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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

Can anyone please tell me something very interesting about maths. It could be an amazing or unbelievable thing, some good shortcut trick or anything which I will like very much. I will also tell you some if you tell me.:)

Practice makes a man perfect.

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,408

This might be interesting to you. I certainly find it interesting.

Start with the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, .....

Divide each term by the one before:

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666

8/5 = 1.6

.......

The numbers converge on the golden ratio (phi) ; oscillating either side of that number.

If you compute 1/1, 1/2, 2/3, 3/5, 5/8, .......

the numbers converge to phi - 1, which is also 1/phi

Apparently, the sequence (of rationals converging on an irrational) has a special property (which I've forgotten for the moment) which means that a plant (such as a sunflower), growing a seed head from the centre, and with angular separation atan(phi), will be least likely to form cleavage planes which might cause the seed head to fall apart. So nature discovered the property first!

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Perhaps the most interesting thing that I ever saw was the Monty Hall problem. Not only does it have mind boggling probability, EM, the worlds highest IQ, tons of differing opinions, it has one more thing...goats! Yes, math ain't nothing without a couple of stinky goats thrown in.

http://www.mathisfunforum.com/viewtopic.php?id=20868

The most amazing things about it, were all personal. I became further convinced that EM was the only way to go if one desired to get the right answer an infuriating amount of times. Also, we were 2 of only 4 people in the world who knew that those Stanford and Duke university guys were not arguing with Marilyn. That would have been difficult enough, they were arguing with their own methods and colleagues who had hashed this problem to death in a statistics journal, many years before. RIPOSTP.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

The most interesting thing I have found in maths till now is the amazing relationship between Fibbonacci Series, The Golden Ratio and Nature. It has really amazed me. I also find pascal's triangle very interesting. There are 7 things that leaves me awestruck about pascal's triangle.

Practice makes a man perfect.

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,408

hi iamaditya

Then I think you'll find this interesting:

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

Yeah I know it.

This is one of the 7 things of the pascals triangle which i find very interesting.

*Last edited by iamaditya (2017-03-12 22:36:40)*

Practice makes a man perfect.

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 26,308

Hi;

325 is the smallest number to be the sum of two squares in 3 different ways:

and .425 also has a similarly property :

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

Hmm, good!

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**Monox D. I-Fly****Member**- From: Indonesia
- Registered: 2015-12-02
- Posts: 1,283

iamaditya wrote:

Yeah I know it.

This is one of the 7 things of the pascals triangle which i find very interesting.

What are the other 6?

Actually I never watch Star Wars and not interested in it anyway, but I choose a Yoda card as my avatar in honor of our great friend bobbym who has passed away. May his adventurous soul rest in peace at heaven.

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

Actually there are 13. They all are listed below in brief.

1. A number of the nth row and nth position getting found out by Combiantion.

2.Existence of Fibonacci Sequence as bob showed above.

3.Shading Odd nos. in a Pascal's Triangle gives Sierpenski triangle series.

4.HCF of any row of a prime no. excluding 1 is that prime no. itself.

5.If you draw a hockey stick starting from an extreme end (that is a 1)and make a Hockey Stick,with only 1 number in the "J", the sum of all nos. in the stick = to the no. in the "J" of the hockey.

6.Adding nos. of the nth row give value as (2^n)

7.The nos. of the 2nd diagonal are the counting nos., the nos. in the 3rd diagonal are triangular nos., in the fourth are tetrahedral nos. and so on

8.The nos. of the nth row are the digits of 11^n.(From 11^5, it begans to overlap itself)

9.The pascal's triangle is symmetrical

10.Existence of catalan nos.

11.Pick any number inside Pascal’s triangle and look at the six numbers around it (that form alternating petals in the flowers drawn above). If you multiply the numbers in every second petal, you’ll end up with the same answer no matter which of the petals you start from.

12.Pick any counting number from along the first diagonal and square it. Then look at its two neighbours that lie deeper inside the triangle – they’ll always add up to that very same square number.

13.If you add up every single number in the first n rows, you’ll get the nth Mersenne number.

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**cmowla****Member**- Registered: 2012-06-14
- Posts: 63

As I discovered, you can also modify Pascal's Triangle to help to construct the positive integer power sum formulas (image, video series) and the Bernoulli Numbers.

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**iamaditya****Member**- From: Planet Mars
- Registered: 2016-11-15
- Posts: 790

Hmm, excellent!

There is no substitute to hard work

All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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**Jabuto****Member**- Registered: 2017-07-29
- Posts: 11

Mandelbrot's set is one of the most known fractals out there.

It really sticks out.

