Ridiculously large numbers,Graham's Number, Infinite and beyond!!!!!

We humans are curious about everything, we want to reach the limits of everything.
We are never satisfied until we reach the thresh hold, our thirst is never satiated.
When we see something good, we want something better. When we have a lot of things, we crave for more. Something like in the case of money.
We we see something is big, we want to see something bigger. There came the empire state building, then the twin towers then Bhurj Khalifa. Probably there are plans to build something bigger.

The same is with numbers!
There was ten, hundred, thousand, million, billion, trillion, zillions squillions and somethingelsions.

Do we stop there???  NNNNOOOOOOPPPPPPEEEEE!!!!!!!!!!!!!!

Lets check out some insanely large numbers.........................and by insane.............
I MEAN INSANE!!!

The Googol:-

A Googol is defined as 10^100 = 10,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000

Its 1 followed by a HUNDRED zeroes. Thats equal to ten thousand trillion trillion  trillion  trillion  trillion  trillion trillion trillion. Thats a lot!!

The Googolplex:-
The googolplex= 10^googolplex = 10^(10^100)
Thats 1 followed by ten thousand trillion trillion  trillion  trillion  trillion  trillion trillion trillion ZEROES!!
Thats a VERY BIG number.

Skewe's number :-
In 1933, Stanley Skewes used the number 10¹⁰¹⁰³⁴ (= 10^10^10^34) in a proof involving prime numbers. G. H. Hardy said it was "the largest number which has ever served any definite purpose in mathematics".

But this number doesnt hold the record...................

Guys

Welcome To

*drum roll*

THE GRAHAMS NUMBER!!!!!!!!!!!!!!!!!!!!!!!!!!!!

what? think im exaggerating??? pfffffffffffffffffffffft.
If u dont know about grahams number, u have absolutely NO idea how big it is. NOPE!

Note:- Serious Math Ahead!!!

first, lets see whats the arrow notation.

 = 3^3=27

 =   =  3^(3^3)=3^27= 7625597484987................yup, one small teeny little arrow made the value BLOW UP!!!!

 =    =      which is UNBELIEVABLY LARGE!!!!!!!!!!!! SO large that  its digits wont even fit in the universe.

is this the grahams number?? hohoho........nope..........nowhere near...........

