HKJSMS Newsle er 2013‐2014 Issue 2
From the Editor Dear Readers, As we enter the year of the Horse, the HKJSMS is once again ready for everyone to enjoy. We wish a frui ul year ahead for everyone. Thanks to all the writers and everyone suppor ng the newsle er. Ma hew Kan
Contents: Tangent Half‐Angle Subs tu on…………………………………………………………………………………………………………………………………..P.2 Interval Bisec on Method—Release your inner Casio…………………………………………………………………………………………………..P.4 Complex Numbers…………………………………………………………………………………………………………………………………………………………P.7 Applica ons of Logarithm……………………………………………………………………………………………………………………………………………..P.8 Geometry……………………………………………………………………………………………………………………………………………………………………P.10 Maths Apprecia on…………………………………………………………………………………………………………………………………………………….P.11 Math Jokes and Puzzles……………………………………………………………………………………………………………………………………………….P.12
Tangent Half-Angle Substitution
d d t (tan ) d d 2 dt 1 2 1 1 t2 sec (1 tan 2 ) d 2 2 2 2 2 2dt d 1 t2
By Helson Go The Tangent Half‐Angle is a subtle but powerful method of Integra on. Its main purpose is to find Integrals that com‐ bine trigonometric func ons and constants, for instance 2 d
1 cos sin
Such integrals could not be solved using simple subs tu on or reduc on formulae due to the constant part in the inte‐ gral.
Deriva on using Ra onal Parameteriza on of a Unit Circle A deriva on method that suits intui on is through parame‐ teriza ons of a circle. Through which, coordinates of each point on the circle can be represented as func on of a pa‐ rameter, i.e. a special variable. This method needs men on‐ ing also because it is arguably the origin of the Tangent‐Half Angle Subs tu on, having first been used as far back as the Classical period.
This is where the Tangent Half‐Angle subs tu on becomes Those who learnt about parametric equa ons may be famil‐ useful. It transforms trigonometric func ons into ra onal iar with the first parameteriza on, func ons to enable usage of other techniques. However, x cos t although Tangent Half‐Angle subs tu on is very versa le, it y sin t can get extremely complicated. Therefore, simpler or more straigh orward methods are always preferred, if available Coordinates of a point (x,y) lying on the circumference of a The Tangent‐Half Angle method, as its name suggests, is circle are related to the angle of eleva on t between a line based on the half‐angle iden ty of tangents, i.e. drawn from that point to the origin and the posi ve x‐axis. When subs tu ng t tan
The second parameteriza on is based on the simultaneous equa ons
x2 y 2 1 y m( x 1)
Then cos( )
1 t2 1 t2
2t 1 t2
2dt 1 t2
Deriva on using Trigonometric Iden
The upper equa on is equa on of a unit circle, and the sec‐ ond is equa on of a line passing through (‐1,0) with a varia‐ ble slope m. Solving the simultaneous equa ons yields
The most straigh orward way to derive these formulae is to use trigonometric iden es taught in Module 2, shown be‐ low.
2t 1 t2
1 t2 1 t2
Deriva on for sine
2t y 1 t 2 2 x 1 t 1 t2
sin 2sin cos 2 2 2t cos 2 2
1 tan 2
2 1 t2
x cos t y sin t
Deriva on for Cosine cos 2 cos
2 1 t 1 1 t2 1 t2
Deriva on for Diﬀeren al
1 t2 1 t2
2t 1 t2
x 3sin sin
Here we use the example shown above,
x sin 2 cos 2 1 cos 2 1 sin 2 1 ( ) 2 3
Using the tangent half‐angle subs tu on (Check page 1 for the formulae), we obtain
1 cos sin
9 x2 3 9 x2 3 x x 3
Thus, the final expression is
in terms of θ, i.e.
makes the final expression wri en
1 dt 2 ln(t 1) c t 1 2
9 x2 x2 9 3
Now we have a much easier and more manageable integral. Using simple subs tu on to deal with this integral gives us:
Subs tu ng t tan
Subs tu ng the values of sine and cosine found into the half angle formula for tangent, i.e.
2dt ( ) dt 1 t2 2 2 2 1 t 2t t 1 1 2 2 1 t 1 t
Using a Pythagorean Iden ty
2 d 1 cos sin
3 4 tan
3 9 x2 4( )2 x
12 4 9 x 2 2 x
Example 3 The second example is an example exhibi ng the ability of the Tangent Half‐Angle subs tu on to deal with complicated This is an exhibi on of the universality of Tangent Half‐ problems Angle. A seemingly simple trigonometric equa on that is actually extremely complicated can be solved via the Tan‐ 9 x 2 dx gent Half‐Angle Subs tu on: 15 3 9 x2 4 x sin 4 cos The first step is use Trigonometric Subs tu on, a standard Due to the 4θ, subs tu on is notoriously rigorous. However, it is not sophis cated, requiring only algebraic work. To pre‐ vent was ng eﬀort describing a very complicated opera on, only the steps that are cri cal to understanding of the solu‐ on will be shown here.
