Transcendental Numbers
History of Numbers
The idea of counting things started in the late ages of hunter-gather societies. They used the tally system to make records of days, lunar cycles, livestock, population, etc. After the agricultural revolution took over scarcity of food was never a problem. As cultivation became easy, people traded grains for other necessities like meat and manpower.
Tally system failed for recording huge numbers. They had to adapt to a new number system for counting which led them to create all kinds of number systems ranging from base 60 (By Babylonians) to base 10 Roman numerals. None of those number systems include zero because the idea of 5 sheep or 1 Meter of cloth made sense but Zero sheep or zero Meter of cloth doesn’t because it is very abstract. So they never included it but as math became more prominent it required a number that has no value. Zero was first adapted into Hindu numerals and later it was integrated into the Hindu-Arabic numerals which we use today.
Types of Numbers
Numbers are classified into many different types like Natural, Whole, Integers, Rational, Irrational, etc but all these numbers are subsets of a bigger set called Complex numbers. Fundamentally numbers fall into two categories:
Algebraic
Numbers that are the roots of any n-degree polynomial with rational coefficients are algebraic. In other words, we can describe them as numbers that can result in zero by performing any arithmetic operations on them with integers (except Zero).
For example, 10/2 can result 0 by multiplying 2 and subtracting 10.
(10/2) * 2 – 10 = 0
Similarly, √2 -1 can result 0 by
((√2 -1)2 -3)2 -8 =0
If we remove our number from the equation by replacing it with x gives:
((x)2 – 3)2 – 8 = 0 => x4 -2.x2 .3 + 9 -8 = 0 => x4 -6.x2 + 1 = 0 is a polynomial of degree 4.
Our number (√2 -1) is the root of the above polynomial.
Transcendental
Numbers which are not the roots of any n-degree polynomial with rational coefficients are Transcendental. The complex plane is largely filled with these numbers. There are infinitely many Transcendental numbers compared to Algebraic.
If you would have to point at the complex plane by closing your eyes, you’d most probably point at a Transcendental number.
The most known Transcendental numbers are π and e.
What is Pi(π)?
Pi(π) is the simplest possible ratio of the simplest possible shape. It is the ratio of the circumference of a circle to its diameter. Pi is an irrational number with an infinite decimal expansion and it was named after periphery(Circumference). It was known for almost 4000 years. Ancient Babylonians tried finding the area of a circle by approximating Pi to 3. When ancient Egyptians started making wheels they required a constant to make rims and spokes of perfect size. So they tried finding its ratio and rounded it to 3.1605. Archimedes and his group of classical Geeks invented the Method of Exhaustion. In this method a circle is inscribed between two polygons and Pi is approximated by calculating the perimeters of those polygons.
Check out the video below to know more about Method of Exhaustion.
Calculate Pi by yourself
Pi using Binomial Expansion
Binomial Theorem
(x + y)n = nΣr=0 nCr xn – r · yr where n ∈ N,x,y,∈ R, nCr = n!/(n-r)!.r!
It’s 1665 and Issac Newton had already invented calculus and working on the Binomial expansion of numbers and he started experimenting with it. Originally it only allowed positive integers as the power but he later tried it with negative integers and fractions. When he came across the unit circle equation (x2 + y2 = 1) he tried solving it with binomial expansion.
y2 = 1 – x2
y = (1 – x2)1/2
According to Binomial Theorem,
(1 – x2)1/2= [1 – 1/2.x2 – 1/8.x4 – 1/16.x6 – 5/128.x8 – ……….]
Integrating on both sides from 0 to 1 gives the area of the quarter circle.
∫_0^1(1 - x^2)^(1/2) = ∫_0^1[1 – 1/2.x^2 – 1/8.x^4 – 1/16.x^6 – 5/128.x^8 – ……….]
∫_0^1(1 - x^2)^(1/2) = π/4
π/4 = ∫_0^1[1 – 1/2.x^2 – 1/8.x^4 – 1/16.x^6 – 5/128.x^8 – ……….]
