIGH or LOW) in which they fall.
  • Better accuracy and precision.
  • Easily Programmable Analog systems can also be programmed, but the available operations variety and complexity is severely limited. Digital systems interface well with computers and are easy to control with software. It is often possible to add new features to a digital system without changing hardware, and to do this remotely, just by uploading new software. Design errors or bugs can be worked-around with a software upgrade, after the product is in customer hands. A digital system is often preferred because of (re-)programmability and ease of upgrading without requiring hardware changes.
  • Digital circuits are less affected by noise, as long as the noise is not large enough to prevent us from distinguishing HIGH from LOW 
  • Cheap Electronic Circuits : More digital circuitry can be fabricated per square millimeter of integrated-circuit material. Information storage can be much easier in digital systems than in analog ones. In particular, the great noise-immunity of digital systems makes it possible to store data and retrieve it later without degradation. In an analog system, aging and wear and tear will degrade the information in storage, but in a digital system, as long as the wear and tear is below a certain level, the information can be recovered perfectly. Theoretically, there is no data-loss when copying digital data. This is a great advantage over analog systems, which faithfully reproduce every bit of noise that makes its way into the signal.

Disadvantages 


of
Digital Circuits

  • Most physical quantities in real world are analog in nature, and these quantities are often the inputs and outputs that are being monitored, operated on, and controlled by a system. Thus conversion to digital format and re-conversion to analog format is needed.     For a digital system to do useful things in the real world, translation from the continuous realm to the discrete digital realm must occur, resulting in quantization errors. This problem can usually be mitigated by designing the system to store enough digital data to represent the signal to the desired degree of fidelity. The Nyquist-Shannon sampling theorem provides an important guideline as to how much digital data is needed to accurately portray a given analog signal.

Wednesday, August 20, 2014

Changing the base of logs in Maxima

From http://math-blog.com/2007/06/04/a-10-minute-tutorial-for-solving-math-problems-with-maxima/,  you can change the base of logs by using the formula below. Log in Maxima is to the base of e. Changing to the log of base 10 is as follows -

log10(x):= log(x)/log(10);

So for logs to the base 9, below is what you should be using for the software

log9(x):= log(x)/log(9.0);

For example,

log9(x):= log(x)/log(9.0);
log9(81.0);

which should give an answer of 2.0. I had to use a decimal point for most of the  numbers in the brackets. 

Number of solutions to linear equations

When x= some number then you have a solution.
If you get something like 5=7 then you have no solution.
If both sides are equal like 3x+5 = 3x+5 then you have infinite number of solutions.



 



Tuesday, August 19, 2014

Recognizing rational and irrational expressions

The sum, difference, product or quotient of two irrational numbers could be either rational or irrational. Therefore, the other expressions could be either rational or irrational, as we see below.

If c=2 and d=21, then cd=1 is rational.
If c=2 and d=2, then cd=22 is irrational.
If c=22 and d=2, then 

c/d=2
 is rational.
If c=2 and d=24, then cd=24 is irrational.