What Is the Biggest Number?
You’ve probably gotten into these contests as kids before, seeing who can spit out the biggest number. Everyone loves to pull out infinity, or the fabled “infinity plus one.” Maybe if you were inclined to do so, you pulled out the googol or the googolplex. Smaller than infinity, but really big numbers each. But as you probably figured, infinity doesn’t really count. So that aside, what is the biggest number?
Why Doesn’t Infinity Count?
Infinity exceeds the boundaries of any real or natural number. That is to say, it’s actually a lot easier to consider infinity as not a number. Better to think about it as an abstract concept–especially since infinity relies on our understanding of how big other numbers are.
“Infinities” can also be different sizes, depending on the infinite set you’re manipulating. Bearing in mind that an expression like “infinity plus one” is still infinity. It sounds weird, but just keep that in mind. We’re going to be toying with infinity in this post, but with the stipulation that the answer to “what is the biggest number” is not necessarily just spitting the word infinity out at you.
We’re also going to set out that when we talk about the “biggest number,” we’re dealing with something called cardinality. It just means numbers you can count to. Like there are 26 letters in the English alphabet. It seems rather pointless to go any further.
Further reading: What Is Infinity?
How High Can A Human Count?
Well, there is an absolute mad lad who spent 89 days straight just counting for 16 hours each day. His name is Jeremy Harper and he made it to 1 million. So if you want to go out and beat a world record, go nuts.
If you take the average human life expectancy of about 78 years and round it up to 80, and assume you count for 24 hours each day every day and each number takes you exactly 1 second to recite, you’d only make it to 2,523,000,000. Which is somewhere in the realm of 2.5 billion seconds. And realistically, you’d never be able to count that high since you have to eat and sleep.
Really all we did was calculate how many seconds the average person lives, so enjoy that existential crisis. Oh, and also the fact that that number is about 1% of Jeff Bezos’ net worth.
So how high can a human count? Well. Less than 2.523 billion.
If you assume wakefulness for 16 hours each day and that you only start counting when you’re, say, 10 years old, you’re counting for about 70 years. You get about 1,471,680,000 if you still assume it takes 1 second to recite each number.
So less than 1.5 billion, really.
How High can the Universe Count?
Okay, we know. Technically, the universe cannot count. At least the way we understand it. We’ll get into it.
So then let’s instead take the smallest unit of something and see how many of that thing fit inside the universe? The smallest unit is the Planck length, which is the smallest something gets before anything collapses in on itself and becomes a black hole.
Trivia Fact: If you wanted to turn Earth into a black hole you’d have to compress it to the size of about a peanut. Take that, peanut allergies?
Anyway, let’s take the Planck length and make it 3D–that gives us a cube with side lengths equal to Planck lengths. Physics already thought of that, that’s a Planck volume. Still the smallest thing. So small, 100 quintillion of these little buggers can fit inside a single proton. Which, mind you, is a single hydrogen atom, or otherwise just a part of a larger atom (like say, uranium).
Alright, so how many of these little Planck cubes could we fit inside the observable universe?
You get 10^185. Which is 1 followed by 185 zeroes.
What Does that Even Mean?
We’ve reached a point where the numbers are so big there’s nothing to compare them to. Luckily, there’s an expression for that. We mentioned it earlier, it’s googol. No, not the company that is collecting all of your data at all times. The number.
Anyway, 1 googol is equal to 10^100, 1 followed by 100 zeroes. Which is big.
But we just spat a number bigger than that out at you.
Thus enters the googolplex. A 1 followed by 1 googol of zeroes. That number absolutely dwarfs 10^185.
So yeah, googolplex is probably the biggest named number we have that will never be relevant.
So you’ve probably been sitting here this whole time going “well we can just add 1 to any of the numbers discussed here and get one bigger.” Which is… fair.
But you can always just keep adding +1 to everything, which is where aleph numbers come in. Because you can always add +1 to every natural number, you get an infinite set. This number therefore contains every number we’ve talked about so far. It also contains those numbers multiplied by those numbers and so on. It’s named aleph-null, and it’s the smallest “cardinal infinity” you can get. Remember that thing about infinities sort of being bigger than each other?
By the way, aleph numbers are different from infinity. To a layperson they might as well be, but honestly, a googolplex is a functionally infinite number for people like us. If we had a googolplex of anything right now, we’d never run out in our lifetimes.
Even if you get semantic and say “I want a googol years of life,” you’d lose. That exceeds the life expectancy of the universe by a country mile. No really, the universe might only go to 10^34 years. We break that down in the post below.
Further Reading: When Will the Universe End?
Aleph null and its siblings are expressions that represent sets of numbers, and thus the cardinal values and sizes of those sets. It’s still meant to represent a value. Infinity is used to express an extreme limit.
Aleph-null is also referred to as aleph-zero, which means you can have different aleph numbers. But aleph-null plus aleph-null is still aleph-null. That’s because aleph-null is still a cardinal number, but it expresses every cardinal number. So every cardinal number plus every cardinal number… is still a cardinal number.
Honestly we should have stopped at aleph-null. But we’re committed now.
If you want to count past aleph-null, you need to exit the realm of natural numbers. We have to go beyond the rules we set out at the beginning of this post. We have to find ordinal numbers. First one of those is omega. After that is omega+1 and so on. That infinite set is aleph-one.
You can also take power sets of aleph-null, which you can also keep doing that will produce more “larger” aleph-nulls.
These numbers are kind of nightmarish. Here’s more nightmare number sequences.