technique. By subs tu ng x 3sin
9 9sin 2 d (3sin )
9 9sin 2 12sin
3d sin 5 3cos 4sin
Now apply Tangent Half‐Angle Subs tu on (The actual oper‐ a on is given to you as an exercise) we obtain
3dt 4t 1
We start with sin 4 cos , sin 4 cos 0
3dt 3dt 3 d (2t 1) 4t 1 (2t 1) 2 2 (2t 1) 2 3 3 c c 2(2t 1) 4t 2 3 c 4 tan 2 2
From here on, using Tangent‐Half Angle subs tu on with
Using simple subs tu on, we have
In order to return the final answer to be in terms of x, we manipulate some trigonometric iden es Given 3
Interval Bisection Method—Release your Inner Casio
2sin 2 cos 2 cos 0 4sin cos (1 2sin 2 ) cos 0 cos (4sin 8sin 3 1) 0 1 t2 2t 2t 3 [4( ) 8( ) 1] 0 1 t2 1 t2 1 t2 8t (1 t 2 ) 8(2t )3 (1 t 2 ) 1 t 2 0 (1 t 2 ) 2 (1 t 2 ) 4 1 t2
By David Chong
and simplifying or factorizing suitably, we obtain, (t 1)(t 1)(t 2 4t 1)(t 4 4t 3 14t 2 4t 1)
Solve this expression gives us several values of t, shown be‐ low t 1
The Mo va on
t 2 3 t 1 5 5 2 5 t 1 5 5 2 5
Applying inverse tangent gives us the values for hence n
2 3 2n 10 7 2n 10
10 9 2n 10
Have you ever stumbled upon an equa on that you have no idea how to solve? Or, have you ever lost faith in your own algebra and wished that there is another way to check your answers? If your answer is “yes”, you’ve come to the right place. But if it’s a “no”, this ar cle is s ll worth reading. Why? It releases the inner poten al of your standard issue Casio calculator.
But what’s more unfortunate is that we aren’t always able to write the unknown as the subject, or at least not in some simple closed form. Don’t believe in me? By all means, try to solve the equa on below:
6 5 2n 6
When we are solving an algebraic equa on of an unknown, we o en tackle them by doing the same thing on two sides – namely transposi on, squaring both sides (at the expense of extraneous solu ons), exponen a on, taking square roots (not always jus fied, at least not in the reals), so on and so forth, in order to make our unknown the subject of the equa on. If this is possible, then we could replace the coeﬃ‐ cients of the powers or elementary func ons of our un‐ known with arbitrary constants, barring the restraints of our opera ons, say, in the quadra c equa on, when b2 – 4ac is unfortunately smaller than 0, we couldn’t get any real solu‐ ons.
This problem is so abstract; it seems there may be no appli‐ ca on in the world. Naturally, be er methods of solving such problems exist at higher levels, and the value of this example lies in it being a problem that forces us to use the Tangent‐half angle subs tu on but also retains a set of solu‐ ons that are not surds.
Where is the familiar (I hope) Euler’s number, 2.71828…
For those who didn’t give up at first sight, I commend your perseverance, but me to take a break and reflect. What are you solving this equa on for? You want to find the value of t. But let’s accept it, we can’t write it in a simple exact closed form. Now take a step back. If we can’t precisely determine it, is an approxima on really that bad? Maybe to you, but not bad at all to me, since you can make it as precise as you like, as long as you can accept the fact that it is not dead‐on, as we will see later, so I will con nue on with my blabber anyways.
If you are interested in this problem, use the link below to inves gate more.
For the sake of my convenience, we will denote g(t) = t ‐ e‐t, and our problem becomes finding t such that g(t) = 0. This does not simplify our problem, but it certainly helps stream‐ lining the nota ons. Now, conjure some deeply burrowed memories from your math lessons. Do you remember how to find zeroes of a func on graphically? Oh yes you do. Draw the graph of the func on and find its intersec on with the x‐ axis. But we’re a li le bit smarter (and regre ably I am only a li le bit smarter) than that. First, we have to no ce that our g(t) is a con nuous func on, since the func on of a straight
and the exponen al func on are both con nuous func ons, (3d.p.), which is a nega ve number, and that is all that and our g(t) is just their diﬀerence. Loosely speaking, a con‐ ma ers. nuous func on is a func on (which automa cally forces us g(t) to draw its graph from le to right, since one input cannot have more than one output for a func on) whose graph can be drawn without li ing the pen. To do the magic, evaluate g (0) and g(1): (1, 0.632) No ce that 1‐1/e is some constant larger than zero, since 1/ e < 1. Now, we have determined two points on the graph paper: g(t)
t (0.5, ‐0.107)
Now we are in the same situa on – we have a root of g(t) in between 0.5 and 1. Had we obtained a posi ve number, we could just do the reverse and discard (1, 0.632) in favour of (0.5, some posi ve number). Assuming that f(a) is nega ve and f(b) is posi ve, our algorithm becomes: If f((a+b)/2) > 0 => take (a+b)/2 as the new b. Then there is a root between a and “new b”.
