Monday, February 28, 2011

The Highest Number


What is the highest number there is? Let's try and find out by working our way up. To save myself from having to write down a lot of digits, let me first explain the scientific notation:
  • 1 × 102 = 1 × 10 × 10 = 100
  • 1 × 103 = 1 × 10 × 10 × 10 = 1000
  • 1 × 104 = 1 × 10 × 10 × 10 × 10 = 10000
Basically, powers of 10 can be used to define the number of zeros that follow the ‘1’. So here's some large numbers you might know: a million, a billion, a trillion, a quadrillion... These can be written respectively as 106, 109, 1012 and 1015. You might knew these first few, but that list actually goes on for a while. Take for example a vigintillion, which is 1063. Are there any higher?

Of course! How about the googol? It is notated as 10100. That's a ‘1’ with a hundred zeros! Is this the largest number? Nah, we can put at least the centillion (10303), septuagintacentillion (10513) and the millinillion (103003) on the table.

We need to do better than this, so let's push it into crazy territory. A googolplex is 10googol — yes, that's 10 to the power of a googol. A ‘1’ with a googol zeros. 1010100. Fine! It is a 1 with 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 zeros. So as you can see I just wrote out a googol. I cannot write out a googolplex: there is not enough room in our universe to do so. See, the number of atoms in the observable universe is estimated to be around 1080. This means that writing down a googolplex requires far more zeros than there are atoms in the universe!


The smallest definable distance is the Planck length: 10-35 or ten undecillionth of a meter. This is many, many times smaller than an atom, even a proton. In fact, anything smaller than the Planck length has no meaning in physics. There's an estimated 1082 cubic meters in the observable universe. Divide that by the Plank length cubed, and you will find there are 10187 Planck volumes in the universe. In comparison to a googleplex, that's still nothing. I can safely say that a googolplex is unimaginably more times larger than the total volume of the observable universe, expressed in the smallest possible unit of space!

If a googolplex is that astronomically big, could it be the highest number? No way — there are far, far higher numbers. Like Graham's Number. That number is so ridiculously large that I cannot even write it down using the scientific notation, like I did a googolplex. Even using ‘power towers’ (10 to the power of 10 to the power of 10 to the power of 10...) is no good. There would be too many levels! Yet, this number was used in a mathematical proof. Not by writing it down though. Even Graham himself does not know how many digits it has. Or what the first digit of it is. Only some of the rightmost digits be can be extrapolated. Still, if you thought the scale of the googolplex was absurd, Graham's Number is exponentially more so.

So is this it then? Hah! Not even close. You might have already realized that if I claimed this to be the highest number, you could just add 1 to it. And ta-da, you would have your new highest number. But we could add 1 to that again as well...

The truth is, there is no such thing as the highest number. Because you can always increase that number and the math would still work. It is this realization that led to the concept of infinity, or ∞.

In the mathematical sense, infinity is the highest number, yet it cannot be assigned a value. Those freaky mathematicians still manage to do calculations with it though!
Infinity has some bizarre properties too. But those are better saved for another post...

10 comments:

  1. Too many numbers are accountable in this universe. Can infinity be multiplied by infinity?

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    Replies
    1. Sure you can multiply infinity by infinity but the answer still infinity

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  2. Why can't numbers be divided by 0?

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    Replies
    1. Technically any number divided by zero is asking how many times can nothing going to that number so the answer would be infinity. 1/0 = ∞

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    2. There is some form of math that allows for dividing by zero.

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  3. Any I n f i n i t y -doesn't exist, happen, become, operate, effect, etc.

    Why? Would take me forever to explain...

    Eternity -for example- has no time, nor anything involving any infinity...except via grand~illusory imaging...

    So...IF there is Eternity -if we can express that... what do we do in Eternity?? Well "doing"
    is another impossible in Eternity! Being IS doable...and doing isn't needed. Just BEING perfect!

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  4. There are numbers larger than infinity such as (omega+1).

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    Replies
    1. Omega+1 just comes after alef null, it isnt bigger, however

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  5. in reality a number can be divided by zero because if you divide something 0 times than the answer would just be the same

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  6. Actually, that is not true. "The answer to a division problem is 'How Much One Gets.'" Think of this concept. 15 divided by 5. If there are 15 cookies divided by 5 people, one person gets 3. Answer: 3. If 8 cookies are divided by 16 people, each person gets half a cookie, so one person gets one half. Sometimes answers are infinitely repeating decimals, but it's still a representation of how much one gets. Dividing by zero does not answer and cannot answer how much one gets. Logically, 5 cookies divided by 0 people... how much does one get? That question cannot be answered. There aren't any people. With no people, one person cannot be represented. As soon as you put in one person and try to say that person gets none, you've changed the basis of the equation. You cannot, cannot, cannot answer the question of "How much does one get" when there is not one to begin with.

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