A student of mine has recently been working on what are called RATIONAL ALGEBRAIC EXPRESSIONs, or when two polynomials are divided by each other. As you may remember, a RATIONAL NUMBER is a number that can be expressed as a FRACTION. So numbers like 3 (or 3/1), 3/4, 22/7 are all rational numbers. If you remember, the word RATIO and RATION are names for fraction, which is why these are called rational numbers.
You may remember that some fractions, when expressed as decimals, involve an endlessly repeating sequence. For instance 1/3 or one-third is equal to 0.33333... and so would need an infinite number of numbers to express exactly. Another example is above, 22/7 which can be represented as 3.1414141414... and is often used as an approximation to PI. But because these numbers can be expressed as a fraction, the decimal sequence will always a nice pattern to it. In the case of 1/3, the number "3" over and over again; in the case of 22/7, the numbers "14" over and over again.
Alternatively, there are numbers which, when expressed as a decimal, are also endless sequences but which CANNOT be represented in a repeatable pattern. The most prominent examples are the square roots of numbers that are not PERFECT SQUARES. So numbers like the square root of 25 can be represented by 5 or 5/1, but a number like the square root of 23 could only be represented in decimal form if we kept an infinite number of decimal places. And so these numbers ARE NOT expressible as a fraction, and are called IRRATIONAL NUMBERS.
An excellent exercise to test your ability to identify rational and irrational numbers, as well as perfect squares is Number Cop, by Hans Software, which I have embedded below . . . Use the drop-down menu to decide which types of numbers you would like to test your knowledge of.
From Rational Numbers to Rational Algebraic Expressions
Similarly, a RATIONAL ALGEBRAIC EXPRESSION is an algebraic formula that is represented as a fraction. So, for example, the following are rational algebraic expressions:
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The techniques that are stressed in teaching RAE's are finding what are called the ZEROS, or the places where the RAE is equal to ZERO, and the POLES, or the places where the denominator is equal to ZERO. Though it may seem like a completely random and useless subject to introduce, it turns out that understanding expressions like these, and the meanings of ZEROS and POLES, is essential to understanding many physical systems from bridges to car suspensions to musical amplifiers. We'll consider an example below from music and electronics, known as a BANDPASS FILTER, once we look at how to analyze expressions like the ones above.
So let's say that we have a typical RAE like:
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If we were able to factor the numerator and denominator, we would be able to determine the values of s that make this expression equal to zero (by finding the ROOTS of the numerator) or which make this expression infinitely large (by finding the places where the denominator is equal to zero. If we were to graph an expression like the one above, we would in general get a graph that looks like:
FILTERS
In mathematical and engineering lingo, a FILTER is an object that selects certain frequencies to pass, while blocking others. (To learn more about frequencies, click here). The most common implementation might be a radio tuner. In general, in any city every radio station is being broadcast all the time to every place in its range. So, the antenna in your car is actually picking up every station that is being broadcast in your area -- if you tried to listen to all of them at the same time, it would likely sound like a whole bunch of static.
When you set the dial of a radio to a particular station (106.1 on FM is my fave in the Bay Area), what you are doing is telling your radio antenna to only listen to a particular frequency, as the number of a radio station corresponds to the frequency of wave that is carrying that station's signal. So, when I set my radio dial to 106.1, I am telling a circuit inside of my radio to FILTER out every other frequency that it hears besides 106.1, and so my speakers only broadcast the music that is coming from that station.
The name of such a circuit is a BANDPASS FILTER, since I am able to set a particular range of frequencies, or FREQUENCY BAND, that are allowed to PASS. (BAND-PASS, get it?). Usually, this is implemented by a circuit as shown below; the mathematics of such a circuit turn out to be controlled by a RATIONAL ALGEBRAIC EXPRESSION.
The diagram above is an example of what is called an RLC circuit. The R, L and C are the names given to electronic components called a resistor, an inductor and a capacitor. When desigining a circuit like the one above, we are able to specify which frequencies are allowed to pass by changing the values of the resistor, inductor and capacitor. If we were to analyze this circuit using techniques specified HERE, we would find that the equation governing its behavior is given by the rational algebraic expression:
For the purposes of this lesson, the most important thing to notice here is that this expression has a polynomial on the top and the bottom of our fraction, and that the coefficients of our polynomials is goverened by the values of the resistor R, inductor L and capacitor C.
When you set the dial of a radio to a particular station (106.1 on FM is my fave in the Bay Area), what you are doing is telling your radio antenna to only listen to a particular frequency, as the number of a radio station corresponds to the frequency of wave that is carrying that station's signal. So, when I set my radio dial to 106.1, I am telling a circuit inside of my radio to FILTER out every other frequency that it hears besides 106.1, and so my speakers only broadcast the music that is coming from that station.
The name of such a circuit is a BANDPASS FILTER, since I am able to set a particular range of frequencies, or FREQUENCY BAND, that are allowed to PASS. (BAND-PASS, get it?). Usually, this is implemented by a circuit as shown below; the mathematics of such a circuit turn out to be controlled by a RATIONAL ALGEBRAIC EXPRESSION.
The diagram above is an example of what is called an RLC circuit. The R, L and C are the names given to electronic components called a resistor, an inductor and a capacitor. When desigining a circuit like the one above, we are able to specify which frequencies are allowed to pass by changing the values of the resistor, inductor and capacitor. If we were to analyze this circuit using techniques specified HERE, we would find that the equation governing its behavior is given by the rational algebraic expression:
For the purposes of this lesson, the most important thing to notice here is that this expression has a polynomial on the top and the bottom of our fraction, and that the coefficients of our polynomials is goverened by the values of the resistor R, inductor L and capacitor C.
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