Category Archives: math

What is a number? Particularly √2

After the previous post, on the reality or otherwise of the square root of -1, I thought that the square root of 2 might benefit from a similar inquiry. After all, what can we actually say about √2 ? The answer to this question is very simple. “Not a lot !”.

In the real world of engineering, architecture, mathematical modelling, business, medicine and so on numbers are either counts (1,2,3,4,…) or absolute or relative measurements (1.20cm, 240 secs, 15 mins, 4096 ft, 35.7 mph,….). The first group is the natural or counting numbers, the second group is the rational numbers, and not so many of them. In practice it is rare for the size of a quantity to be expressed with more than four significant figures. So every practical quantity has a measurement in the form of a rational number, and most importantly IT CAN BE WRITTEN DOWN. I am going to call this the VALUE of the number.

The only thing is an assertion that there is a sort of number which when multiplied by itself produces the value 2.

So where does that leave √2 ?  It cannot be written down in the form of a rational number, so it has NO value in the above sense.
Ok, I can write  1.41422 < 2 < 1.41432 but neither of the two values shown is the value of √2. I could go on and get more digits in the two numbers and this would still be true.

This all started with the ancient Greeks, who found out that the length of the diagonal of a unit square was a quantity very different from quantities which could be measured using the side length of a unit square as the measurement unit. They described this state of affairs as “The side length and the diagonal length of a square are incommensurable”, which is a nice long word.

In passing I have to say that the Common Core math makes a real pig’s ear of this stuff.

So the Greeks were happy with the idea that every line segment has a length, and that the length is expressed as a number, but this wasn’t good enough for the nineteenth century mathematicians. I may write about this later, but for now we should be seeing if √2 can reasonably be “joined ” to the rational number system in a non magical, non wishful thinking way.

Let’s pretend that √2 is a sort of number, and that new numbers can be formed by a rational number “a” plus a rational amount “b” of √2, and write this as a + b√2

Then the sum of two these comes in as
(a + b√2) + (p + q√2) = (a + p) + (b + q)√2

and the product comes in as 
(a + b√2)(p + q√2) = (ap + 2bq) + (aq + bp)√2

In each case we have another of the “new” numbers.

One tricky question remains. What about division ?

If I multiply a + b√2 by a – b√2 I get a2 – 2b2 which has no √2 in it, it is a normal rational number, and it is only zero if BOTH a and b are zero.
This is called the root(2) conjugate.
In a division, if the divisor has its b not zero then I can multiply the top and the bottom (the divisor and the dividend) by the conjugate of the bottom, and the only √2’s are then on top.

(3+2√2)/(4-√2) = (3+2√2)(4+√2)/((4-√2)(4+√2)) = (16+11√2)/(16-2) …

As with the square root of -1 we can see that this is all about pairs of rational numbers, and the √2 symbol just keeps the members of each pair in order.

So rewriting the multiplication we get (a,b)(p,q) = (ap + 2bq, aq + bp)
and all the rules for operations can be expressed in this way and be seen to work.

We have ended up with a totally valid extension of the rational numbers by √2.
It is quite amusing to represent these pairs on an xy grid, and see the effect of multiplication.

But √2 still does not have a value ! ! ! ! !
 

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Filed under extension, irrational numbers, math, square roots

MUSIC: tuning,harmonic, equal temper, beat frequency :MATH – part trois !

So, what is the difference in musical effect between the harmonic scale and the equal temper scale ?

Whenever two notes are played together the ear “hears” the two notes both separately and together. The “togetherness” is a consequence of the perception by the ear of a third note, usually quite faint, the beat note with associated “beat frequency”. The effect is very noticeable when two recorders are played, as the notes are very “pure”. A pure note is one which consists of vibrations at exactly one frequency, and this is described by a sine wave or sine function y = sin(2πkx) where x is time and k is frequency.

Below, and with many thanks to DESMOS, which made the job almost painless, are plots of the sum of a frequency 5 wave and a frequency 5+b/2 wave, for various decreasing values of b. (The first has b = 0 to show the sum of the frequency 5 wave with itself, giving a reference point of 2sin(5*2πx)

beatpic1a

Now with frequencies 5 and 10
beatpic2a

Now with frequencies 5 and 9beatpic3a

Now with frequencies 5 and 7, something going on herebeatpic4a

Now with frequencies 5 and 5.75, notice the appearance of the beat in the signalbeatpic5a

Now with frequencies 5 and 5.5beatpic6a

Now with frequencies 5 and 5.25beatpic7a

And finally with frequencies 5 and 5.05beatpic9a

The first thing to note is that the beat frequency is the difference between the two “added” frequencies. This can be seen by seeing the period of the beat in this last one as 20, which is a frequency of 1/20 or 0.05 (= 5.5 – 5). (The math for all of this will be in another post). See London police whistle

The second thing to note is that in the case of 5 and 5.75 the frequency ratio is 5.75/5 = 23/20, and this is greater than the relative frequency of the 9/8 whole tone (C to D), and a bit less than 6/5, one of the estimates for the interval C to Eflat. This one has quite a large effect on the perceived sound of the whole tone interval, and is one reason why it is difficult to hear the two notes separately.