π/4 = [1 – 1/2.x^2 – 1/8.x^4 – 1/16.x^6 – 5/128.x^8 – ……….]_0^1
π = 4[x – 1/2.x3/3 – 1/8.x5/5 – 1/16.x7/7 – 5/128.x9/9 – ……….]
By Putting x=1,
π = 4[1 – 1/6 – 1/40 – 1/112 – 5/1152 – ……….]
By this method it was very easy and quick to calculate Pi.
Since the digital revolution, we calculated Pi for knowing how powerful one’s computer is. By our modern computers, we calculated more than 31 Trillion digits of Pi.
Pi is approximately 3.141592653589793238…..
Euler’s Number(e)
e was first came into light when swiss mathematician Jacob Bernoulli was working on compound interest. He did a thought experiment which led to the discovery of e. Here it goes…
The formula for compound interest is P(1+r)n. where P is the principal amount,
r is the rate of interest, n is the time period.
Occurrence of e in Compound Inerest
Assume you have 1€ in your bank account and your bank is soo generous that it gives you 100% interest every year. This means your 1€ doubles every year. It yields 2€ in the first year 4€ in the second year 8€ in the third year and it goes on. It is an exponential function of 2 (2x).
Bernoulli thought “What if the bank gave an interest of 50% every 6 months?”. Let’s try it, if we had 1€ in 6 months it becomes 1.5€(1 already existing + 0.5 interest). After a year the balance becomes 2.25€(1.5 already existing + 0.75 interest).
It is a profit of 0.25€ compared to the first method. Then he thought “What if it was 1/12% every 12 months?”. This model yields 2.613€ in a year. It is an improvement from the previous model. Then he thought “What if it was 1/365% every 365 days?”. It results 2.71€. What if we can get interest for every second? For every millisecond? What if it was for every instant? Spoilers! the number converges.
Later Leonhard Euler calculated it to approximately 2.71828182…. He found that e is irrational and invented other ways to calculate it. Some of which are…
e = 2 + 1/(1+ 1/(2 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + 1/(1 + 1/(6 + 1/(1 + …))))))) ))
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + 1/8! + …
Usage of e
The importance of e is that it is the natural log of all exponential functions. Derivative of an exponential function has a proportionality constant. e is the only function that has the proportionality constant of 1. Which means applying derivatives to e results e. If we would have to plot a graph of ex, it results in a curve where for any given value of x the height equals to ex. The amazing thing is that the area under the curve and slope also equals ex. We use ln which is loge as the standard for the natural log of all exponential functions.
Euler’s Identity
Even though Pi and e come from two different parts of mathematics, they strangely have a relationship. The Euler’s identity states that e to the power of i (√(−1)) times Pi results in -1(eiπ = -1). It is very contradictory that two different irrational numbers somehow result in an integer. Let’s validate the identity.
Consider eix,
eix = 1 + (i.x)/1! + (i.x)2/2! + (i.x)3/3! + (i.x)4/4! + (i.x)5/5! + (i.x)6/6! + (i.x)7/7! + (i.x)8/8! +……
eix = 1 + i.x/1! – x2/2! – i.x3/3! + x4/4! + i.x5/5! – x6/6! – i.x7/7! + x8/8! +……
eix = 1 – x2/2! + x4/4! – x6/6! + x8/8! + i.x/1! – i.x3/3! + i.x5/5! – i.x7/7! +……
eix = 1 – x2/2! + x4/4! – x6/6! + x8/8! …… + i(x/1! – x3/3! + x5/5! – x7/7! +……)
cos(x) = 1 – x2/2! + x4/4! – x6/6! + x8/8! ……
i.sin(x) = i(x/1! – x3/3! + x5/5! – x7/7! +……)
eix = cos(x) + i.sin(x)
Replacing x with π gives,
eiπ= cos(π) + i.sin(π)
eiπ = -1
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