let  33= G1

G₁ = 3^^^^3
G₂ = 3^...(G₁ arrows)...^3
G₃ = 3^...(G₂ arrows)...^3
G₄ = 3^...(G₃ arrows)...^3
G₅ = 3^...(G₄ arrows)...^3
G₆ = 3^...(G₅ arrows)...^3
G₇ = 3^...(G₆ arrows)...^3
G₈ = 3^...(G₇ arrows)...^3
G₉ = 3^...(G₈ arrows)...^3
G₁₀ = 3^...(G₉ arrows)...^3
G₁₁ = 3^...(G₁₀ arrows)...^3
G₁₂ = 3^...(G₁₁ arrows)...^3
G₁₃ = 3^...(G₁₂ arrows)...^3
G₁₄ = 3^...(G₁₃ arrows)...^3
G₁₅ = 3^...(G₁₄ arrows)...^3
G₁₆ = 3^...(G₁₅ arrows)...^3
G₁₇ = 3^...(G₁₆ arrows)...^3
G₁₈ = 3^...(G₁₇ arrows)...^3
G₁₉ = 3^...(G₁₈ arrows)...^3
G₂₀ = 3^...(G₁₉ arrows)...^3
G₂₁ = 3^...(G₂₀ arrows)...^3
G₂₂ = 3^...(G₂₁ arrows)...^3
G₂₃ = 3^...(G₂₂ arrows)...^3
G₂₄ = 3^...(G₂₃ arrows)...^3
G₂₅ = 3^...(G₂₄ arrows)...^3
G₂₆ = 3^...(G₂₅ arrows)...^3
G₂₇ = 3^...(G₂₆ arrows)...^3
G₂₈ = 3^...(G₂₇ arrows)...^3
G₂₉ = 3^...(G₂₈ arrows)...^3
G₃₀ = 3^...(G₂₉ arrows)...^3
G₃₁ = 3^...(G₃₀ arrows)...^3
G₃₂ = 3^...(G₃₁ arrows)...^3
G₃₃ = 3^...(G₃₂ arrows)...^3
G₃₄ = 3^...(G₃₃ arrows)...^3
G₃₅ = 3^...(G₃₄ arrows)...^3
G₃₆ = 3^...(G₃₅ arrows)...^3
G₃₇ = 3^...(G₃₆ arrows)...^3
G₃₈ = 3^...(G₃₇ arrows)...^3
G₃₉ = 3^...(G₃₈ arrows)...^3
G₄₀ = 3^...(G₃₉ arrows)...^3
G₄₁ = 3^...(G₄₀ arrows)...^3
G₄₂ = 3^...(G₄₁ arrows)...^3
G₄₃ = 3^...(G₄₂ arrows)...^3
G₄₄ = 3^...(G₄₃ arrows)...^3
G₄₅ = 3^...(G₄₄ arrows)...^3
G₄₆ = 3^...(G₄₅ arrows)...^3
G₄₇ = 3^...(G₄₆ arrows)...^3
G₄₈ = 3^...(G₄₇ arrows)...^3
G₄₉ = 3^...(G₄₈ arrows)...^3
G₅₀ = 3^...(G₄₉ arrows)...^3
G₅₁ = 3^...(G₅₀ arrows)...^3
G₅₂ = 3^...(G₅₁ arrows)...^3
G₅₃ = 3^...(G₅₂ arrows)...^3
G₅₄ = 3^...(G₅₃ arrows)...^3
G₅₅ = 3^...(G₅₄ arrows)...^3
G₅₆ = 3^...(G₅₅ arrows)...^3
G₅₇ = 3^...(G₅₆ arrows)...^3
G₅₈ = 3^...(G₅₇ arrows)...^3
G₅₉ = 3^...(G₅₈ arrows)...^3
G₆₀ = 3^...(G₅₉ arrows)...^3
G₆₁ = 3^...(G₆₀ arrows)...^3
G₆₂ = 3^...(G₆₁ arrows)...^3
G₆₃ = 3^...(G₆₂ arrows)...^3
G₆₄ = 3^...(G₆₃ arrows)...^3    (source http://www.susans.org/forums/index.php?topic=99242.0)

G64 is grahams number!!!!!!!!!!!!! Phew!!!!!!!!!!!!!!!!

its something like this

Source- WIkipedia

Dont worry................my mind nearly imploded trying to comprehend this.
This is UNBELIEVABLY huge.

THIS IS ABSOLUTELY VERY CRAZILY RIDICULOUSLY 

INSANELY MIND NUMBINGLY HUGE!!!!!!!!!!!


and you know what??
this was ACTAULLY used in a serious mathematical proof........... ( ~OO__OO~ )


*splurge* was that ur brain???? ahhhh, i wont blame you, thinking about this will almost turn your brain into a blackhole
(having a blackhole in a brain is not pleasant :P )

so now lets go to infinity............


INFINITY:-

Grahams number is HUGE

is it near infinity?? is it infinity?????

no.......
the interesting thing is that, Grahams number is as near to infinity as zero is.
everyone is at the same distance from infinity.

Infinity is NOT a number.
Never ever treat is as a number.
Why isnt a number???
if it was a number.......
infinity + 1= infinity
infinity + 0= infinity
therefore 1=0
yaaay!!! i just proved something that literally obliterates the foundation of math!!!!!
nope, infinity is not a number!


Infinity is an idea! its just an idea about endlessness.
Infinity is unfathomable, depths unknown, and would never be known.