Now we can see what this all means in the harmonic tuning system. Let us take the notes C and E, frequency ratios 1 and 5/4.

The difference is 5/4 – 1 = 1/4. What note, if any, is this?

We have seen that multiplying by 2 doubles the frequency and produces a note one octave higher, so dividing by 2 produces a note one octave lower. Do it again and we get a note two octaves lower.

So the beat frequency for the pair CE is the C two octaves down. I will write this as C,, and the C two octaves up will be C”.

This means that each (in this case) of the two notes C and E is a harmonic of the beat frequency.

Consequently the CE interval will appear to have more “body” than might be expected. Just try playing the C on its own and then with a quieter 2 C’s below added.

This can be done for all the intervals on the harmonic scale (results shown in the table below).

The main conclusion I have come to is that the reason a choir trained without a piano accompaniment has a fuller sound than one used to singing with a piano, or other musical “backing” is that the first type automatically tunes itself in the harmonic system, and consequently has the benefit of beat frequencies which are in tune with the notes being sung.

This is particularly noticeable with barbershop singing.

Let us compare equal temper with harmonic on the major third interval CB

Harmonic: interval is 1 to 5/4, beat frequency is 1/4 or 0.25 , nice !

Equal temper: interval is 1 to cube root of 2 (twelfth root of 2 raised to power of 4, 4 semitones from C to E), and this is 0.260. So when we look at the second harmonic of this we get 4*0.260 = 1.04, which is NOT C. It is a bit less that a semitone above, and creates a low volume buzz that causes the major third to feel harsh.

The fifth, C to G, is not as much affected as the two systems give almost the same frequency.

Table coming

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Filed under a cappella, barbershop, harmonic, math, music, piano, singing, tuning

MUSIC: tuning, scales, fractions, ratio, harmonics :MATH ! Part deux.

Recap from previous post. here is a readable table of the notes and relative frequencies for the scale of C:

beat harmonic scale
Notice that the whole tone intervals are not all the same size. There are two distinct sizes, with frequency ratios 9/8 and 10/9. Only the two semitone intervals EF and BC’ are the same. Check the others ! Use the slightly not obvious relationship between ratios  P/Q = (P/R)/(Q/R)

beat tone semitone

This was the situation faced by J. S. Bach when he wanted to write music in any key. He figured that if two intervals had the same relative size then the two notes from the first, played together, would have the same musical “feel” as the two notes from the second interval. He wanted the major chord Bflat-D-F to sound like the major chord C-E-G, and so on. Not only that, but the scale devised so far is lacking four of the five “black” notes, and hitherto these had been fudged in. If one pursues the harmonic approach one finds that for example F-sharp and G-flat are different. See later for details.

So he figured that as there were 12 semitones in the full octave the frequency of each note would have to be a fixed multiple of the previous note, so multiplying 1 by this fixed number 12 times must get you to 2. In other words the magic number is the twelfth root of 2. Its value is 1.059463094 from my calculator.

The table below shows the comparison between the new Equal Temper scale and the harmonic scale.

beat pitch comparison

In 1722 Bach composed twelve preludes and fugues for keyboard called Das Wohltemperierte Klavier

If the link doesn’t work here it is:

https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier

Part three will look at beat frequencies and reasons for preferring the harmonic system, though not for keyboard instruments.

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MUSIC: tuning, scales, fractions, ratio, harmonics :MATH !

beat spinet 1704A spinet, from1704

Nobody knows when or how humans began to make music, but as soon as they could make things that could be used to play “nice” sounds (call them “notes”) they wanted to make musical instments, objects that could produce a number of different notes. The ocarina was an early one of these, and a strange collection of notes it makes. Eventually it was found that some sequences of notes of increasing pitch sounded “better” than others, and the result was a “scale”. The one I am considering here is the well known eight note scale, which runs from a start note to a finish note one octave higher. Found on a piano as the “white” notes, they are C D E F G A B and C’. As usual, it’s one thing knowing what you want, it’s often quite another to find a way of achieving it. In this case it is “string to the rescue”.

