Your Favorite Mathematical Formulas

[GameSetAndMath]

I will just post one to begin with.

e^{ i*pi} = -1.

Jump in with yours.

 

[Kieran]

I don't know what that means, but it looks spectacular! I remember a few years back in a pub in NY we met these mathematical geniuses and they showed me the squiggles and hieroglyphics and symbols and tried to explain.

Two things: it got me drunk quicker.

And I was amazed.

It was like listening to a virtuoso pianist or a master of philosophy. Complex, but extremely ordered...

 

[shawnbm]

Wow! No idea what all that means, but here's mine: 2+2=4

 

[GameSetAndMath]

Wow! No idea what all that means, but here's mine: 2+2=4

That is good.

Formula from any branch of Mathematics (algebra, analysis, geometry, calculus,
trigonometry, discrete math etc) is welcome. Formula at any level (from most elementary
to most advanced) is welcome.

I always wondered why people say "I put two and two together", when they
want to say their reasoning is pretty simple. Why don't they use "one plus one"?
Is not that simpler than 2+2?

 

[shawnbm]

I agree with you, gamesetandmath. I also dig the idea of pi going on ad infinitum. I am now at an age that I only recall 3.14 and that is it (once upon a time I knew another three or four digits, but it has been almost forty years). Also, I dig this whole string theory thing and what I read abour quantum mechanics (for lay folks like me) gets the brain going.

 

[Asmodeus]

Here's the worhorse of linear regression analysis (in matrix form):

b = (X'X)^-1X'y

Please excuse the ^ sign as the font does not seem to allow superscript.

 

[GameSetAndMath]

Here's the worhorse of linear regression analysis (in matrix form):

b = (X'X)^-1X'y

Please excuse the ^ sign as the font does not seem to allow superscript.

That is a non-trivial one. I am sorry that I forgot to mention stat formula
are also welcome.

Hopefully, britbox or someone will load math fonts sometime. Till then
get by with standard symbols and conventions.

 

[GameSetAndMath]

I agree with you, gamesetandmath. I also dig the idea of pi going on ad infinitum.

Such number are called irrational numbers . At one time people believed allnumbers were rational. Hippasus proved in 5th century B.C. that sqrt(2) is irrationalwhile on a ship sailing over a sea. The scholars in the Pythagorean school checked his proof and found there is nothing wrong with the proof and Hippasus is correct. Nonetheless, it was like a heresy as everybody in the Pythagorean school at that time believed all numbers are rational and math was almost like religion. So, they got together and pushed Hippasus over the board and into the sea. Talk
about occupational hazards.

 

[ashwin#1]

(a + b)2 = a2 + 2ab + b2 .

 

[Kieran]

For further info, see

http://en.wikipedia.org/wiki/Irrational_number

That's a really interesting article. How advanced were the Greeks, eh? Incredible the depths they went to with these things, and philosophy, poetry, architecture, art, etc. A great pity for me is that none of their music survives, since that was also considered to be as great as everything else.

But in mathematics, they were certainly advanced...

 

[GameSetAndMath]

(a + b)2 = a2 + 2ab + b2 .

That is nice. But use ^ for exponentiation so that people do not confuse
a2 to be a * 2.

(a+b)^2 = a^2 + 2ab + b^2.

 

[GameSetAndMath]

Here is another interesting and important formula

e^{i * theta} = cos(theta) + i * sin(theta)

 

[Kieran]

Here is another interesting and important formula

e^{i * theta} = cos(theta) + i * sin(theta)

In brief layman terms, can you explain why that's important, or interesting? I'm curious, not cynical!

Cheers...

 

[GameSetAndMath]

Here is another interesting and important formula

e^{i * theta} = cos(theta) + i * sin(theta)

In brief layman terms, can you explain why that's important, or interesting? I'm curious, not cynical!

Cheers...

This formula is called Euler's Formula as it was first discovered by Leonard
Euler in 1743. It is interesting because it connects trigonometric functions
(sin, cos etc) with complex exponentiation function (note that the exponent of
e is a complex number). It is important because it has lots of applications
in physics and engineering, not to mention other branches within Mathematics itself.

Incidentally, Euler is from Switzerland and is considered a master of Math,
just like another Swiss fellow is considered master of Tennis.

 

[Kieran]

Great stuff, GS&M, thanks!

 

[GameSetAndMath]

Although, Euler is considered a great master of Math, to be fair I must mention that the
German Mathematician Gauss is widely considered to be the GOAT of Mathematics.

 

[Kieran]

Greater than the Geeks?

I mean, the Greeks!

 

[GameSetAndMath]

Greater than the Geeks?

I mean, the Greeks!

Well, the Greeks were good but not great. One of the important reasons for
the GOAT status of Gauss is the era in which he worked. That was the period
after Newton and Leibinitz have invented
Calculus
and so was a hectic period
of activity with lots of discoveries happening. From the year minus infinity to
around 1650, there was no calculus an important tool for mathematical
advancement. Once Calculus was invented, it opened a flood gate of
discovery/invention in Mathematics and its applications to Physics and
Engineering.

This is much like the Graphite Racquets coming into scene around
mid 1980s and revolutionizing the way Tennis was played.

 

[GameSetAndMath]

Greater than the Geeks?

I mean, the Greeks!

Well, the Greeks were good but not great. One of the important reasons for the GOAT status of Gauss is the era in which he worked. That was the period after Newton and Leibiniz have invented Calculus and so was a hectic period of activity with lots of discoveries happening. From the year minus infinity to around 1650, there was no calculus, an important tool for mathematical advancement. Once Calculus was invented, it opened a flood gate of discovery/invention in Mathematics and its applications to Physics and Engineering.
With the aid of Calculus, Gauss could go lot further than Greeks.

This is much like the Graphite Racquets coming into scene around mid 1980s and revolutionizing the way Tennis was played compared to the Wooden Racquet Era.

 

[Fiero425]

(a + b)2 = a2 + 2ab + b2 .

Pythagorean Theorem was the one I had locked in my head along with 2TTR2 which gives us the area of a circle! Just for the hell of it, y-intercept = m+b where b is the slope! MATH was my thing back in school, preferring to work a problem long-hand over a calculator or slide-rule!