(1, 1‐1/e) O
If f((a+b)/2) = 0 (exactly) => this is the root you want. What more are you asking for?
(0, ‐1) Now try to draw a line that goes from le to right. It is bound to intersect with the horizontal axis no ma er how you draw it. That could only mean one thing: g(t) has a zero lying be‐ tween 0 and 1. If we flip over the points about the horizontal axis, the situa on is s ll the same, and the con nuous func‐ on in considera on has a zero in the interval. To write this down more generally:
If f((a+b)/2) < 0 => take (a+b)/2 as the new a. Then there is a root between “new a” and b. Did it improve? Definitely! We have successfully increased our accuracy by cu ng down half of the range! And since the situa on is essen ally the same, we can apply the meth‐ od repeatedly un l the interval is as small as we want. Bril‐ liant, right? This method of approxima ng a zero is called the interval bisec on method.
Given a con nuous func on f(x), if f(a) and f(b) has diﬀerent The only pi all is that we cannot construct a and b. We have signs, there is a zero of f(x) lying between a and b. to be observant and guess them. As in our example, 0 and 1 The Theory are just wild guesses. But since they happen to be of oppo‐ The above fact enables us to confirm the existence of a zero site signs, we can just proceed. You can write a systema c of f(x) between a and b by considering f(a) and f(b). At first guessing programme if you like, but that’s just brute force, glance, this fact might look useless for our purpose, since it and uses up computa onal me unnecessarily. My advice is tells us nothing about how to construct a and b, let alone the to rely on our own brains for the ini al guess, since the hu‐ value of the zero! But look more closely. The zero is between man guessing ins nct is much be er than the computer’s a and b. How would this help us to determine the value of guessing ins nct (which assumes the value of 0). the zero approximately? The execu on: WARNING – INCOMING GEEK LANGUAGE Let’s look at what is meant by “approximately”. Normally, Now, how to we write a programme on our standard issue we said that something is approximately something in the Casio calculators so that it will loop the algorithm for us? sense that e is approximately 2.718 to 3 decimal places. There are many advanced loops available in Casio 50FH, but However, saying this is equivalent to saying “the value of e is let’s not forget about simpler (and faster) yet commonplace in between 2.7175 and 2.7185”. See? Specifying the range 3650P, in which we could only construct loops “by hand” that a constant falls into is also an approxima on of it. The with label and goto commands. Now, open the programme narrower the range, the more accurate our approxima on is. wri ng window, which is mode‐6‐1‐1 (programme no. 1) for So, if, magically, we could find a and b such that f(a) and f(b) 50FH, and mode–mode–mode‐1 for 3650P. are of diﬀerent signs, and we have a way to narrow down A good habit (or, just my weird habit) to start a programme the range, we can claim that we have a way to approximate is to write clear memory by shi ‐9‐1 (shi ‐mode‐1 on the zero of f in between a and b. Now, what’s the best way 3650P). This clears the memory of your stored variables to narrow down an interval? Cut it into half! (ABCDXYM), and you can safely assume that the variables Let’s look back to our numerical example. Let’s cut the inter‐ start with a value of 0. val into half by evalua ng g(0.5). It’s me to take out your To separate commands, we use the colon : . This can be calculator if you haven’t already, since this ar cle is about typed by the EXE bu on. To type variables we use alpha (one the applica on of calculators in mathema cs. I got ‐0.107 of the 4 bu ons at the top) ‐‐> the bu on with a variable wri en on the upper right corner. This works just like the 5
shi bu on. To input values to variables, we use the single‐ lined arrow —> which can be found in the programme bu on (orange one on 50FH, shi ‐3 for 3650P). So, we start And give an answer to 3 significant figures (telling them a our programme with 9d.p. answer would give out your calculator method, so I wouldn’t advise it). ClrMemory:0—>A:1—>B: This bit determines our star ng points, since we have evalu‐ And… The a er”math” ated that g(0)<0 and g(1)>0. Now into the mathema cs of the algorithm. In the previous Now, to write a loop, we need a label (Lbl) – goto (goto) parts, I have men oned that we could repeat the procedure clause, which can be found in the programme bu on. We as many mes as we like, so that the approxima on can get start our algorithm by defining C as the midpoint of the in‐ as accurate as we like, meaning that the size of the error interval can be as small as we like. So how exactly does this terval: work? ClrMemory:0—>A:1—>B:Lbl 1:(A+B)/2—>C: Take a closer look to the algorithm. We start with an error of And we evaluate g(C), which is C ‐ e‐C. Let’s call the value of g (b‐a), and we halve the error a er each step. So, a er n (C) D. steps, the error interval would be: ClrMemory:0—>A:1—>B:Lbl 1:(A+B)/2—>C:C‐e^(‐C)—>D: Now, the sign of D dictates what we do on our next step. If D>0, then we replace B with C; if D<0, we replace A with C. If D=0, then it doesn’t hurt for either clause to happen. You could write a separate clause for D=0, but I deem it as an unwanted waste of programing space (even the Lbl‐goto formula on of the loop is, but it is for applicability to 3650P. As if I have a choice!), since we are only allows to store 680 characters in 50FH and 360 for 3650P. So, how to put the above condi onal statement into Casio language? We can use the double‐lined arrow => for these simple condi onals. You can find it in the programme bu on. Basically statement P=>ac on Q means that if statement P is sa sfied, then do ac on Q. So:
Now we rearrange this expression and write n as the subject:
No ce that n is posi ve since (b‐a)>ϵ so the term inside the logarithm is larger than 1. Also, if the n that follows from the formula is not an integer, round it up, because we can only perform integral number of procedures, and rounding up gives and error smaller than our requirement, which is okay (rounding down, however, does not work, since we end up with an error larger than our tolerance).