A stretched string can be made to vibrate by plucking it. The result is a “nice” sound. This sound will have a “pitch”, which is musical jargon for the frequency of the vibrations, which is physics/engineering jargon for the rate at which the string repeats its vibrations. This rate is expressed in repeats or cycles per second (or other unit of time).
For the record the pitch of the middle C on a piano is 256 (movements of the string up and back down
per second).

Now it was observed millenia ago that if a finger is placed on a string halfway along and the string is then plucked the vibration rate or pitch is doubled, and the sound is described as one octave higher than from the unfingered string. This new note is also known as the first harmonic of the original. The second harmonic is what you get if the finger is placed one third of the way along the string, and its pitch is three times the original. The third harmonic, with pitch four times the original, two octaves up, comes from a quarter of the original string length, and so on …….
Taking the second harmonic, and halving the frequency, or pitch, we get a new note which is between the original and the first harmonic, and either of these played at the same time as the new note gives a “pleasant” sound.
Doing this for the the fourth harmonic (one fifth of the original string length) we get another new note and the original plus the two new notes together produce what is known as the “major chord”.
On the piano, with C the original note, the new notes described above are the G and the E.
Now we can do some math !
If the pitch of C is 256 then the (harmonic) pitch of G is 256 times 3/2. Why? Because the pitch of a harmonic from a plucked string is inversely proportional to the fraction of the string used to make the harmonic.
First harmonic – half the string – pitch 2 times 256 (one octave up)
Second harmonic – one third of the string – pitch 3 times 256 (to get the G)
Third harmonic – one quarter of the string – pitch 4 times 256 (two octaves up)
Fourth harmonic – one fifth of the string – pitch 5 times 256 (to get the E)
and continuing..
Fifth harmonic – one sixth of the string – pitch 6 times 256
Sixth harmonic – one seventh of the string – pitch 5 times 256
The fifth gives a note one octave up from the second, and the sixth gives a note which we call B-flat,
with pitch 256 times 7 divided by 4

The result of all this is that we now have five notes in the scale:
C 256 … E 320 … G 384 … B-flat 448 … C’ 512
but it is easier to see what is going on if we just look at these as proportions of the “home” note C
Then we get
C 1 … E 5/4 … G 3/2 … B-flat 7/4 … C’ 2
or even better
C 4/4 … E 5/4 … G 6/4 … B-flat 7/4 … C’ 8/4

There do seem to be some holes in this, when comparing with the piano.
Where are D, F, A and B ?
We could go up in eighths, which does produce good values for D and B:
C 8/8 … D 9/8 … E 10/8 … G 12/8 … B-flat 14/8 … B 15/8 … C’ 8/4
but F at 11/8 and A at 13/8 don’t look good.
So we look at G, the first of the “new” notes, and think of it as a new home note. Then we can figure out
the B and the D in relation to the G.

The three notes C, E and G form a major triad (the chord of C major), and have the ratios 1, 5/4, 3/2
So to get the ratios of B and D to G just divide all three by the G to C ratio:
(3/2)/(3/2), (15/8)/(3/2) and (9/4)/(3/2) (using 2 times 9/8 as the ratio for D ‘)
and get 1, 5/4 and 3/2, which shows that G B D’ forms a major triad, just like C E G

We still have a problem with F and A, so let us try to set up F A C’ as a major triad by using some simple fraction manipulation.
F to C’ is to be the same ratio interval as C to G, so using the letter F to stand for its ratio to C we want
C’/F = G/C = 3/2
but C’ stands for the ratio of C’ to C, which is 2
So 2/F = 3/2, and solving we get F = 4/3

You can now do the calculations for A, which ends up as A = 5/3, so our full scale, including B-flat, is
C 1 … D 9/8 … E 5/4 … F 4/3 … G 3/2 … A 5/3 … B-flat 7/4 … B 15/8 … C’ 2

Observe that these frequency ratios determine the notes for the scales of C and F, and the scale of G without the F-sharp.
More ratios of interest can be found, as we can see that the intervals CD, DE, FG, GA, AB are whole tones, and EF and BC’ are half or semitones.

Up to the time when Johann Sebastian Bach intruduced the equal tempered scale together with a set of pieces written in each of the possible keys the methods of tuning keyboard instruments (harpsichord, clavichord, spinet) were based on some variant of the harmonic approach.

