Skip to main content

Important Mathematical Constants!

Important Mathematical Constants

Mathematical constants are those numbers that are special and interesting because they come up in the various fields of mathematics like geometry, calculus etc. These mathematical constants are usually named after the person who discovered it and they are represented by a symbol that is usually picked up from the Greek alphabet. Mathematical constants are by definition very important. In this article we will take a look at certain mathematical constants that are more commonplace than others.

1.     π (pi) or Archimedes constant (~3.14159):

 

π is defined as the ratio of the circumference of a circle to its diameter. This is probably the most popular mathematical constant. So π is the circumference of the circle whose diameter is 1 unit. You might have seen it popping up when calculating the area of a circle (πr2) or the circumference of a circle (2πr). It has many uses throughout mathematics from calculating the area of certain shapes to the Gaussian integral in complex analysis. π is irrational and is represented by the small letter π in the Greek alphabet.

2.     e or Euler’s constant (~2.71828):

 

e is also called the exponential growth constant. It is a very interesting number that shows very interesting properties. e is used in calculus. It is a very amazing number in general. The value of e can actually be calculated with compound interest. Imagine you have one rupee in the bank. The bank gives a compound interest of 100% per annum. So at the end of the year you will have 2 rupees. Imagine the bank, instead of giving you 100% per annum, gave 50% per six months. This is not equal to the above value. Therefore the final amount will be equal to 2.25 rupees (do the math). The value received has increased even though the principal is same. Imagine that for 33% per 4 months. The final value would be 2.36 rupees. Therefore it keeps on increasing. Then imagine how much you could get from 1/infinity % per 12/infinity months. You don’t have to. That number is e.

 

The formula is as follows:





e is also used in many other mathematical fields. It is used in Bernoulli trials, derangements, asymptotics, standard normal distribution and calculus.

 

If a graph is made with y=ex, it will show some unique properties too. At any point on the curved line formed on the graph, the y value of the point and the area under it are equal.

 

3.     i or the imaginary number (-10.5):

 

i is a very interesting variable. It’s value is the square root of -1. It falls into the category of imaginary numbers, in contrast to real numbers. Being a quadratic equation with a multiple root, i is not calculable. But other relations for i can be seen.

 

i2 = -1

i3 = i2i = (-1)i = -i

i4 = (-1)2 = 1

i5 = i3i2 = (-i)(-1) = i

 



It follows a certain pattern. The pattern is (-1, -i, 1, i). This is only applicable if we start with i4n+2. This leads to the conclusion that in = in mod 4 . ‘Mod’ refers to the modulus function.

 

 

 4.     √2 or “root 2” or Pythagorean constant (~1.414213):

√2 is a very well known mathematical constant. It is the ratio between the diagonal and side of a square. √2 is calculated by square rooting 2. It is based around the Pythagoras’ Theorem.

 

Let us assume x represents length of one side of a square. Let y assume the length of the side adjacent to it. x = y. Let z represent length of diagonal. x2 +  y2 = z2. This simplifies into 2x2 = z2. Therefore z = √2x. So the ratio z : x becomes √2x : x, which leaves you with √2.

Therefore √2 can be very easily proved geometrically. It is also a number whose infinite tetrate is equal to its square. This means:

√2 raised to the power to √2 raised to the power of √2 raised to the po……. infinitely = 2 or √22.


Comments

Popular Posts

Anode Ray Experiment

→Anode ray experiment was conducted by E Goldstein. →These rays are also known as canal rays. →This experiment helped in the discovery of the proton. Apparatus Used A discharge tube  was taken in which there were 2 electrodes i.e. Anode(+ve) and the cathode (-ve). The tube was filled with an inert gas. A perforated or porous cathode was used. A layer of zinc sulphide was placed at the back of the cathode. There was a vacuum pump in the tube. High voltage (5000v-10000v) was allowed to flow through the system. It was observed that when the gas was at 1atm(atmospheric pressure ) no change was seen in the tube.  When the   pressure   was decreased inside the tube, a glow could be seen at the back side of the cathode.

