-By Yagnamurthi Pooja

### Introduction

What if I say people of 21 century are still trying really hard to figure out the unsolved riddle of “Mathematical constants”.

In 2010 a Japanese engineer and American Computer enthusiastically broke the record by calculating nearly 5 trillion pi digits.

It turns out there are so many constants that are significant and interesting which makes problem understanding and solving much easier.

The concept of having a definite value irrespective of its parameters is not only fascinating but also a little puzzling, well in this article we are going to talk about the five most interesting Mathematical constants that legitimately changed and solved the most mysteries in the math world

So, let’s explore these magical mathematical values that changed the perspective of many great mathematicians.

**What is the exact definition of a mathematical constant?**

Well, the answer to your question lies in the question itself, A mathematical “constant” is a constant value or a fixed value that is denoted by a symbol or named after a mathematician, it is widely used to solve multiple mathematical problems.

Although there are so many mathematical constants we picked the well known and mostly used constants that are worth knowing and appreciating. keep reading to know all the constants and surprising facts.

**1.Archimedes’ constant, or “Pi**

- Definition:

The ratio of circles circumcircle to its diameter is defined as “pi”. pi is denoted by a greek symbol “π”

what makes pi so interesting is that it is an irrational number the decimal value doesn’t settle and never goes on with a similar pattern so there is no exact value of pi however 3.14159 is the approximate value of it.it also fractionally expressed as 22/7 which is said to be approximate to pi.

- Significance of the constant:

If you think pi is mostly seen in geometry and trigonometry and physics where you study concepts related to circles then you are probably wrong!!

That’s the amazing thing about pi, even it is more related to the circles it goes beyond the limit by making an appearance where it is not expected

you can find pi in so many other concepts which doesn’t have much to do with circles like the sum of the infinite series, the theory of statistics, collection of random whole numbers, it also has its significance in physics as you can see pi in many concepts that tell about waves, ripples of light and sound.it is also made its space in the equation that defines the state of the universe named as Heisenberg’s uncertainty principle.

- A little History:

let’s start with a little know fact, pi is also called Archimedes constant as the value 3.14 is first calculated by Archimedes in the 18th century

In 1706, British mathematician Athanasian William johns gave the greek name and symbol for “pi”. However, it is established by a Swiss mathematician Leonhard Euler.

- Intriguing fact :

As we all know the number of the pi goes up to infinity but somehow it is shocking that the combination of 123456 never appeared even a single time in the first million digits combinations. hence the combination 123456 is said to be a unique combination.

**2.Euler’s number**

- Definition:

Euler’s number is a mathematical constant that plays a significant role in both mathematics and physics.

The value of e=2.71828

Irrespective of other constants” e” has multiple definitions.

some of them are,

- The sum of infinite series 1+1/1!+1/2!+1/3!+1/4!+…. is e.
- if e is mentioned as the base of a logarithm then that particular logarithm is said to be a natural logarithm and it is written as ln(x).

NOTE: The log of e is equal to 1 and the log of 1 is equal to zero

ln(e)=1 and ln(1)=0

- Significance of the constant

Euler’s number is popularly known as the exponential king as it plays a vital role in many mathematical concepts like logarithms, it is also seen in some physics-related concepts too.

you can see Euler’s numbers in algebra, calculus, component interest, probability theory, problems related to exponential growth, standard normal distribution, and various mathematical problems. It is also mentioned in many physics-related topics like fluid flow dynamics, light waves, sound waves, and quantum waves.

- A little history

The irrational number e is named after the Swiss mathematician Leonhard Euler in the 18th century. but, the number was discovered by mathematician Jacob Bernoulli while studying compound interest.

The Euler constant is also known as Napier’s Constant as it is used in the work of John Napier the inventor of logarithms in 1614.

- Intriguing fact

Leonhard Euler in 1736 first used e to represent the number whose hyperbolic algorithm is equal to 1.