So we see that as long as we are okay with an error larger ClrMemory:0—>A:1—>B:Lbl 1:(A+B)/2—>C: C‐e^(‐C)—>D:D than zero, we can take as many steps we need to produce ≧ 0=>C—>B:D<0=>C—>A: the approxima on. Thus we can say that the method is ro‐ bust, or is guaranteed to converge, since nothing (apart from The strict/not strict inequali es can also be found in the pro‐ compu ng me) can stop us from ge ng as close as we like. gramming bu on. Now here comes the crucial part. How This can be credited to the fact that the error a er n steps accurate do we want the answer to be? Or, how small do we has no dependence on f(x), since the method does not de‐ want to interval to become? As much as the calculator can pend on the nature of f(x) except a requirement on its con ‐ display is the usual answer. For Casio is should be 9 decimal nuity. This is not so for other methods, some of which con‐ places (check my coun ng). So, for simplicity, let’s take a verges much more rapidly i.e. given the same tolerance we ‐10 tolerance of 10 : need less steps to reach our desired accuracy, such as the ClrMemory:0—>A:1—>B:Lbl 1:(A+B)/2—>C: C‐e^(‐C) Newton‐Ralphson method. Consider it a trade‐oﬀ. —>D:D ≧ 0=>C—>B:D<0=>C—>A:B‐A>E‐10=>Goto 1:A
And this is about it for this method and its use on a standard scien fic calculator. Exploring mathema cs with your devic‐ The choice of 10‐10 is also because of the small amount of es does pay you serendipitous dividends – and once your characters it occupies, by the way. Typing Exp – 1 0 will give soul is at one with your calculator, like mine did, good things ‐10 a display of E‐10, which means precisely 10 . The clause will happen, I assure you… basically means that if the diﬀerence between (new) B and (new) A is larger than our tolerance, loop back to label 1. If not, the loop will terminate and the “A” at the back is just a way to display it. Now press “On” to leave the programming window. Press programming bu on, then 1 to run the first programme, which is the slot we have wri en the programme. What val‐ ue of A did you get? I got 0.56714329. Now check g (0.56714329). I got, ‐6.42192 x 10‐10, which is (I hope) rea‐ sonably close to 0. Now, brag to your friend about how you can solve “the un‐ solvable equa on” 6
Addi on and subtrac on are quite trivial:
By Frankie Lam
which is similar to the vector addi on and subtrac on. Solve
Mathema cians in the 16th century encountered similar Mul plica on: seemingly “unsolvable” quadra c, as well as cubic and quar‐ c equa ons. For example, by subs tu ng a = 1, b = 0, and c= 1 into the quadra c formula Division is a bit tricky:
to obtain the solu on, one concludes which simply can’t be represented on the real number line.
In complex numbers, there are also some special opera ons Therefore, in order to resolve the problem, mathema cians unseen in the real number system, such as the complex con‐ addressed a new type of numbers, called complex numbers jugate and the aforemen oned argument and norm. with C as the symbol for its set. These numbers can be ex‐ pressed in the form of a + bi, where and i is called the imagi‐ equa on nary unit which sa sfies the Complex conjugate:
The complex conjugate of a complex number z is usually de‐ noted as or For z = a + bi, its conjugate = a ‐ bi ; we could then see that a real number’s conjugate is essen ally itself.
In other words, oﬀering the solu on needed for the problem stated in the beginning of this ar cle. In the expression men oned above, a is called the real part, denoted as Re(Z); and b is called the as if b = 0 imaginary part, denoted as Im(z). Note that , then the number is regarded as a real number; on the con‐ trary, if a = 0 then the number is referred to as “purely imag‐ inary”.
