Very Large Numbers - some way bigger than the Universe itself

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Very large numbers often have to be used when describing the vastness of the Universe. Some are very famous and others you may not be so familiar with, or may need explaining.

Yet other numbers exist that are literally far bigger than the Universe. "Literally" is an appropriate word here, because attempting to write down some of these would need more space than the Universe has to offer. Yes, really! Let's start at the most basic.

The Astronomical Unit

Abbreviated as AU, this is simply the average distance from the Earth to the Sun. It equals 92,955,807.273 miles.

When you consider the scale of the Universe, this distance is very small indeed and therefore is used mainly to describe distances within our Solar System. Thus, the planet Jupiter is approximately 5.2AU from the Sun; the Kuiper Belt begins at roughly 30AU; the NASA space probe Voyager 1 had, by the end of 2013, travelled 125AU from the Sun and is continuing to move away at a rate of 3.5AU per year.

The Light Year

This is the one most people are familiar with. It is the distance that light travels in one year.

In 1976, the International Astronomical Union laid down a System of Astronomical Constants, which are used in this calculation. Thus, one year is defined as exactly 365.25 days - or 31,557,600 seconds. The speed of light is set at 186,282.397 miles per second.

This works out at one light year being 5,878,625,541,248 miles and 722 yards. A light year equals 63,241.077AU.

The Parsec

This unit of distance is the one favoured by astronomers. Calculated using trigonometry, the name is an abbreviation of parallax of one arcsecond. The method used need not be outlined here; suffice it to say that it equals 19,173,511,445,225.5 miles.

It is the equivalent of 3.2615638 light years and 206,264.81 astronomical units.

How Many Stars In The Universe?

Here's where we start getting to very large numbers, so, in order to avoid huge strings of integers, we have to adopt a shorthand, which involves using powers of 10.

Everybody, from their maths lessons at school, is probably familiar with the concept of the "squared" number. Thus 10² is ten multiplied by itself - 100. So, the little number represents how many zeros you put after the 1.

Now, current observations by the Hubble Space Telescope suggest that there are over 100 billion galaxies in the observable Universe. If we suppose that, on average, there are, similarly, 100 billion stars per galaxy, that sets the number of stars in the entire observable Universe as anything between 10²² and 10²⁴.

How Many Atoms In The Universe?

When you consider that a dot like this . contains roughly 125 million atoms, you'd expect the total atoms in the observable Universe to be a very large number indeed. Believe it or not, scientists have been working on estimating what this total might be.

The answer they came up with was anything between 10⁷⁸ and 10⁸². Remember, the little numbers are how many zeros go after the 1.

Yes, these are very large numbers, but what about those that go a step further. I promised you one that was actually too big to fit into the Universe. Well, read on...

The Googol

For those of you who have never heard of this, it is, as you might imagine, the inspiration for the naming of the Internet company, Google. It is a very large number proposed by American mathematician Edward Kasner and is 10¹⁰⁰, or 1 with 100 zeros after it.

As for the name, Kasner asked his nine-year-old nephew Milton Sirotta for a suggestion and this is what the child came up with.

As you can see, by comparing the googol to the numbers of stars and atoms in the Universe in the previous two paragraphs, it is far larger. Is that the limit, though? Of course not!

The Googolplex

Once again, Google took their inspiration from this, calling their main headquarters in California the "Googleplex". "Very large number" seems a little inadequate to describe the googolplex. It is worked out simply as 10 raised to the power of a googol.

This can be represented as 10^googol, or 10^10¹⁰⁰

This is the first example of a number that's too big for the Universe. Carl Sagan has noted that the space required to write down a googolplex would require more room than the Universe has to offer.

If a person were to start to write out a googolplex at the rate of two zeros per second, it would still take far, far longer than the age of the Universe to do it.

The Googolplexian

Yes, it's been taken one step further. Represented as 10^googolplex, or 10^{10^10¹⁰⁰}, it is a number so large as to be pretty meaningless.

Graham's Number

Well, it's about to get even more meaningless. This very large number, discovered by American Mathematician Ronald Graham, dwarfs even the googolplexian and such is its size, can't be represented by regular mathematical notation. Instead, something called "Knuth's up-arrow notation" is used. It goes like this:

Starting with the simplest example - 3 ↑ 3 just means 3 multiplied by itself three times, i.e. 3 cubed, which equals 27.

Then, 3 ↑ ↑ 3 indicates 3 multiplied by itself 27 times, which equals 7625597484987. You see, with just two arrows, how big the number is already.

Now with 3 ↑ ↑ ↑ 3, it involves multiplying the 3 by itself 7625597484987 times and by now, we already have a number whose digits would fill more than the entire observable Universe. But things don't stop there.

3 ↑ ↑ ↑ ↑ 3 is the next step, using the same procedure, producing a result many times larger than the previous one (which already fills the Universe). This is known as g1.

If we then make g1 the number of arrows between the 3s, the result is called g2. Keeping going in this way, we finally arrive at g64, where the number of arrows is g63.

This is Graham's number.

TREE(3)

Now, this is a number that is supposed to dwarf even Graham's Number. I've looked into it and, not being a mathematics professor, can't make head nor tail of it.

Expressions such as "recursive functions" and "homeomorphically embeddable" are banded about, so you see what I mean. Given that, apparently TREE(1)=1 and TREE(2)=3, it seems quite surprising that TREE(3) should suddenly be far greater than Graham's Number, but there it is.

Just so you know, one way of expressing it is:

TREE(3)>{3,6,3[1[1¬1,2]2]2}

That's enough of that, eh?