Two Lesson Samples Mathematics and Science (Physics)mirnova.org/papers-public/sample-lessons-math-physic… · · 2017-11-11Two Lesson Samples – Mathematics and Science (Physics)

Two Lesson Samples – Mathematics and Science (Physics) Intended Students: Middle School, grades 7-8, ages 12-13, “IB” Program [1] Mathematics Explorations of Interesting Numbers and Sequences [2] Physics Waves, Signals, Electromagnetism and some interesting Phenomena Martin Dudziak (Мартин Дудзяк) Zelenograd and Moscow, Russia ~~~~ Ann Arbor and Traverse City, Michigan, USA

Copyright © 2016 Martin Joseph Dudziak, PhD, All Rights Reserved

[1] Mathematics Explorations of Interesting Numbers and Sequences How many kinds of NUMBER are there? We know about NATURAL Numbers – you can count with them – and historically, these are the first numbers people used. (0, 1, 2, 3…) [but some would exclude 0]


We know about INTEGERS ---- these are the “whole numbers” and we can do anything with Negative numbers as well as Positive numbers

We know about RATIONAL Numbers (they can be a RATIO between two integers) (any number expressed as p/q, p and q being integers) (e.g., 1/3, 2/4, 7/9, 9/7, etc.)


Then there are REAL Numbers which include Rationals + Irrationals What are Irrational Numbers? Here are three examples – they do not “end” in any definite way (e.g., with a “0” or with repeating digits (e.g., 1/3 = decimal 0.33333…)


But there is nothing “irrational” about Irrational Numbers!

Irrational Numbers are very important in science, technology, and even the simplest things in everyday life!


How are some of these numbers special in Everyday Life? Π = fundamental to the CIRCLE and to ANGLES and ROTATION Π ● d --or-- 2 Π r = circumference 1 rad ≈ 57.296° Π rad = 180° Π r2 = area of circle (4/3) Π r3 = volume of sphere

e = natural logarithm (x = log(n) For example, ln(7.5) is 2.0149..., because e2.0149... = 7.5)

√2 = diagonal of a square (do you know about the Pythagorean Theorem?)

Φ = “Golden Ratio” = approx. 1.618033988749894848204586834.


??? “All square roots of integers not divisible by 2 are irrationals” “The square roots of all natural numbers which are not perfect squares are irrational” “Every even integer greater than 2 can be expressed as the sum of two primes.”

What do you think ???

Can some Numbers be connected with How We Think about Beauty in Art and Proportion?

Now this may at first look like just (yada yada, yawn, yawn) some interesting arithmetic…


But think about Art, and Architecture, and the way Nature has designed things!



Leonardo da Vinci certainly understood the principles of

Golden Ratio!

All together, these make up the REAL Numbers.


That’s Not All – but I think we’ll stop here for now, because… if we start talking about COMPLEX Numbers, things will get Very Complex Very Fast! TODAY we will explore a very special class of numbers and these are a SUBSET of the Positive Integers. (We can talk about SETS later, but not today.)

PRIME Numbers are not only “interesting” in mathematics, but we will see that they are extremely important in everyday life including the ways we protect our information with Encryption, with CYBERSECURITY. A PRIME Number is a positive integer that can be divided only by 1 and by itself. Not by any other number.

What are some Prime Numbers?


1 ? --- No, because it is excluded, being a “special case” number. 2 ? 3 ? 4 ? (Why not?) 5 ? 6 ? (Why not?) 7 ? Here’s the first prime numbers ≤ 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 What’s so Special about these numbers?


https://en.wikipedia.org/wiki/2




https://en.wikipedia.org/wiki/11_%28number%29





















What is the simplest way to determine if a number is a PRIME Number? Here are the first Prime Numbers up to 1000: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997



























Here’s one way that the ancient Greek mathematician Erathogenes developed:

How does it work? Can you tell why it is easy and why it is limited for larger numbers to explore?


But Primes are not just “interesting numbers” and interesting only for mathematicians!

Encryption --- Cryptography --- Cybersecurity The whole world of personal, financial, and National Security depends upon Very Intelligent Ways to Keep Data Secure and Secret Prime Numbers are very important these days in some of the most common and most secure ways of encrypting data --- • Passwords • Signatures • Messages • Whole Documents We will not explore this right now, but you can read about “RSA” and “Public Key Cryptography” and find out more on your own!


Here’s the standard way: TRIAL by DIVISION Is n prime? Divide n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if n=ab is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most sqrt {n} . For example, for n=37 , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to n is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Finally, some Fun exploring SEQUENCES of Numbers.

Leonardo Bonacci of Pisa, known as Fibonacci, lived @ 1175 – 1250. He discovered some interesting relationships between numbers: See if you can find what is interesting here: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … What comes after 144? Why? Fn = Fn-1 + Fn-2

With the “seed values” F0 = 0, F1 = 1 But these sequences show up a lot in Nature!


Natural Law? Or a very interesting Tendency and Disposition? And if so, why?!?!


