Raul Chapter 3 Into the Infinite – A Gentle Entry into Advanced Mathematics

by Nelly Kouessan

English Version
Version française

English Version

By : Raul Garcia
IT developer, Mathematician and Yogi
September 2022 | Isabela, Puerto Rico

Into the INFINITE – A Gentle Entry into Advanced Mathematics

HOW MUCH IS INFINITE?

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Note: I know that many of us are afraid of maths! I was once too – believe me.

For a lot of us this is the result of poorly prepared teachers – how can you teach what one barely understand?

My Experience is that one must be a level higher to teach something – that is to say in this particular case at least a BA or Masters is necessary. Your confidence and mastery will then shine and students get it by osmosis.

The ‘LAW OF THREE’:

To study some new subject – 1/first do a review –2/ then a break – then go into it with effort – 3/third study it line by line and do all exercises.

Try to write all major facts or formulas BY MEMORY, and make corrections then or later try again.

So bear with me, make the effort, fire up a few sleeping nerve cells, and get some grasp of real maths!

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One of the wonders of early maths was the discovery that there are many possible infinities – not only in the large direction but also in the small – or infinitesimals.

The one we all know is the Natural numbers or positive integers:

1, 2, 3, 4, …N, N+1,…

A definition is the set N where 1 belongs to it and if n belongs then n+1 also belongs to N, or

N = {1,2,3,….,n, n+1,…}

Now the simplest case of large numbers is the sum from 1 to N, how to get a formula for this?

Say Sum = 1+2+3+ ….N, then by the commutative law Sum= N+…3+2+1, and add the corresponding numbers (second in reverse order) and (asterisk is multiplication):

Sum = 1+2+3+ ….N

Sum = N+ …3+2+1

———————–add then – since the numbers all add to N+1 (why?):

2*Sum = (1+N) + …. (N+1) for n times = (1+N)*N, divide by 2 and get: Sum = N*(N+1)/2.

Let’s try a few examples: 1+2+3 = 6, the formula says for N=3 then N*(N+1)/2 = 3*4/2 = 12/6 = 6 and note that it matches.

There is an alternate more precise proof using ‘maths induction’ ….

It is believed that Gauss used the above method as a child in Germany: as an elementary student in the late 1700s and amazed his teacher by the speed with which he found the sum of the integers from 1 to 100 to be 5,050. Ref: https://www.nctm.org/Publications/TCM-blog/Blog/The-Story-of-Gauss/

Ref: https://letstalkscience.ca/educational-resources/backgrounders/gauss-summation

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Now towards infinity – the sum 1 to N grows slow, kind of close to N*N or N squared.

Note: In computing this is the O notation = O(N*(N+1)/2) = O(N^2 +N) = O(N^2) that means the sum basically grows ‘as fast’ as N*N. This O notation is used widely in the analysis of algorithms.

— Let’s try next with MULTIPLICATION – and now the fun begins:

THE FACTORIAL:

What happens IF we multiply all these numbers: 1*2*3*4 …. *N ?

Well, this is such an important value that there is a name and symbol for it – the FACTORIAL:

N! = 1*2*3*4 …. *N

Examples:

1!=1, 2!=1*2=2, 3!=1*2*3 = 6, and when N=100??

For N=100: N! = 100! =

93,326,215,443,944,100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Wow – that is over my bank account balance!!

(Exercise: why has it so many zeroes at the end? Hint: write down the meaning of 100!)

Here is a calculator: Factorial Calculator n! (calculatorsoup.com)

Exercise: try It with N=1,000? What about 10,000? Must try – you will be amazed!

Note1: In C# and most computer languages you need to use a special type of ‘big numbers’ for this, else you will get an overflow at about N=30. Even excel will replace with zeroes all digits after 30 digits.

The max double size floating point in computers is about: 2.2250738585072014 E – 308) or 300 digits approx. but this is not at full precision after 15 digits.

So there are two concepts here: the size of the number and the precision of the number.

Note2: That is why in blockchain smart contract programming using solidity language, all numbers are integers – since in finance floating points can lose precision with size…and you can lose millions!

WHY N-FACTORIAL GROWS SO FAST?

Observe: (N+1)! = 1*2*3…*(N+1) = (1*2*3…*N) * (N+1) = N! * (N+1)

So the factorial of N+1 is the factorial of N times N+1. Example: 100! = 99! * 100 = 5,050.

Therefore, the value of each successive term of the series is N-times the previous factorial, and as N gets bigger this factor grows correspondingly then it Explodes!

These two formulas for sum and multiplication of integers are FUNDAMENTAL in many, many fields:

computing – analysis of algorithms,
quant finance,
statistics,
probability,
economics,
marketing,
physics,
chemistry, biology etc.