**3.Pythagoras’ constant**

- Definition

The square root of 2 is called the Pythagoras constant and it is written as √2.this positive algebraic number(√2)is represented as the hypotenuse of a right-angle triangle where the other sides are considered as unit 1 which further leads us to the Pythagoras theorem. the approximate value of √2=1.14142… the digits goes on, The same case as you have seen pi and e.

- Significance of the constant

It is said to be the first irrational number ever discovered. The square root of two is occasionally called as Pythagoras constant or Pythagoras number.

- A little history

Hippasus, the one who discovered this very first irrational number was brutally murdered and the crime is finding √2.

It all started in the school of Pythagoreans they believed that everything is made of whole numbers and the concept of irrational numbers doesn’t make sense to them that led to the horrible act of killing one of the greatest mathematicians.

- Intriguing facts

On June 28, 2016, Ron Watkins broke his own record by calculated 10 trillion digits for √2

(On April 3rd of 2016, he calculated 5 trillion digits for √2)

**4.The golden ratio**

- Definition

The golden ratio is found if a line divides into two unequal parts with one longer than the other the long part divided by the short part(a/b) is equal to the sum of both parts divided by the long part(a+b/a)

which is,

a=longest part

b=short part

then

a/b=a+b/a=golden ratio

The value of the constant number is 1.6180…..(goes on) and is denoted by the Greek symbol “*φ” and pronounced as “phi”*

- Significance of the constant

This irrational constant is used in many geometric calculations and the properties of the golden ratio were explored by many great mathematicians like Luca Pacioli.

The golden ratio is used by many profound mathematicians like Abu Kamil, Fibonacci.

Leonardo da Vinci once called this ratio “*sectio aurea” which means golden section.*

- A little history

Michael Maestlin gave the first approximate decimal value for the golden ratio.

In 1910 Mark Barr a Mathematician started using the symbol “φ” to represent the golden ratio.

Many ancient greek Mathematicians studied this golden rule including Euclid and Pythagoras. but further, it was studied by Leonardo of Pisa

- Intriguing facts

Many artists and architects believe that constructions that are built by using the golden ratio look amazing and they also consider the golden rule before starting their work.

**5.Apéry’s constant**

- Definition

Apéry’s constant is defined as the sum of the reciprocal of the positive cube at the intersection of number theory and special functions.

The value of this constant is *ζ*(3) = 1.20205…(goes on)

ζ is the Riemann zeta function

- Significance of the constant

The constant *ζ*(3) is proved as an irrational number as per the result of Apéry’s theorem.

*ζ*(3) is seen in many concepts like the second-and third-order terms of the electron’s gyromagnetic ratio computed using quantum electrodynamics.

- A little history

French mathematician Roger Apery discovered the value of *ζ*(3) hence it is named after him.

Bernhard Riemann somewhere in the middle of the 19th century mentioned *ζ*(3), specific value for 3 in the*ζ*(3) The Riemann Zeta Function is said to be the extension of the function of a series.

- Intriguing facts

The Riemann Zeta Function is one example of an infinite series,*ζ*(3) is derived from the infinite series.

#### Conclusion

Irrespective of above discussed constants there are many mathematical curiosities that might leave you speechless such as Conway’s constant λ,Khinchin’s constant K,The Glaisher–Kinkelin constant A,Chaitin’s constant Ω,The Euler–Mascheroni constant γ,The Feigenbaum constants α and δ,The imaginary unit i(which is more or less a constant)

Being irrational constants may never really have an end but our article does, there are many mysterious constants with amazing derivations and backstories as we picked the top 5 that are more intriguing and astonishing.

#### Reference:

https://mathworld.wolfram.com/Pi.html

http://www.math.com/tables/constants/pi.htm

https://www.toppr.com/guides/maths/value-of-e/

https://www.nature.com/articles/s41567-019-0655-9

https://oeis.org/wiki/Pythagoras%27_constant

https://mathworld.wolfram.com/PythagorassConstant.html

https://www.britannica.com/science/golden-ratio

https://www.livescience.com/37704-phi-golden-ratio.html

https://www.mathematics.pitt.edu/sites/default/files/abstracts/Slides_1.pdf