Proper es of Complex Numbers
For any two complex numbers, z1 and z2, we CANNOT say whether z1 is larger or z2 is larger. It means we cannot direct‐ ly compare two complex numbers, while we could easily dis nguish a “larger” one from two real numbers.
The argument of a complex number z is usually denoted as arg(z). It is defined as the angle measured counterclockwise from the posi ve real axis on the argand diagram to the line connected from the origin the number. It can be calculated by:
The set of complex number has a field structure, implying closure under addi on and mul plica on, obeying the asso‐ cia ve and commuta ve property of addi on and mul plica‐ on, existence of addi ve and mul plica ve iden ty ele‐ ments (0 and 1 respec ve‐ ly), as well as the existence of addi ve and mul plica ve inverses (ie. subtrac on and division). It also comes with a well‐defined “distance from origin” called the “norm” and an “angular distance” called the “argument”, of which we will further discuss later. Opera ons on Complex Numbers From now on, it is useful to think of a complex plane, with the x‐axis being the real part and the y‐axis being the imaginary part. Such la‐ beling on the coordinate system is called an Argand diagram.
Norm: The norm of a complex number z is usually denoted as or a bit like the For z = a + bi, its norm is defined as magnitude of a vector. Note that
Note that One could see a couple of interes ng iden quence of the opera ons listed above.
es as a conse‐
1. The product of a complex number and its complex conju‐ gate is equal to its norm squared. 2.
Exponen als: As complex numbers are also dimensionless, they can also be an exponent of some other complex numbers. The expo‐ nent is widened into the complex number set by the iden ‐ ty: eix = cos x + i sin x, where x is conven onally a real num‐ ber from 0 to 2π . One could verify that this is indeed consistent with the defi‐ ni on of e from the Taylor expansion:
plica on and division process is quite cumbersome as shown before, this form makes the process a lot easier:
That’s it! Apart of simpler calcula on process, it also allows us to view complex mul plica on and division in an illumi‐ na ng way. We see that in mul plica on, the norms of the two complex numbers are mul plied, and their angles of rota on are added up. Division is the opposite process.
Some interes ng problems le for you to speculate:
Subs tu ng ix in x,
1. What is
2. What is log(i)? 3. What is sin i, cos i, tan i ?
5. Are there any algebraic opera ons on complex numbers s.t. another type of numbers outside can be yielded? (Is the set closed?)
By subs tu ng π in x and rearranging the terms, one yields to famous Euler’s formula,
Point of Interest: Quaternions, Octonions, Sedenions, Cayley ‐Dickson Construc on
We also see that:
Application of Logarithm
By Yoyo Lam
Which can be used to deduce various trigonometric iden ‐ es, for example:
Mathema cs is closely linked with Science. Common logrithm, which is commonly used in Science, is a good ex‐ ample. It is applicable in diﬀerent area. Introduc on
If x= by, then y= logb x
Complex numbers represented on a polar coordinate sys‐ tem: Just as when we do coordinate geometry, it is some mes convenient to use the Cartesian system and some mes the polar coordinates; we can also represent the complex num‐ bers on a polar coordinate system. A complex number, in this system, would be represented as er+iθ , where er is the norm of the complex number and θ is the argument of such. The form er+iθ is also called the exponen al form. You might think it is unreasonable to use such nota on due to the irra onality of the natural constant; and the complica‐ on of computa on that the exponent would bring. Howev‐ er, this is exactly the strength of the form. Whereas the mul‐
which means y is the logarithm of x with base b. The loga‐ rithm with base 10 is called the common logarithm. Com‐ mon logarithm is commonly used in science and engineering. Natural logarithm has e as its base, which mainly used in pure mathema cs. The binary logarithm has 2 as its base, which is used in computer science. In this ar cle, we will focus on the applica on of common logarithm in natural science. History A Scotsman, John Napier brought up the method of loga‐ rithm publicly in 1614, in a book called Descrip on of the Wonderful Rule of Logarithms. Mathema cians from diﬀer‐ ent countries helped spread the concept to other places. Therefore, logarithm become well‐known in the world, and con nued to be used nowadays. The inven on of logarithm is convenient to many scien sts as they can perform compu‐ ta ons more easily.