Here is a link for some heavier stuff:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Next post: The equal tempered scale and some more math.

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Filed under fractions, math, music, musical scales, teaching

Real problems with conic sections (ellipse, parabola)

So there is an oval hole in a metal casting. It’s supposed to be an elliptical hole. Is it ????? How can we find out ?????
A good start would be to find the line which would be the major axis if it was elliptical. This turns out to be an engineering problem, not a mathematical one (I cannot see a way!). If the oval curve has an axis of symmetry then the method below will find it:

Firstly, get a computer picture of the oval.
Take two circles, of different radii, and push them along until each one touches the oval in two places.

ellipse12

The line joining the two centers will be the axis of symmetry if there is one (this can be shown mathematically).

ellipse34

ellipse5

The equation of an ellipse uses the lengths of the major and minor axes. Do it !

The closeness to elliptic can be assessed in various ways. Think of one.

next…..finding the focus of a parabolic shape

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GEOSTRUCT, a program for investigative geometry

I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.

It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.

Just try it out, it’s free.

Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html

.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.

help pic 1
Two lines, with points placed on them
help pic 3
Three random lines with two points of intersection generated
help pic 6
Five free points, three generated circles and a center point
help pic 7
Three free points, connected as point pairs, medians generated
help pic 5
Two free circles and three free points, point pairs and centers generated
gif line and circle
GIF showing points of intersection of a line with a circle
hypocycloid locus
Construction for locus of hypocycloid
circle in a segment
gif002
GIF showing a dilation (stretch) in the horizontal direction
gif piston cylinder
Piston and flywheel
gif touching2circles
Construction for circle touching two circles
gif parabola
Construction for the locus of a parabola, focus-directrix definition.

 

 

 

 

 

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Duality, fundamental and profound, but here’s a starter for you.

Duality, how things are connected in unexpected ways. The simplest case is that of the five regular Platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. They all look rather different, BUT…..

take any one of them and find the mid point of each of the faces, join these points up, and you get one of the five regular Platonic solids. Do it to this new one and you get back to the original one. Calling the operation “Doit” we get

tetrahedron –Doit–> tetrahedron –Doit–> tetrahedron
cube –Doit–> octahedron –Doit–> cube
dodecahedron –Doit–> icosahedron –Doit–> dodecahedron

The sizes may change, but we are only interested in the shapes.

This is called a Duality relationship, in which the tetrahedron is the dual of itself, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other.

Now we will look at lines and points in the x-y plane.

3x – 2y = 4 and y = (3/2)x + 2 and 3x – 2y – 4 = 0 are different ways of describing the same line, but there are many more. We can multiply every coefficient, including the constant, by any number not 0 and the result describes the same line, for example 6x – 4y = 8, or 0.75x – 0.5y = 1, or -0.75x + 0.5y + 1 = 0

This means that a line can be described entirely by two numbers, the x and the y coefficients found when the line equation is written in the last of the forms given above. Generally this is ax + by + 1 = 0

Now any point in the plane needs two numbers to specify it, the x and the y coordinates, for example (2,3)

So if a line needs two numbers and a point needs two numbers then given two numbers p and q I can choose to use then to describe a point or a line. So the numbers p and q can be the point (p,q) or the line px + qy + 1 = 0

The word “dual” is used in this situation. The point (p,q) is the dual of the line px + qy + 1 = 0, and vice versa.

dual of a rotating line cleaned up1

The line joining the points C and D is dual to the point K, in red.  The line equation is 2x + y = 3, and we rewrite it in the “standard” form as  -0.67x – 0.33y +1 = 0  so we get  (-0.67, -0.33) for the coordinates of the dual point K.

A quick calculation (using the well known formula) shows that the distance of the line from the origin multiplied by the distance of the point from the origin is a constant (in this case 1).

The second picture shows the construction of the dual point.

dual of a rotating line construction1

What happens as we move the line about ? Parallel to itself, the dual point moves out and in.

More interesting is what happens when we rotate the line around a fixed point on the line:

gif duality rotating line

The line passes through the fixed point C.  The dual point traces out a straight line, shown in green.

This can be interpreted as “A point can be seen as a set of concurrent lines”, just as a line can be seen as a set of collinear points (we have fewer problems with the latter).

It gets more interesting when we consider a curve. There are two ways of looking at a curve, one as a (fairly nicely) organized set of points ( a locus), and the other as a set of (fairly nicely) arranged lines (an envelope).