That is why all such fields like ‘biology’ now have specialties like ‘computational biology’ … which consist of a merger of advanced maths and computing applied to that field quantitative problems.

Long ago I took an advanced course using Knuth analysis of algorithms book – and we were in factorials etc up to our necks …Ref: Design and Analysis of Algorithm (tutorialspoint.com)

All the above was leading to this: as N! becomes large also 1/N! becomes very small – awesome!

It becomes so small that YOU CAN add all (infinite) ‘small’ numbers:

1 + 1/1! + ½! + 1/3! + 1/4! …. + 1/N! …. And the sum is a FINITE NUMBER!

How comes?

An infinite set of numbers can add up to a finite number! YES – because the N! is so large and so quick that 1/N! will become VERY small very quick too, so adding 1/N! for say N>30 is like adding a zero: 1/30! = 3.76998763E-33.

If your bank account had this balance – try to even get a penny!

Like PI it is a fundamental part of advanced maths and science, called ‘e’ in honor or Euler.

e = 2.71828182845904523536028747135266249775724709369995… or just 2.718 for friends…

Ref: https://en.wikipedia.org/wiki/E_(mathematical_constant)

Leonhard Euler

Leonhard Euler was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory and complex analysis.

THE HARMONIC SERIES:

Now let’s compare when we add just the reciprocals of n- called the HARMONIC SERIES – all 1/n’s:

1 + ½ + 1/3 + ¼ + 1/5 + …. 1/N +…

Since the ‘1/N’ gets small too then is the sum finite? NO – it does not – it is an infinite result also called unbounded in maths or a divergent series.

That is because these numbers do NOT get small fast enough and each one just keeps the series growing too fast – if we add enough numbers – like 2 to the N each time.

The amazing details are in: http://dictionary.sensagent.com/Harmonic_series_(mathsematics)/en-en/

Next let’s jump a bit and consider when we raise ‘e’ to any power ‘x’ – or e^X also denoted by Exp(x):

You cannot imagine how essential this formula is in all fields of science or maths!

Like radioactivity and bacterial growth! It is truly all over the place in science, engineering, applied maths and statistics…

A simple example : it is almost exactly the formula for RADIOACTIVE decay (half live) – so important now due to Tchernobyl, A-bombs, reactors and the current danger in Ukraine with its Zaporizhzhia reactors – the largest in Europe – turned into nuclear blackmail by Russia: https://en.wikipedia.org/wiki/Zaporizhzhia_Nuclear_Power_Plant

Zaporizhzhia Nuclear Plant in Ukraine – now under Russian control

Radioactive decay law: R = N*Exp(-λt) where N and lambda are constants and it says it is Exponential in time, the – t.

Ref: https://www.nuclear-power.com/nuclear-power/reactor-physics/atomic-nuclear-physics/radioactive-decay/radioactive-decay-law/radioactive-decay-equation-formula/

Recall that an Exponent, like 2 here means e² = e*e = approx. = (2.718)*(2.718) = 7.3890 …

Next we state without proof (it uses Taylor series and the fact that Exp(0)=1):

Using again the Big-O notation – saying the remainder of the sum is more than x to the fifth power.

REF: https://www.wolframalpha.com/input/?i=Exp%28x%29

Expanded the exact formula for Exp(x) is:

Exp(x) = 1 + x¹/1! + x²/2! + x³/3! +… + x^N/N! + ….

Where we recall that N! = 1*2*3…*N and x^N = x*x*x…*x N-times. Note 0!=1 and x⁰=1

Now the natural question: how can this converge? How can the above sum become a number? Asterisk * denotes multiplication.

Consider a term: x^N/N! = x*x*x…*x N-times / = 1*2*3…*N = separate the factors to get:

x/1 * x/2 * x/3 …. * x/N

Next since x is a real number (can be complex too) and N is an integer, eventually for ANY x, there are endless integers greater than x ([x] is denoted as the integer part of x). Then x/N will become smaller and smaller since N can be FAR greater than x and then the sum above:

Exp(x) = 1 + x¹/1! + x²/2! + x³/3! +… + x^N/N! + ….

Will be adding smaller numbers as N exceeds x, then almost adding zeroes!

Next we give an amazing derivation of why in Calculus the formula is correct:

d/dx( Exp(x) ) = Exp(x)

That is that the derivative (the slope of the tangent at x) of Exp(x) = Exp(x).