5.9 is moderate earthquake; 6.0‐6.9 is a strong earthquake; 7.0‐7.9 is a major earthquake. If the magnitude on the Rich‐ Logarithm is used in Chemistry to find out pH value of a solu‐ ter scale is 8.0 or above, it is a great earthquake. It may on. pH is the measure of the acidi‐ cause severe damages to buildings and roads, serious casual‐ ty or alkalinity of a solu on. Solu‐ es occur. Earthquakes occur day and hour in diﬀerent plac‐ on with pH lower than 7 is said to es. be acidic, while solu on with pH higher than 7 is said to be alkaline. Solu on with pH 7 is said to be neutral. The equa on to calculate the pH value is Chemistry
–log[ (H+)] (H+) is the concentra on of Hydrogen ions in a solu on. For example, a solu on has 0.001 M of Hydrogen ions. The pH value of the solu on will be –log (0.001)= 3. Solu on with pH 1,2 is a strong acid, such as Hydrochloric Acid ; while pH 5,6 is a weak acid, like Ethanoic Acid . Solu‐ on with pH 8,9 is a weak alkaline, for example ammonia ; while pH 13,14 is a strong alkaline, such as Sodium Hydrox‐ ide. Strong acid and base is compound that completely dis‐ sociate in water, while weak acid and base is compound that partly dissociates in water.
There was an earthquake happened on May 12, 2008, which is known as Sichuan earthquake or Wenchuan earthquake. Logarithm is used in Phys- The magnitude on the Richter Scale of this earthquake meas‐ ics to calculate the sound ured at 8.0, which means it caused a lot of casual es and intensity level. Sound Inten- destruc ons. sity level is used to describe sound level. The equation Astronomy to calculate the sound intensity level is
β = 10 log (I/I0) where I0 is the lowest sound intensity that can be heard by human. I0 is found to be 10-12 W/m2 by scientists. Sound intensity level is measured in decibels (dB).
In Astronomy, people use a scale called apparent magnitude For example, the sound intensity produced by a MP3 player is 10-4 W/m2. The sound intensity level will be 10 log (m) tomeasure brightness of a star seen by observers on the earth. It is measuredon a logarithmic scale. The dimmer the (10-4/10-12)= 80 dB. star, the larger is the magnitude. A bright star may have a If sound intensity level is lower than 10-12 W/m-2, people nega ve apparent magnitude. cannot hear thesound. If sound intensity level is higher than that value, there will be discordant sound. The apparent magnitude can be calculated by the formula Geography
m= C‐2.5 log b
Richter Scale in Geography is a logarithmic scale, which use where C is a constant and b is the brightness of a star. to quan fy the energy released during an earthquake. Rich‐ For example, the sun has an apparent magnitude ‐26.7, full ter Scale was developed by an American scien st called moon ‐12.6, Venus at it brightest ‐4.7. The faintest star ob‐ Charles Francis Richter in 1935. It is widely used in servable with the naked eye has an apparent magnitude 6. the world. The formula is Common logarithm is widely used in science. Logarithm is log E=1.5M+K also applicable in many diﬀerent aspects. Can you find out where M is magnitude of an earthquake and K is a constant. other examples, such as applica on in engineering? When magnitude on the Richter Scale is less than 2.0, it is said to be a Microearthquake, most people cannot felt the earthquake and it is recordedby seismographs. When it is between 2.0 and 3.9, it is a minor earthquake, it rarely caus‐ es damage, but there may be shaking of indoor objects. If magnitude is 4.0‐4.9, it is said to be a light earthquake; 5.0‐
Geometry By Joyce Tam Geometry is a large branch of Mathema cs, including Euclid‐ ean geometry, Diﬀeren al geometry, Topology and geome‐ try and Algebraic geometry. It covers a wide area, from points to planes and from planes to space. In the following, I am not going to discuss detailed mathema cal knowledge of geometry, instead I would mainly explore how it contributes to the development of science. History Geometry has a long history as it first appeared in ancient Egypt and Mesopotamia in the 2nd millennium BC. It was also a major component of Mathema cs in ancient Greece. Classic geometry was focused in compass and straightedge construc ons, which was o en set as mathema cal ques‐ ons in ancient Greece. There were quite a number of fa‐ mous Greek mathema cians who have great achievements in geometry, such as Thales, Pythagoras and Plato. Geometry was then revolu onized by Euclid, a student of Plato, who introduced mathema cal rigor and the axioma c meth‐ od s ll in use today. 1.Riemannian geometry and General rela vity
2. Conic sec on in Astronomy and Physics Conic sec on was studied deeply and named by an ancient Greek mathema ‐ cian, Apollonius of Perga. However, the applicability of conic sec on in Astronomy was discovered by Johannes Kepler about 2000 years later. The Kepler's laws of plan‐ etary mo on states that the orbit of every planet is an el‐ lipse with the Sun at one of the two foci and was proved to be applicable in the solar system by Isaac Newton in 1687. Kepler's laws challenged the long‐accepted geocentric mod‐ els of Aristotle and Ptolemy, ensured the heliocentric theo‐ ry of Nicolaus Copernicus by asser ng that the Earth orbited the Sun, proving that the planets' speeds varied, and using ellip cal orbits rather than circular orbits with epicycles. Furthermore, Edmond Halley, a Bri sh scien st, accurately predicted the moment of which the distance between Hal‐ ley's Comet and the Earth was the shortest by using the New‐ ton’s Law and the concept of conic sec on. This raised peo‐ ple’s interest in having a deep‐ er explora on in conic sec‐ ons. On the other hand, diﬀerent conic sec ons have dis nct and interes ng op cal proper es. We make use of their proper‐ es to improve our lives. For example, scien sts improved the design of astronomical telescopes by using the property of hyperbola. Moreover, the most commonly used solar wa‐ ter heaters consist of parabola minor surfaces, in order to ensure heat energy from the sun is well collected. 3.Knot theory in chemistry Another big step of modern science is the discovery of DNA, which is the gene c materi‐ al of all cells. This is related to geometric to‐ pology. Scien sts unknot DNA by using princi‐ ples of topology so as to analyze the infor‐ ma on of it more easily. Principles of knot theory have helped explain the structure of how enzymes unpack DNA. In addi on, to determine the le handed wind‐ ing of DNA around histones, topological methods have been influen al. Nowadays, DNA is useful in crime inves ga on and parentage tes ng.