Isotopes, Isobars and Isotones

Isotopes These are elements which have the same atomic number but different atomic mass . They have the same atomic number because the number of protons that are inside their nuclei remains the same. But, they have different atomic mass because the number of neutrons that are also inside their nuclei is different. As the number of protons inside nuclei remains same, therefore the overall charge of the elements also remains same as in isotopes: no of protons = no of electrons . Hence, as isotopes overall charge remains neutral, therefore their chemical properties will also remain identical.   Therefore, Isotopes are chemically same but physically different.

The Inverse & Reciprocal TRIGONOMETRIC Functions

So, this is my second post on trigonometry. In this post we're gonna cover the reciprocal and the inverse Trigonometric functions. If you haven't seen my first post you should definitely view it as it covers the basics of Trigonometry The Reciprocal Trigonometric Functions The reciprocal Trigonometric function of Sine is Cosecant, of Cosine is Secant & for Tangent it is Cotangent. Cosecant (Csc θ = 1/Sin θ) or (Hypotenuse/Opposite) Secant (Sec θ = 1/Cos θ) or (Hypotenuse/Adjacent) Cotangent (Cot θ = 1/Tan θ) or (Adjacent/Opposite) We can also represent Tan θ in another way. As Tan θ = opposite/adjacent  & Sin θ = opposite/hypotenuse  & Cos θ = adjacent/hypotenuse ∴ Tan θ = Sin θ/Cos θ (The hypotenuses cancel out) As Cot θ = 1/Tan θ  So, we can also represent Cot θ as Cos θ/Sin θ.

Rutherford Alpha Ray Scattering Experiment

Rutherford Alpha Ray Scattering Experiment Hey, Guys, most of you might have heard about the alpha ray scattering experiment and if you want to know in detail about Rutherford's model and the experiment he conducted, this is the right place for you... But first: Things You Must Know Some basic information that will help you understand rutherford experiment properly: Proton is a sub-atomic particle which is positively charged and has a mass of 1u. Alpha particles are helium atom with a charge of +2 as they have lost 2 electrons. Alpha particles have an atomic mass  of 4u. Gold is highly malleable and can be beaten into very thin sheets. Experiment Rutherford conducted his experiment in the following way: Rutherford took a very thin gold foil and bombarded it with high energy alpha particles. He placed a layer of zinc sulphide on the walls where the experiment was taking place because when alpha particles strike zinc sulphide layer, it results i

Cathode Ray Experiment

This experiment was conducted by J.J. Thomson (Sir Joseph John Thomson) in the year 1897. This experiment proved that atom is made up of fundamental particles which are much smaller than the smallest atom 'hydrogen' This experiment helped to discover electron. According to J.J. Thomson, the cathode rays consisted of very light, small and negatively charged particles. He named the particles "corpuscles" which were later known as electrons

UEFA CHAMPIONS LEAGUE

UEFA CHAMPIONS LEAGUE Europe's biggest club soccer competition, The UEFA Champions League, takes place every year. The competition begins in September and ends in May. It is a classic tournament following a round-robin format after which the top teams qualify to the knockout stages. It includes 32 best clubs in different countries of Europe. Obviously teams from major countries like England, Spain, France, Germany and Italy qualify easily while teams from smaller countries like Ireland, Austria, Netherlands, Denmark have to go through a qualifying round. The teams who win the leagues in their countries qualify directly while in some countries even the second, third and fourth placed teams also qualify. Sometimes a team may qualify in the Champions League because they finished in the top-four. These clubs may or may not qualify through their leagues but their good performance may help them qualify. Even the champions of The UEFA Europa League, another prestigious league

Important Mathematical Constants!

Important Mathematical Constants Mathematical constants are those numbers that are special and interesting because they come up in the various fields of mathematics like geometry, calculus etc. These mathematical constants are usually named after the person who discovered it and they are represented by a symbol that is usually picked up from the Greek alphabet. Mathematical constants are by definition very important. In this article we will take a look at certain mathematical constants that are more commonplace than others. 1.       π (pi) or Archimedes constant (~3.14159):   π is defined as the ratio of the circumference of a circle to its diameter. This is probably the most popular mathematical constant. So π is the circumference of the circle whose diameter is 1 unit. You might have seen it popping up when calculating the area of a circle (πr 2 ) or the circumference of a circle (2πr). It has many uses throughout mathematics from calculating the area of certain shap