Universe Word Problem: How big is "10^10^123 power"?

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Steelrails

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I used to go to a Catholic School, and my Religion teacher told us that the probability of there being another planet sustainable for life (as a Harvard Profecer worked out, or so she claimed) is 10^10^123 power. Everyone in the class knew it was a big number, but no one knew how big. I have always wondered what that number was, but no calculator (that I can find, both physical, and online) can calculate it. So, the challenge I propose to anyone who reads this is to solve 10^10^123 power.
 
There's nothing to "solve" here. \(10^{10^{123}}\) is a number, and its value is \(10^{10^{123}}\). If you mean you want to see it written out fully as a looooong string of digits, well you're out of luck. That particular number has so many digits that it literally can't be written down. Even if we write it so small that each digit takes up a single atom of space, and convert every atom in the visible universe to ink, there still wouldn't be enough ink.

You can get a grasp of how big the number is by learning more about how exponents, and specifically how powers of 10, work. \(10^{\text{(number)}}\) is the same as saying write a 1 and then write (number) zeros.

So \(10^{123}\) is 1 followed by 123 zeros. That's already an insanely big number, but then it gets way worse because \(10^{10^{123}}\) is 1 followed by \(10^{123}\) zeros.
 
Steelrails said:
I used to go to a Catholic School, and my Religion teacher told us that the probability of there being another planet sustainable for life (as a Harvard Profecer worked out, or so she claimed) is 10^10^123 power. Everyone in the class knew it was a big number, but no one knew how big. I have always wondered what that number was, but no calculator (that I can find, both physical, and online) can calculate it. So, the challenge I propose to anyone who reads this is to solve 10^10^123 power.
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A probabilty is a number between 0 and 1. Thus, either the professor from Harvard was grossly wrong (quite probable) or your teacher of religion was mistaken about the statement's provenance (not inconceivable).
 
Personally I think that the probability is a number in between 1 over that number and 1 - 1 over that number
 
Well... I’m not good at this but I will try. Ten to the tenth power is 10000000000 (with ten zeros) the that to the 123 power is (I think) is 1 and hundred thirty three zeros. Please don’t hate if I’m wrong I just wanted to try.
 
Narwalalpaca12$ said:
Well... I’m not good at this but I will try. Ten to the tenth power is 10000000000 (with ten zeros) the that to the 123 power is (I think) is 1 and hundred thirty three zeros. Please don’t hate if I’m wrong I just wanted to try.
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Almost but not quite. 10123 is 1 followed by 123 zeros so 10,000,000,000123is NOT 1 followed by 123 zeros.
You also have another problem. You tried solving (1010)123 which is not the same as 10(10123)
Let's look at an easier case: (102)3 = (100)3 = 1,000,000. While 1023= 108 = 100,000,000.
Note that 1010123 = (10)(10123)
 
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To make sure we are all working with the same number,
Let's say 10^123 = d then 10^10^123 = 10^d

If exponentiation is indicated by superscript notation, the usual rule is to work from the top down:
a^b^c = a^(b^c)

Equations of the form a^b^c are done by evaluating b^c =d and THEN determining a^(d)

2^3^4 COULD be evaluated as 8^4 or 4,096
However, if we evaluate as 2^81 then it equals 2.42 x 10^24 - a GIGANTIC difference!!

As already stated 10^123 equals 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

So, the unappealing 10^10^123 could better be displayed as 10^d
 
"10^10^123" is, strictly speaking, ambiguous. But since (10^10)^123 could be written more simply as 10^1230, it is most commonly accepted as meaning 10^(10^123).
 
HallsofIvy
you said "(10^10)^123 could be written more simply as 10^1230"
I think that is incorrect.
(10^10)^123 would equal 10,000,000,000 ^123 or
(10 billion)^123 or
(10^10) ^ 123
 
wolf said:
HallsofIvy
you said "(10^10)^123 could be written more simply as 10^1230"
I think that is incorrect.
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No, look at this rule:

(a^b)^c = a^(b*c)

Ex. \(\displaystyle \ \ \) (4^3)^2 = 4^(3*2) = 4^6 = 4,096

or

(4^3)^2 = (64)^2 = 4,096
 
Yes, that is one rule of exponentiation.
Couldn't 4^3^2 also be interpreted as 4^(3^2) or 4^9 or 262,144?

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Getting back to what HallsofIvy stated:
(10^10)^123 could be written more simply as 10^1230, it is most commonly accepted as meaning 10^(10^123).
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I cannot see how (10^10)^123 equals 10^1230.
 
wolf said:
Couldn't 4^3^2 also be interpreted as 4^(3^2) or 4^9 or 262,144?
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Yes, that is the standard interpretation. But that is not what lookagain was talking about.
wolf said:
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Getting back to what HallsofIvy stated:
(10^10)^123 could be written more simply as 10^1230, it is most commonly accepted as meaning 10^(10^123).
**************************************************************************************
I cannot see how (10^10)^123 equals 10^1230.
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You don't see that (10^10)^123 = 10^(10*123) = 10^1230? Even after agreeing that (a^b)^c = a^(b*c) ?
 
Dr.Peterson stated:
You don't see that (10^10)^123 = 10^(10*123) = 10^1230? Even after agreeing that (a^b)^c = a^(b*c) ?
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Yes, I made a mistake!!!
Interpreting these phenomenally huge numbers does get confusing (to say the least)!!!
 
Steelrails said:
I used to go to a Catholic School, and my Religion teacher told us that the probability of there being another planet sustainable for life (as a Harvard Profecer worked out, or so she claimed) is 10^10^123 power. Everyone in the class knew it was a big number, but no one knew how big. I have always wondered what that number was, but no calculator (that I can find, both physical, and online) can calculate it. So, the challenge I propose to anyone who reads this is to solve 10^10^123 power.
Click to expand...
10^79 is about the number of all of the atoms in the known universe. Some physicists believe there is evidence to substantiate a theory for a infinite multiple universes. With a finite universe and infinite universe things must repeat.
So in order for 10 to the 79 atoms to arrange in exactly the same order (meaning somewhere out there would be a second you reading this answer) would be about 10 to the 10 to the 123.
meaning every possible arrangement of 10 to the 79 atoms would repeat at least once.
Roger Penrose also stated that the chance of this low level of entropy required to sustain human life is 1 in 10 to the 10 to the 123. (Not in anyway to be confused with 10 to the 124)
 
Take a look at the Fermi paradox and the Drake equation (google it). I think you will find it relevant.
 
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