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

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 2^{64}.

2^{64} is a doubling of 2, 64 times.

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

2^{0} =
1

2^{1} =
2

2^{2} =
4 = 2 x 2

2^{3} =
8 = 2^{2} x 2 = 2 x 2 x 2 x 2

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

2^{8} =
256

2^{10} =
1024

2^{16} =
65536

and now some larger powers of two.

2^{32} = 4,294,967,296
This is in the same ballpark as the human population of the
Earth which is currently approaching 7 billion.

2^{42} = 4,398,046,511,104
This is in the range of the number of cells in the human body.

2^{64} = 18,446,744,073,709,551,616

2^{96} = 79,228,162,514,264,337,593,543,950,336

2^{128} = 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 2^{64} does not look much different in symbols to 2^{128}, 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.

a^{b} where b=c^{d} where d=e^{f}...

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 2^{128}. Hopefully when you see this number you will have a reasonable idea what it means.

## Comments

**bilal** on February 27, 2020:

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.

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**Binh Nguyen** on March 02, 2017:

Wow, thanks for the answer!

**i luv maths** on May 22, 2013:

amazin I LUV POWERS IN MATHS SOOOOOOOOOO FASANATING

**Manna in the wild (author)** from Australia on March 06, 2011:

You are welcome.

**Dan Harmon** from Boise, Idaho on March 06, 2011:

Interesting hub - thanks!

**daniel steve** from India on January 11, 2011:

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

**diogenes** from UK and Mexico on January 10, 2011:

"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** on January 10, 2011:

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