And what do we have here, also, with FIBONACCI Numbers? The Golden Ration of 1.618…


SPECIAL PROJECT for “Extra Credit” Homework Explore Famous Russian IKON paintings See if you can find the Golden Ratio or some other Special Proportions. Do you find some mathematical ratios that are common in Ikons? Do you find a special proportion that evokes Special Spiritual Qualities, Moods, Feelings?


SPECIAL PROJECT for “Extra Credit” Homework Explore Cryptography and the Use of Prime Numbers. Why are Prime Numbers so important for hiding the “key” used to encrypt and decrypt messages?


SPECIAL PROJECT for “Extra Credit” And TONS OF EXTRA CASH $$$$ Prove or Disprove “Goldbach’s Conjecture” ---- Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been shown to hold for all integers less than 4 × 1018 but no one has been able to produce a definitive proof in over 270 years!

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII)[6] in which he proposed the following conjecture: Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units. Euler replied in a letter dated 30 June 1742: "Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." ("That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.")


[2] Physics Waves, Signals, Electromagnetism and some interesting Phenomena


What is in common between

Visible Light Music Infrared Sonar Ultraviolet Radio and television signals Microwaves Radar X-Rays Gamma Rays Cosmic Rays


All of these involve WAVE phenomena. All of these are forms of ENERGY. All of these involve either Photons or Other Particles. Those which we call ELECTROMAGNETICS involve PHOTONS and there are some basic relationships that we know about these, whether they are Radio or Television signals,

Or Visible Light

Or Microwave

Or X-Rays


When we talk about WAVE phenomena, with sound, with light, with other electromagnetic energy, we are talking about Oscillations that behave (typically) as SINE waves – so Trigonometry is and will be very important after all:


FREQUENCY is what we usually hear about with respect to any waves and it is a measure of how many CYCLES there are (usually measures “per second”) It is important to know some of the important terms. Can you see the relationships between the different terms? Try to learn and remember your “nanos” and “petas” !!


Three Important Properties that can be measured: frequency f, wavelength λ, and photon energy E

c = 299792458 m/s is the speed of light in a vacuum h = 6.62606896(33)×10−34 J·s = 4.13566733(10)×10−15 eV·s is Planck's constant


https://en.wikipedia.org/wiki/Speed_of_light

https://en.wikipedia.org/wiki/Planck%27s_constant

https://en.wikipedia.org/wiki/Planck%27s_constant

There are Electrical and Magnetic Fields and they are in special relationships with each other, at right-angles, as the wave travels…


Some of the most interesting Waves, whether they are Electromagnetic or Acoustic or Hydrodynamic, or in other media, have very unique and valuable properties. Most waves normally dissipate, sometimes quickly. Think about throwing pebbles or rocks into a pond:


Or how about the waves that come in from the Ocean or a big lake like Baikal:


Even light or radio or radar waves - all of those dissipate and pretty quickly. Do you know why? But some waves are NON-Dissipative, sometimes for a surprisingly long time! They can last and last and last…

These are known as SOLITONS


Solitons are not only interesting but they are very important in all aspects of “STEM” and this includes our Everyday Life and how we communicate – and it even includes our Life Itself and how our proteins and DNA behave inside our cells and how our neurons send signals! So let’s take a short look at Solitons – first, how they were discovered:


John Scott Russell observed solitons from the movement of barges in the Union canal in Scotland in 1834. Later he noted from his home-built “wave tank” experiments: • The waves are stable

• They can travel over very large distances (normal waves would tend to

either flatten out, or steepen and topple over)

• The speed depends on the size of the wave

• The width depends on the depth of water

• Unlike normal waves they will never merge—so a small wave is overtaken by a large one, rather than the two combining

• If a wave is too big for the depth of water, it splits into two, one big and one small


Now, let’s watch a short video, and you can practice both your physics and your English. https://www.youtube.com/watch?v=w2s2fZr8sqQ


https://www.youtube.com/watch?v=w2s2fZr8sqQ

Now, you have just watched a video because of the wonders of modern technology, using high-speed broadband internet, and Wi-Fi transmission, and high-performance computing in your desktop computer or laptop or tablet or phone. All of these work because of (among other technologies) high-speed GigaHertz fiber-optics – and that depends upon sending signals as solitons through optical and copper cables, and without that technology we would be unable surely to have the speeds and bandwidths of today.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ But we have a famous Russian physicist to thank for opening the doors to thinking about solitons in other ways, namely, Inside our bodies, inside our brains, inside our cells For the ways that energy can be transferred and signals can be propagated along the chains of proteins and nucleic acids.


Alexander Sergeevich Davydov


FOOD FOR THOUGHT! [1] Simpler “snack” --- What is the difference between the “rays” of “cosmic rays” and the “rays” of X-rays and Gamma rays? [2] A bigger “meal” --- How might the future of computing be changed through new understanding about waves and signals in macromolecules like Protein and DNA? Is biological life the most sophisticated computer in the universe? What can Biology and Water and other everyday phenomena tell us about the Deep Cosmos or the Deep-Down Quantum World?
