What are some commonly used mathematical constants and their purposes?

[pairofstrings]

Hi. π is a constant. The purpose of π is to use it to find circumference of a circle or area of a circle. Correct? Can you please elaborate?
Can you point out any other two constants and its purpose?
Thanks.

 

[jedishrfu]

There are many constants in math which you can find via google search. One such constant is Euler's number.

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

Is this for a school report?

 

[Svein]

Well - not exactly. π occurs almost everywhere in mathematics, the ratio between the circle diameter and its circumference is just one example.

Other constants are (for example)

• e = 2.7182818..., the basis for natural logarithms etc. Also part of the famous formula e^iπ=-1.
• γ=0.57721... (the Euler–Mascheroni constant). Check out https://en.wikipedia.org/wiki/Euler–Mascheroni_constant

Then there are the sequences of constants like

• The Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, ... (where each term is the sum of the two preceding terms)
• The Bernoulli numbers - not integers but rationals (check out https://en.wikipedia.org/wiki/Bernoulli_number)

There is a website dedicated to such sequences: https://oeis.org/.

 

[pairofstrings]

Is this for a school report?

Looking for a new constant.

π occurs almost everywhere in mathematics, the ratio between the circle diameter and its circumference is just one example.

What are the uses of constants?
Web does not show satisfying answers.

 

[fresh_42]

Looking for a new constant.

I'm not sure whether this makes any sense. In the end a mathematical constant is simply a number. The only reason why it can be called a mathematical constant, which I personally would not, is the fact that it occurs on various seemingly independent occasions and formulas. So first are the principles and then comes the constant, not the other way around. Or you just can pick any number.

What are the uses of constants?

Usually to write down formulas in a short and precise way. Otherwise the circumference of a circle with radius $1$ couldn't be handled nicely. Carpenters might be satisfied with $3.14$, mathematicians can not. Or how would one write the normal distribution, which is basically all around in this universe?

Web does not show satisfying answers.

Here's a nice, but by no means complete list:
https://en.wikipedia.org/wiki/Mathematical_constant
If you want to find usages, then pick one and start searching. E.g. for $\pi\; , \;e$ or the golden ratio you will find more webpages than you can read.

 

[jedishrfu]

The constants come out solving some arcane math problem. As an example, 'e' came out of work done on the compound interest problem and then went on to become fundamental for other areas of math.

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

Graham's number came out of work on Ramsey theory:

https://en.wikipedia.org/wiki/Graham's_number

 

[Khashishi]

1 is also a constant.

 

[jbriggs444]

In the end a mathematical constant is simply a number.

In this context, I would say that a mathematical constant is a named number. We give the number a name for the same reason that we give anything a name -- to make it easy to refer to. It is a lot easier to write down $\pi$ than to write down "the ratio of the circumference of a circle to its diameter"

 

[jedishrfu]

Except we have also named numbers like ten, a hundred, a thousand... or a googol for the convenience of talking and writing about them in a common sense way.

 

[mfb]

Looking for a new constant.

Here you go: 464.36466905383373880693157320423356352311325674
I am sure no one ever used this number before.
Is there any application for this number? Probably not.

You don't start with "I want a new constant". You start with a mathematical question. e. g. "what is the ratio between circumference and diameter of a circle", and you discover that it is the same for every circle, and determine properties of this constant ratio.

 

[jedishrfu]

I like the story of 1729. Ramanujan was sick in the hospital and GH Hardy went to see him. Hardy remarked

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

https://en.wikipedia.org/wiki/1729_(number)

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

More recently, mathematicians found that Ramanujan had been investigating other numbers like these and realized that he was looking into the Fermat's little theorem:

https://www.zmescience.com/science/math/taxi-number-ramanujan-03213/

 

[pairofstrings]

I have another question: π is a constant - has value 3.14 (roughly), if I measure the area of circle in mm² and area of square in mm² and obtain two values - area of circle in mm² and area of square in mm² then how would I know that this number is talking about area of circle and this number is talking about area of square because the area of both shapes is in mm²? Assume that area of square is 201.06 mm² and area of circle is 201.06 mm².

Is it possible to distinguish area of circle from area of square and so that only by looking at the units or perhaps result I can say that the person is talking about area of square or area of circle? Is it possible to talk about the shape just by looking at units or perhaps result?

πr²: we know what the value of constant π is. If I multiply π with 25: 3.14 x 25 mm² then the result is 78.5 mm². What influence does π or 3.14 have on answer? Will this π say that we are talking about circle? But the end result is a number with units mm² - we could not distinguish area of square with area of circle if the units are same!

 

[Svein]

Is it possible to distinguish area of circle from area of square and so that only by looking at the units

No. An area is an area. The shape of an area tells you what kind of formula you need to use to calculate the area. Of course, you are free to use the raw force method - divide the area into very small squares and count them.

What influence does π or 3.14 have on answer?

π is part of the formula to calculate the area of the circle, just as ½b⋅h is the formula for calculating the area of a triangle. Those formulas are there to make things easier for you, you do not have to use them.

Look at the pentacle above. There is a formula for calculating the area, I am not going to tell you what it is. How are you going to find the area and how do you know that the area is the correct one?

 

[FactChecker]

Is it possible to distinguish area of circle from area of square and so that only by looking at the units or perhaps result I can say that the person is talking about area of square or area of circle? Is it possible to talk about the shape just by looking at units or perhaps result?

You seem to keep underestimating the importance of how something can be used. Out of all the mathematics and arithmetic that has been done throughout history, only a tiny, tiny fraction is worth becoming part of standard mathematics. Any single number is a "constant" but only a few have a name.
The concept of area, independent of shape is very useful. It helps to abstract many calculations and concepts without regard to shape. Abstraction of area/volume is extremely important in mathematics and physics.

 

[fresh_42]

My most favorite constant: $O(1)$.

 

[jedishrfu]

Thread closed, because the question has been answered.