Formulae
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Written by Bryce Ringwood   

Algebra and Formulae

The previous article was about doing arithmetic, rather than abstract thought. The pages on this web-site have quite a few formulae and we need to know how to manipulate them and eventually turn them into an arithmetic problem.  The objective being to order and use the correct parts (electronic components) for our projects - its always good to bear in mind the end result!

In this article, we are interested in representing numbers as symbols in a formula, so that we can re-use a calculation over and over again. This is "elementary algebra" as taught in schools, as opposed to "abstract algebra" and the other algebras of pure mathematics. Closely allied to what we want to do, is the idea of an algorithm.

Let's take the idea of finding the current through a resistor of known value. You may know the formula, but if algebra hadn't been invented, you would learn the method instead. You would take the applied voltage - say 5 Volts and divide that by the resistance, say 10Ω to give you a current of 0.5 Ampères.

In ancient Babylon (circa 1000 BC ?), someone came up with the idea of substituting characters for numbers, but the introduction of Algebra has been credited to the Greek mathematician Diophantus in the 3rd century AD. Prior to that time the Greeks represented abstract thought through the medium of geometric descriptions. The word "Algebra" comes from an Arabic word, and Arab/Muslim scholars are  credited with many algebraic proofs. We can now, therefore, describe Ohm's Law with the following familiar equation.

{\displaystyle V = I \times R}
Where V is the applied voltage, R is the resistance and I is a symbol representing the current. In our example, we have done some manipulating: Divide each side by of the equality by \displaystyle R then
{ V \over R} = {{I \times R} \over R}
So
{{V \over R} = I}
 or, clearly,
 I = {V \over R}

Now we substitute 5 for V and 10 for R to get our value of 0.5 for V. Note how we can manipulate formulae to give us what we want. As long as we do to the left side of the equation waht we do to the right - everybody is happy.

Another "calculation rule" goes something like this:
"In a right angled triangle the square of the length on the hypotenuse equals the sum of the squares of the lengths on the other two sides". This is actually rather unhelpful if we want to perform a calculation, but then, I think the ancient egyptian temple builders really did use squares, not numbers to make their right angles for setting out the base of a pyramid. If the length of the hypotenuse is designated as h, and the length of the other two sides is a and b then:

h^2 = a^2 + b^2; \quad a^2 = h^2 - b^2 = (h+b)\times(h-b); \quad h = \sqrt{(a^2 + b^2)}
and so on ...

So, here are two very well known formulae. You can manipulate the symbols the way you would manipulate numbers (the symbols are really placeholders for numbers that you intend to put in later), and the symbols can be intermingled with numbers, as in the second formula describing Pythagoras' theorem. Pity poor Pythagoras - he never got to see his theorem written out as a formula. Pythagoras lived from about 530 - 495 BC.

Now we have jogged the old memory about formulae - we can ask from whence they cometh?

The Basis of Formulae

Formulae can be derived from pure intellectual thought  ("theoretical") and then tested against experiment, or they can be "empirical' formulae derived from many sets of measurements. 

Tycho Brahe made numerous observations of planetary motion, leaving copious notes of all of his observations. It was Kepler who was able to convert these observations to formulae ("The planets sweep out equal areas in equal times" and so on). It was Sir Isaac Newton who eventually provided the theoretical basis for Kepler's laws. In a later era Einstein developed more refined laws of motion that would allow all observers to have equal status.

In similar vein, Georg Ohm used pieces of wire of different lengths to determine empirically that the voltage across an electrical circuit is proportional to current:

\displaystyle  {V \propto I}

We call the "constant of proportonality "Resistance (R)". The funny symbol is what we used to call "lobster", so we would subvocalise "V lobster R". Nowadays we have a theoretical basis to explain Ohm's law in terms of the drift of electrons through conducting materials, but poor Georg just had bits of wire, Leyden Jars and meters ....

Sometimes things are complicated - such as the relationship between turns on a coil and its inductance.  Fortunately, a gentleman called Wheeler did hundreds of experiments to find the relationship between inductance and coil geometry.

Talking of geometry, lets brush up on some simple trigonometry, but before we do this we need to understand how to measure angles and what π is all about.


