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how big is 2 to the power 128? i.e. 2^128

Updated on March 23, 2012

When you see notation like this:

2^64

What does it mean?

Why is 2^64 radically different to 2^128 ? It looks similar. 

First let's dissect the notation.

2^64 is another way to write 264.

264 is a doubling of 2, 64 times.

Below, is a table showing 2 raised to the power of exponents from 0 to 128.

20 = 1

21 = 2

22 = 4 = 2 x 2

23 = 8 = 22 x 2 = 2 x 2 x 2 x 2

We can skip a few and state the results of some common low-value results.

28 = 256

210 = 1024

216 = 65536

and now some larger powers of two.

232 = 4,294,967,296  This is in the same ballpark as the human population of the Earth which is currently approaching 7 billion.

242 = 4,398,046,511,104  This is in the range of the number of cells in the human body.

264 = 18,446,744,073,709,551,616

296 = 79,228,162,514,264,337,593,543,950,336

2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 This would be something like the number of cells in 77,371,252,455,336,267,181,195,264 human bodies.




Tetration

You can read more about tetration here or here for a more accessible treatment.

So although 264 does not look much different in symbols to 2128, the former number is easy to comprehend, and the latter pushes the limits of your imagination.

Yet this number is comprehend-able with a little effort. There are other numbers that are mind-numbing in concept. A number called Graham's number is the largest number ever used in a mathematical proof. It was a proof involving a branch of mathematics as an upper bound in a problem presented in Ramsey Theory.

Graham's number is so large that an ordinary digital representation requires so many bits of information that there are not enough states in the observable universe for it. It cannot even be represented with a 'tower' of powers.

ab where b=cd where d=ef...

and we need to use Knuth's up arrow notation or something equivalent. This is fodder for another article but it's useful to know that the up arrow notation uses iterated exponentiation. This is tetration.


The theoretical (but not practical) upper possible number of internet addresses available in an IPv6 packet header is 2128.  Hopefully when you see this number you will have a reasonable idea what it means.

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    • profile image

      bilal 

      4 weeks ago

      The cryptographic algorithms used in Advanced Encryption Standards are more secure due to 128-bit symmetric keys, if someone sets a password containing both letters and symbols it is very hard for any hacker to find out the code. I use a 128 bit key size password on our workflow management systems and I am sure no one will break it, for a better security I use a random password generator that maximizes the security of the password.

      Best computer ide Borland C compiler ide

    • profile image

      Binh Nguyen 

      3 years ago

      Wow, thanks for the answer!

    • profile image

      i luv maths 

      6 years ago

      amazin I LUV POWERS IN MATHS SOOOOOOOOOO FASANATING

    • Manna in the wild profile imageAUTHOR

      Manna in the wild 

      9 years ago from Australia

      You are welcome.

    • wilderness profile image

      Dan Harmon 

      9 years ago from Boise, Idaho

      Interesting hub - thanks!

    • daniel steve profile image

      daniel steve 

      9 years ago from India

      wow nice article every software engg should know about it man fantastic

    • diogenes profile image

      diogenes 

      9 years ago from UK and Mexico

      "I say twoooo, I say twooo, just me and you!"

      Wasn't that a song? Something like that; maths was my weak subject all my life, I expect that this is a brilliant hub for those with brilliant minds, Well done,,,Bob

    • simeonvisser profile image

      simeonvisser 

      9 years ago

      Every computer programmer should be familiar with the powers of two! They appear in so many places that you should just memorize them.

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