As John Green said, some infinities are bigger than the others.
There are infinities inside infinities.
But they are infinite.
No limits
No bounds
No ends
Just endless.

Fermi Problems

I thought this will be my first post just because of one reason...................... THIS GUY IS AWESOME!!!!!!

I mean, he is like one of the golden nuggets, one of the rare assets Science could have. Fermi problems give us a glimpse on how brilliant this person was.

Fermi sitting on a chair, using a piece of paper, a pencil , and his brain could calculate stuff like the circumference of the Earth,  what is the weight of the air above lake superior, how many hair are their on a human head, how many molecules are there in a standard classroom, how many piano tumers are there in Chicago and............

*deep breath*                                                                                                                                       How many atoms of your great^n-grandfather who lived about 2000 years ago do you ingest with each meal ............... wait WHAT!!!!!! O_O                                                                                                                                                                                                                                                                                          
yup, thats the wonder of approximations. Lets dive into some of them..........

Q How many Jelly Beans fit in a one Liter Jar?

First, what will the size of a normal bean be? 

We can think of a jelly bean as a small cylinder, 2 cm long and 1.5 cm  long

If we fill the whole bottle with jelly beans. the complete jar wont be filled, and there will be some air spaces left. So we can assume that about 80% of the bottle is filled.

what is the volume of a single Jelly Bean?

Volume of 1 Jelly Bean = h(pi)(d/2)^2  where h=2cm is the height of the cylinder and d=1.5 is the diameter

thus volume= = 2cm * 3.14* (1.5cm/2)^2 = 3.53 cubic cm

The number of jellybeans in a one liter jar is Volume of the Jar/Volume of one bean

= ((8/10) x 1000 cubic centimeters)/(3.53 cubic cm)= 227 jelly beans

Voila! you can check this too! the answer may not be exact, but it will be very near to this!

Q How many piano tuners are there in Chicago???

The question seems very absurd at first and impossible to answer, but hey.............there is a solution to everything!!!

Each individual may not own pianos for their own. A Piano is like a family property, a family will have one piano.

Lets take the number of families in Chicago to be 2,000,000. (its crowded out there)

Of course, every family will not have a piano. we can assume that one out of 5 families have a piano.

Thus the number of pianos is  400,000

Suppose a piano tuner can tune 5 pianos a day, and supposing the number of working days is 200, he can tune 1000 pianos a year.

Frankly speaking, i dont know how much time it takes, till a piano needs tuning. I assume it to be five year.

So in one year, 400,000/5= 80,000 pianos need tuning.

Thus the number of piano tuners is 50,000/1000= 80 piano tuners in the city.

Q How many kernels of popcorn will it take to fill a classroom?

WE can assume that in a cubic inch, there will be nearly 4 popcorn kernels.

 There are 12*12*12 inch cubes in a cubic foot. This value is near 1500. Thus there are 1500*4= 6000 kernels in a cubic foot..

A standard classroom can be taken as 70 feet long and 40 feet wide and nearly 10 feet high.

So the volume of the classroom will be  28000 cubic foot.

Thus the number of kernels in a standard classroom is

28000*6000= 168,000,000.  Phew! thats a LOT!

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You might be thinking that they answers may be waaaay off the actual value as we are taking  huge approximations.

Even though the approximations seem random, the answer is near to the actual values.

The approximation will be of the same order of magnitude as that of the actual answer.

so why does this happen??

in simple words, the errors tend to cancel each other.

The over approximations and the under approximations cancel each other, thus the answer tends to be near the actual value.

During a testing of the nuclear bomb, Fermi was sitting in a lab, near to the bomb site.

When the bomb exploded, he started dropping small pieces of paper to the ground.

When the shock-wave arrived, the small pieces got displaced.

Then on the back of an envelope, he calculated the strength of the bomb to be 10 kilo tonnes, which is quite close to the actual value!!!!!

Thats how  brilliant this method was!!!!!!!!