π  

The symbol for pi is used to represent the ratio of the circumference to the diameter of a circle. This ratio, approximately 3.1415926535...., is an Description f Piirrational number. There is no fraction that can be used to represent it accurately, although 355/113 is good enough or most purposes. The ancients were aghast that the deity could do such a thing, and even in fairly recent times there has been legislation decreeing that π should have the value "3".

The value of π is known to around ten trillion places of decimals, and you might reasonably ask what the point of this is (it probably will give you the circumference of the universe with less than 1 mm error, assuming you know the radius. AND assuming the universe to be spherical. I believe some unfortuate has memorised the value to 67000 decimal places. One answer is that the numbers trailing the decimal point are truly random. The other is that calculations like this test a computer's processing speed.

The ancients only had simple computing devices and it is instructive to see how they approached the problem by inscribing geometrical figures inside a circle. In this example, I used an octagon. We calculate the length of an individual side, then multiply by the number of sides to arrive at a (fairly inaccurate) value for π.

We can refine this by dividing the chord into two and inscribing the circle with a 16 sided figure. The steps in the calculation are outlined as follows:

approximating pi

 (You can follow me on the HP-35, or using the Traycalc - which is what I used here.)

 Divide the chord length (0.7653668647) by 2.

Square the result.    = 0.1464466094

Subtract it from 1,0 (The unit radius). = 0.8535533906

Square root it = 0.9238795325

Subtract from 1 = 0.0761204675

Square the result = 0.0057943256

Add the square of half the original chord length = 0.1522409350

Take the square root to get 0.3901806441 - now multiply by 8 to get 3.1214451528 - a better value. You can perform this process repeatedly with 32, 64 ... etc sided figures, but your calculator will soon run out of accuracy. Now imagine performing this sort of calculation with an abacus, or on your fingers.

This (hopefully) illustrates that pi can be calculated by successively refining a simple calculation.


Radians - measuring angles

You may be familiar with measuring angles in degrees, but in 1714 a mathematician called Roger Cotes conceived the idea of measuring angle Definition of 1 Radianbased on the angle subtended at the center fo the circle by an arc length equal to the radius. I think it must befairly obvious that there are 2π radians in a whole circle, and that a right angle is π/2 radians.

Here, we use radians because the electrical formulae we use later on all express values in radians - not degrees. (We are doing this so we can use the formulae to get the right bits and bobs for our projects - right?).

The trouble is, we are so familiar with degress - 360 degrees in a full circle = 2π radians and so forth,that we can easily get confused.

SO - to convert from degrees to radians - multiply by 2π then divide by 360

To convert from radians to degrees - divide by 2π.then multiply by 360.

(The ancient Babylonians used a base 60 number system - from which we get our 60 seconds in a minute - etcetera. I would have had insufficient fingers to get me out of primary school)


Trigonometrical Ratios

The once-feared trig is now just a few simple keystrokes on your calculator. No more "Peters 12 place tables". trig ratios

The small diagram on the right gives you the meaning. The triangle is right -angled with θ being the angle between the adjacent side and the hypotenuse of length r. Notice that sin(\theta) can take on a maximum value of 1 when \theta ={\pi/2} and 0 when \theta = 0. The cosine also takes on values of 0 and 1 at maximum, but is{\pi/2} out of phase with the sine function, or in plain English, when the sine is 1 the cosine is zero and vice versa.

Of course, the sine and cosine can be negative, with a minimum value of  {-1}.

The tangent is the slope of the hypotenuse, and can take on values between minus infinity and plus infinity.

A simple rule for remembering the ratios is the nonsense word OHSAHCOAT - opp/hyp = sin; adj/hyp=cos; opp/adj=tan.

Here are some trig formulae that you can use. You can amuse yourself by attempting to prove them. Some are pretty obvious:

sin^2\theta + cos^2\theta = 1
sin(\theta + \phi) = sin\theta cos\phi + cos\theta sin\phi
cos(\theta + \phi) = cos\theta cos\phi - sin\theta sin\phi

You can also learn something by visiting this web site.


Graphs and coordinates

Nowadays most people are familiar with GPS and GPS coordinates, so I'm not sure very much explaining needs to be done.