As everyone knows, physics has the closest rela on with Mathema cs among diﬀerent branches of natural science. General rela vity is a theory of gravita on that was devel‐ oped by Albert Einstein between 1907 and 1915, sta ng that the observed gravita onal eﬀect between masses results from their warping of space me. The core of it is the Ein‐ stein's equa ons. Einstein was able to formulate the equa‐ on with reference to the concepts of Riemannian geome‐ try, in which the geometric proper es of a space (or a space me) are described by a quan ty called a metric. The metric encodes the informa on needed to compute the fun‐ damental geometric no ons of distance and angle in a curved space (or space me). Einstein's equa ons provide a precise formula on of the rela onship between space me geometry and the proper es of ma er, using the language of mathema cs. General rela vity not only implies the exist‐ ence of black holes, it has also become the basis of cur‐ rent cosmological models of a consistently expanding uni‐ verse. Therefore, the discovery has brought a big step for‐ Geometry requires imaginary and s ll has a lot of room for ward of the development of Astrophysics. discovery. It is appearing everywhere in our everyday life. Apart from science, it can also be found in art work, architec‐ ture in art work and so on. Let’s look for more use of it in our daily lives!
The Swiss architect Le Corbusier, famous for his contribu‐ ons to the modern interna onal style, centered his design philosophy on systems of harmony and propor on. Le Cor‐ busier's faith in the mathema cal order of the universe was By Irene Lam closely bound to the golden ra o and the Fibonacci series, which he described as "rhythms apparent to the eye and A perfect body figure has long been pursued by people re‐ gardless of gender, age and race in the course of history, of clear in their rela ons with one another. And these rhythms this, golden ra o plays an important role in concre zing the are at the very root of human ac vi es. They resound in man by an organic inevitability, the same fine inevitability concept of “perfect”. Nevertheless, golden ra o does not only contribute to the defini on of a perfect body shape, it which causes the tracing out of the Golden Sec on by chil‐ dren, old men, savages and the learned." exists in nature where the arrangement of branches along stems of plants, veins in leaves, to the propor on of chemi‐ Le Corbusier explicitly used the golden ra o in his Modulor cal compounds, and even the branches of animals nerves. In system for the scale of architectural propor on. He saw this our daily live, golden plays an important role in aesthe cs, system as a con nua on of the long tradi on of Vitruvius, pain ngs, design and music, while its contribu on to archi‐ Leonardo da Vinci's "Vitruvian Man", the work of Leon tecture is the most significant as seen from our magnificent Ba sta Alber , and others who used the propor ons of the though mysterious ancient edifice‐‐‐ the Egyp an Pyramids human body to improve the appearance and func on of and the Parthenon. Let us start apprecia ng the beauty of architecture. In addi on to the golden ra o, Le Corbusier nature and the ancient’s intelligence with the guide of math‐ based the system on human measurements, Fibonacci num‐ ema cs. bers, and the double unit. He took sugges on of the golden
The Parthenon The Parthenon's façade as well as elements of its façade and else‐ where are said by some to be circumscribed by golden rectan‐ gles. Other scholars deny that the Greeks had any aesthe c associa‐ on with golden ra o. For example, Midhat J. Gazalé says, "It was not un l Euclid, however, that the golden ra o's mathe‐ ma cal proper es were studied. In the Elements (308 BC) the Greek mathema cian merely regarded that number as an interes ng irra onal number, in connec on with the mid‐ dle and extreme ra os. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahe‐ dron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, con‐ trary to genera ons of mys cs who followed, would soberly treat that number for what it is, without a aching to it other than its factual proper es." And Keith Devlin says, "Certainly, the o repeated asser on that the Parthenon in Athens is based on the golden ra o is not supported by actual meas‐ urements. In fact, the en re story about the Greeks and golden ra o seems to be without founda on. The one thing we know for sure is that Euclid, in his famous textbook Ele‐ ments, wri en around 300 BC, showed how to calculate its value." Near‐contemporary sources like Vitruvius exclusively discuss propor ons that can be expressed in whole numbers, i.e. commensurate as opposed to irra onal propor ons. A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent applica on of the golden ra o throughout the design, according to Bous‐ sora and Mazouz. They found ra os close to the golden ra o in the overall propor on of the plan and in the dimensioning of the prayer space, the court, and the minaret. The authors note, however, that the areas where ra os close to the gold‐ en ra o were found are not part of the original construc on, and theorize that these elements were added in a recon‐ struc on.