A circle is a set of points equidistant from a central point, but it is also the envelope of a set of lines equidistant from a central point (the tangent lines).

So what happens when we look for the dual of a circle? We can either find the line dual to each point on the circle, or find the point dual to each tangent line to the circle. Here’s both:

dual of a circle4

In this case the circle being dualled is the one with center C, and the result is a hyperbola, shown in green.  The result can be deduced analytically, but it is a pain to do so.

dual of a circle3

The hyperbola again.  It doesn’t look quite perfect, probably due to rounding errors.

The question remains – If I do the dualling operation on the hyperbola, will I get back to the circle ?

Also, why a hyperbola and not an ellipse ? Looking at what is going on suggests that if the circle to be dualled has the origin inside then we will get an ellipse. This argument can be made more believable with a little care !

If you get this far and want more, try this very heavy article:

http://en.wikipedia.org/wiki/Duality_(mathematics)

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What is Algebra really for ?

An example tells a good tale.

Translation of a line in an x-y coordinate system:

Take a line  y = 2x -3, and translate it by 4 up and 5 to the right.

Simple approach : The point P = (2, 1) is on the line (so are some others!). Let us translate the point to get Q = (2+5, 1+4), which is Q = (7, 5), and find the line through Q parallel to the original line.  The only thing that changes is the c value, so the new equation is  y = 2x + c, and it must pass through Q.  So we require  5 = 14 + c, giving the value of c as -9.

Not much algebra there, but a horrible question remains – “What happened to all the other points on the line ?”

We try a more algebraic approach – with any old line  ax + by + c = 0, and any old translation, q up and p to the right.

First thing is to find a point on the line – “What ? We don’t know ANYTHING about the line.”

This is where algebra comes to the rescue. Let us suppose (state) that a point P = (d, e) IS on the line.

Then ad + be + c = 0

Now we can move the point P to Q = (d + p, e + q)  (as with the numbers earlier), and make the new line pass through this point:  This requires a new constant c (call it newc) and we then have  a(d + p) + b(e + q) + ‘newc’ = 0

Expand the parentheses (UK brackets, and it’s shorter) to get  ad + ap + be + bq + ‘newc’ = 0

Some inspired rearranging gives  ‘newc’ = -ap – bq – ad – be, which is equal to -(ap + bq) – (ad + be)

“Why did you do that last step ?” – “Because I looked back a few lines and figured that  (ad + be) = -c, which not only simplifies the expression, it also disposes of the unspecified point  P.

End result is:  Translated line equation is  ax + by + ‘newc’ =0,  that is,  ax + by + c – (ap + bq) = 0

and the job is done for ALL lines, even the vertical ones, and ALL translations. Also we can be sure that we know what has happened to ALL the points on the line.

I am not going to check this with the numerical example, you are !

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Rigid motions are actually useful !

I am currently reconstructing my geometrical construction application, Geostruct, to run in a web page using javascript.

One of the actions is to find the points of intersection of a straight line with a circle. Here is a gif showing the result:

gif line and circle

The algebra needed to solve the two simultaneous equations is straightforward, but a pain in the butt to get right and code up, so I thought “Why not solve the equations for the very simple case of the circle centered at the origin and the line vertical, at the same distance (a) from the centre of the circle

line vertical circle at origin

Then it is a simple matter of  rotating the two points (a,b) and (a,-b) about the origin, through the angle made by the original line to the vertical, and then translating the circle back to its original position, the translated points are then the desired points of intersection.

The same routine can be used for the intersection of two circles, with a little bit of prior calculation.

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Subtraction and the “standard” algorithm

CCSSM talks about “the standard algorithm” but doesn’t define it – Oh, how naughty, done on purpose I suspect, since there are varieties even of the  “American Standard Algorithm”. Besides, if it is not defined it cannot be tested (one hopes!). I checked some internet teaching stuff on it, and as presented it won’t work on for example 403 – 227 without modification.

Anyway, I was thinking about subtraction the other day (really, have you nothing better to think about?), and concluded that subtraction is easiest if the first number ends in all 9’s or the second number ends in all 0’s. So, fix it then, I thought, change the problem, and here are the results

.Two simple algorithms for subtraction

I am quite sure that some of you can extract the general rule in each case, and see that it works the same in all positions.

While I am going on about this I would like an answer to the following-

“If I understand subtraction, and can explain the ideas to another, and I learn the standard algorithm and how to apply it, and I have faith in it based on experience, WHY THE HELL DO I HAVE TO BE ABLE TO EXPLAIN IT?”

I guess this post counts as a rant!

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