This means that the function y = Exp(x) is the IDENTITY FUNCTION (LIKE 1 in multiplication or zero in addition) of the DERIVATIVE operation…

REF: https://en.wikipedia.org/wiki/Exponential_function

If the word ‘derivative’ or ‘calculus’ makes you cold – even with all the global warming – this will set you free – an excellent smooth intro: https://www.mathssisfun.com/calculus/derivatives-introduction.html

Next we will use the basic formulas of differential Calculus: d( x^N) / dx = N * x^(N-1)

Also we need: d (constant ) /dx =0 (slope is zero at all points) and

d( c * f(x) ) = c* ( f(x) ) where c is a constant – then it can go out of the derivative, a special case

is division: d( f(x) / c ) = f(x) /c where c is not zero. It is a sin to divide by zero …

REF: https://tutorial.maths.lamar.edu/classes/calcI/DiffFormulas.aspx

In human-speak: the derivative of X to the N is N times x to the N-1 power ! wow…

Example: d( x⁵) / dx = 5 * x^(5-1) = 5 * x⁴

Now let’s derive for Exp(x) using the fact that the derivative of a convergent sum is the sum of the derivatives of EACH term (* means multiplication):

d ( Exp(x) ) /dx =

d ( 1 + x¹/1! + x²/2! + x³/3! +… + x^N/N! + ….) /dx =

d(1)/dx + d(x¹/1!)/dx + d(x²/2!)/dx +d( x³/3!)/dx +… +d( x^N/N! ) + ….

Using the above formulas, since N! is constant (does not depend on x):

d ( Exp(x) ) /dx =

d(1)/dx + d(x¹/1!)/dx + d(x²/2!)/dx +d( x³/3!)/dx +… +d( x^N/N! ) + …. =

d(1)/dx + (1/1!) * d(x¹)/dx + (1/2!)* d(x²)/dx + (1/3!) * d( x³)/dx +… + (1/N!) * d( x^N ) + …. =

Now let’s look at the general term for simplicity:

+ (1/N!) * d( x^N ) + =

+ (1/N!) * N * x^(N-1) + = (using calculus formula)

+ (1/(123…N-1N) N * x^(N-1) + =

Now the N factor cancels with the 1/N factor and we have 1*2*3…(N-1) and that is precisely (N-1)! <– the key idea

+ (1/(123…N-1) * x^(N-1) + =

+ (1/(N-1)! * x^(N-1) + =

+ * x^(N-1) /(N-1)! + =

AND THIS IS THE PREVIOUS TERM FOR THE SUM!

THEN WE HAVE THE SAME SUM AS BEFORE – SO THEN:

d( Exp(x) )/dx = Exp(x)

The key here is that:

N!/N = (123…N-1N) / N = 123…*N-1 = (N-1)!

Since the last N/N is 1 or it cancels out.

From this we will understand NEXT the amazing equations, were:

e is Euler’s number, the base of natural logarithms, approx. 2.71828
i is the imaginary unit, which by definition satisfies i² = −1, and
π is pi, the ratio of the circumference of a circle to its diameter, approx. 3.1415

Coming soon :

In the next lessons we will cover what amazing things happen when we raise

Exp(x) to a complex number…. And what are complex numbers?

We will also reveal the amazing relation of Exp(x) to trigonometry sine and cosines.

And dealing with infinities: the proof that the number of fractions is the same as the number of integers (ALEPH ZERO) and some unsolved problems of maths!

Version française

Par : Raul Garcia
Développeur informatique, mathématicien et yogi
Septembre 2022 | Isabela, Puerto Rico

Vers l’INFINI – Une entrée en douceur dans les mathématiques avancées

QUANTIFIER L’INFINI ?

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Note : Je sais que beaucoup d’entre nous ont peur des maths ! Je l’ai moi aussi été à une époque!

Cet état de fait est à mon sens en partie la conséquence d’enseignants mal préparés – comment enseigner ce que l’on ne maîtrise pas soi-même ?

D’expérience, il faut un niveau d’études supérieur pour enseigner une matière – c’est-à-dire que pour les enseigner, il faut au moins une licence ou une maîtrise en mathématiques. Votre confiance et votre maîtrise s’expriment librement et les élèves reçoivent votre perspective par osmose.

La « LOI DES TROIS » :

Pour étudier une nouvelle matière 1/ La réviser 2/ faire une pause nécessaire avant de s’y plonger avec effort – 3/ puis l’étudier point par point et s’y exercer.

Tentez d’écrire les principaux faits / formules DE MÉMOIRE, faites les corrections à ce moment-là ou essayez à nouveau plus tard.

Soyez indulgents avec moi et essayons ensemble de réveiller quelques cellules nerveuses endormies : allons à la rencontre des véritables mathématiques !

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L’une des merveilles des premières mathématiques a été la découverte qu’il existe de nombreux infinis possibles – vers la grandeur ET de la petitesse (infinitésimaux).

Nous connaissons tous est le nombre naturel ou les nombres entiers positifs :