ra o in human propor ons to an extreme: he sec oned his model human body's height at the navel with the two sec‐ ons in golden ra o, then subdivided those sec ons in gold‐ en ra o at the knees and throat; he used these golden ra o propor ons in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's applica‐ on. The villa's rectangular ground plan, eleva on, and inner structure closely approximate golden rectangles. Another Swiss architect, Mario Bo a, bases many of his de‐ signs on geometric figures. Several private houses he de‐ signed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ra o is the propor on between the central sec on and the side sec ons of the house. In a recent book, author Jason Elliot speculated that the golden ra o was used by the designers of the Naqsh‐e Jahan Square and the adjacent Lo ollah mosque. Egyp an pyramids In the mid‐nineteenth century, Röber studied various Egyp‐ an pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors iden ‐ fied as the Kepler triangle; many other mathema cal theo‐ ries of the shape of the pyramids have also been explored. One Egyp an pyramid is remarkably close to a "golden pyra‐ mid"—the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclina on of 51° 50' and the π‐based pyramid inclina on of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47') are also quite close. Whether the rela onship to the golden ra o in these pyramids is by design or by accident remains open to specu‐ la on. Several other Egyp an pyramids are very close to the ra onal 3:4:5 shape.
Maths Jokes and Puzzles
Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathema cian and historian, claimed in 1950 that Egyp an mathema cs would not have supported the ability to calculate the slant height By Veronica Lau of the pyramids, or the ra o to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only Q: Why do they never serve beer at a math party? right triangle known to the Egyp ans and they did not know the Pythagorean theorem, nor any way to reason about irra‐ A: Because you can't drink and derive... onals such as π or φ. Q: What happened to the plant in math class? A: It grew square roots. Michael Rice asserts that principal authori es on the history of Egyp an architecture have argued that the Egyp ans were well acquainted with the golden ra o and that it is part of mathema cs of the Pyramids, ci ng Giedon (1957). Histo‐ rians of science have always debated whether the Egyp ans had any such knowledge or not, contending rather that its appearance in an Egyp an building is the result of chance. In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ra o is represented by the ra o of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ. The above two lengths were about 186.4 and 115.2 meters re‐ spec vely. The ra o of these lengths is the golden ra o, ac‐ curate to more digits than either of the original measure‐ ments. Similarly, Howard Vyse, according to Ma la Ghyka, reported the great pyramid height 148.2 m, and half‐base 116.4 m, yielding 1.6189 for the ra o of slant height to half‐ base, again more accurate than the data variability.
Q: Why don't you do arithme c in the jungle? A: Because if you add 4+4 you get ate! Q: Why is 6 afraid of 7? A: Because 7 8 9 Q: How do you know when you've reached your Math Pro‐ fessors voice‐mail? A: The message is "The number you have dialed is imaginary. Please, rotate your phone by 90 degrees and try again..."
Teacher: Why are you doing your mul plica on on the floor? Student: You told me not to use tables. History vs Math Once a math teacher and a history teacher had a fight wheather maths is be er or history........... History teacher: I will call all of Stalins army and kill you. Math teacher: Then I will put all the army in the bracket and mul ply it by zero.
Adolf Zeising, whose main interests were mathema cs and philosophy, found the golden ra o expressed in the arrange‐ ment of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the propor ons of chemical compounds and the geometry of crystals, even to the use of propor on in ar s c endeavors. In these phenomena he saw the golden ra o op‐ era ng as a universal law. In connec on with his scheme for golden‐ra o‐based human body propor ons, Zeising wrote in 1854 of a universal law "in which is contained the ground‐ principle of all forma ve striving for beauty and complete‐ ness in the realms of both nature and art, and which perme‐ ates, as a paramount spiritual ideal, all structures, forms and propor ons, whether cosmic or individual, organic or inor‐ ganic, acous c or op cal; which finds its fullest realiza on, however, in the human form." In 2010, the journal Science reported that the golden ra o is present at the atomic scale in the magne c resonance of spins in cobalt niobate crystals. Since 1991, several researchers have proposed connec ons between the golden ra o and human genome DNA.
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However, some have argued that many of the apparent man‐ ifesta ons of the golden ra o in nature, especially in regard to animal dimensions, are in fact fic ous. 12
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