Typically, you might want to plot a graph of battery voltage against time, or maybe plot FET characeristics for an amp. At one time things weren't easy - take the National NC 100 receiver, for example. It was supplied with graphs of dial reading vs frequency (or, more likely, wavelength.) To tune to 1485 kHz, you would read 1.485 on one axis and read the dial setting on the other. To produce the graphs, you would use an accurate signal generator and make a table of frequencies against dial reading: (Click on the image to enlarge it)

Here we see the dial has been set to 278 and the frequency is 7.168.

You can download graphcalc to plot functions. This works like many graphing pocket calculators. There is also dreamcalc and a TI85 emulator - which will only work on Windows 7 and above.

None of these are particularly useful, although they may be educational. For practical purposes, its best to learn the graphing capabilities of Microsoft Excel or Open-Office's equivalent (free!).

Graphs are abundant in data sheets for valves, transistors and various ICs. The manufacturers of these products want you to use them, so they make everything as simple as they can.


Polynomials

Polynomials are algebraic expressions in one or more variables raised to a non-negative power with constant coefficients. The powers may not be fractional. Rather than provide a sophisticated formula, I will just give some examples:

f(x) = a_0 + a_1x +a_2 x^2 + a_3x^3 + ...  ... (1)
f(x) = 3x^2 + 5x - 10 ... (2)
f(x,y) = x^2 +y^2 + 2gx + 2fy + c ... (3)
f(x) = 3 ... (4)
The following is NOT a polynomial
f(x) = 3x - 0.2x^{0.5} ...( 5)

The last good example is a polynomial with only an absolute term. The bad example has a square root.Cubic Curve

If you get a piece of graph paper, you can plot these functions by substituting values for x (and y). For example, equation 3 is the equation of a circle. The expression (x-3)(x-2)(x-1) \equiv (x^2 -5x +6)(x-1) \equiv (x^3 - 6x^2 + 11x -6) can be plotted, and you will see the curve cuts the x-axis at three points. (1,2 and 3).  These are the roots of the equation:

{x^3 - 6x^2 + 11x -6} = 0
The funny equals sign with three bars reads "is equivalent to".

This leads us to the problem of solving polynomial equations. Lets look at the simplest possible polynomial equation :

x^2 - 1 = 0

Fairly obviously, the roots are 1 and -1, since if you substitute either of these values for x, the equation is "satisfied". Other equations which can be factorised:

 

What about:

x^2 + 1 = 0 ?
W will be going ino equations in more detail in a later article.

Hmmm. Now is the time to introduce i. and j (not the fish fingers).


i and j

i, j and sometimes ι (iota) are used to denote the imaginary unit. i is favoured by mathematicians, but everywhere except in the maths review, we will use j - simply to avoid confusion with current. Just as we have so-called real numbers - the ones of our everyday life extending from - infinity to plus infinity and passing through zero on the way, we also have "imaginary" numbers extending from -i times infinity through zero (hmmm) and up to plus i times infinity.

Simply put:

{i = \sqrt{-1}}

Where does i come from? Well in the previous section we saw that polynomial equations in x have n solutions, where n is the degree of the polynomial. The degree of a polynomial is the degree of the term with the highest degree, thus  x^5 -1 = 0 is a fifth degree polynomial equation. Before we attempt to solve it, let's look at something simpler, namely x^2+1 = 0. Clearly, this equation has two roots, best expressed by\sqrt{-1} and   {\displaystyle {-\sqrt{-1}}}. These are written as i  and -i respectively.

This is all well and good, but can be confusing. For example: what does \sqrt{-7} mean? To be honest, I am not sure. I think it is probably best to think of i as being an operator (even if it isn't) and not try to muddle it up with arithmetic, so rather write it {\displaystyle {i\sqrt{7}}}. The problem is that the identities:

{\displaystyle {  {\sqrt{a}}\times{\sqrt{b}} \equiv {\sqrt{a\times b}}    }}
and
{\displaystyle {  {\sqrt{a}}\over{\sqrt{b}} }\equiv {\sqrt{a\over b}}    }
do not work for anything other than real numbers.


Summary

In this section, we looked at:

  • The basic idea behind formulae
  • The life of pi
  • Graphs and polynomials
  • The imaginary unit

The next section deals with equations and more advanced algebraic expressions.

 

 

 
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