The set contains all complex numbers, which if infinitely raised to power of 2 and added the initial value to, don't blow up to infinity but converge to a value.

You can also create another fractal, or halo, around the set if you plot the last breakpoint value.

If you think that every function transforms 2D plane's every point, the act of complex exponentation will bend the straight lines invards infinitely. With negative values the numbers close to zero explode close to infinity and wise versa.

You can also create other fantastic fractals with similar equations, namely, ** {iterated = iterated ^ start + start}, {iterated = start ^ initerated}, {iterated = -iterated ^ iterated}, {iterated = sqrt(iterated)+initial} **

just to name a few. With some of these the halos are more incredible than the actual set.

*Last edited by Jabuto (2017-09-27 21:13:45)*

* You will die knowing being right. My only cost is your fullfilment. ** --- The Devil *

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**Jabuto****Member**- Registered: 2017-07-29
- Posts: 11

A square has many symmetries. It stays same if you turn it 90 degrees, 180 degrees, to other direction of -90 degrees, rotate it from corner to corner and so on.

Cube has even more symmetries.

Hypercube has oh so many... and the number of symmetries rise exponentially.

But mathamaticans have calculated that if you have some 200 000 -dimensional shape (just a big number idk), it doesen't follow the regular symmetry logic so far. It form's its own group of symmetry, sometimes referred as monster group, and is not the only instance when it happens.

It is really hard to think what is really going on there, but John Convay himself thinks (somewhere in lines of) these obejcts like a christmas tree decorations; like this some regular shape, with some faces some going that way, some horizontally, some verticlaly, some in high-dimensional directions, but also unexpectelly in THIS direction.

* You will die knowing being right. My only cost is your fullfilment. ** --- The Devil *

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**Βεν Γ. Κυθισ****Member**- Registered: 2018-10-09
- Posts: 16

I have a piece of paper that has good exponentiation equations to memorize.

This is what is written on it at the time of this post (edited ones have an underline).

a^n=a multiplied by itself

~~n times~~n-1 timesIf n is even then (-a)^n=a^n

(abc⋯)^n=a^n×b^n×c^n×⋯

(1÷a)^n=1÷a^n

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

a^m×a^n×⋯=a^(m+n+⋯)

a^n÷a^m=a^(n-m) (Makes sense right?)

((a^m)^n)^⋯=a^(mn⋯)

If n>0 then (a^m)a^n=a^(m+n)

2^n-2^(n-1)=(2^n)÷2

*Last edited by Βεν Γ. Κυθισ (2018-10-10 21:04:22)*

Profile image is from public domain.

1+1=|e^(π×i)-1|

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Hi Βεν,

Welcome to the forum! Thanks for your contribution. That looks like a nice list. Some comments:

a^n=a multiplied by itself n times

If n is even then (-a^n)=a^n

a^n÷a^m=a^(m-n) (Makes sense right?)

If n>0 then (a^m)na=a^(m+n)

What did you mean here?

**LearnMathsFree: Videos on various topics.New: Integration Problem | Adding FractionsPopular: Continued Fractions | Metric Spaces | Duality**

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**Βεν Γ. Κυθισ****Member**- Registered: 2018-10-09
- Posts: 16

zetafunc wrote:

Hi Βεν,

Welcome to the forum! Thanks for your contribution. That looks like a nice list. Some comments:

I know what you mean, but this can be misleading: is multiplied by itself times. For example, , where gets multiplied by itself one time.a^n=a multiplied by itself n times

The brackets should go around the here, i.e. for even .If n is even then (-a^n)=a^n

On one side of the equation, the and are the wrong way round. It should read:a^n÷a^m=a^(m-n) (Makes sense right?)

If n>0 then (a^m)na=a^(m+n)

What did you mean here?

I fixed my errors and typos. My answer to

zetafunc wrote:

What did you mean here?

is that I messed up; it shouldn't be (a^m)na, it should be (a^m)n^a. I feel so stupid for not checking with a calculator.

Profile image is from public domain.

1+1=|e^(π×i)-1|

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Βεν Γ. Κυθισ wrote:

Do you mean rather than ?
it should be (a^m)n^a.

**LearnMathsFree: Videos on various topics.New: Integration Problem | Adding FractionsPopular: Continued Fractions | Metric Spaces | Duality**

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**Βεν Γ. Κυθισ****Member**- Registered: 2018-10-09
- Posts: 16

zetafunc wrote:

Βεν Γ. Κυθισ wrote:Do you mean rather than ?it should be (a^m)n^a.

DANG IT! I forgot to move the variables around! And I forgot to confirm it with a calculator or *my miiiiind*.

Profile image is from public domain.

1+1=|e^(π×i)-1|

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