Determining the biggest number in the universe is a fascinating question that has intrigued mathematicians and scientists for centuries. While we may never know the absolute largest number, we can explore some extraordinarily large numbers that have been conceived within mathematics and physics to get a sense of the vast scales involved.
What does “biggest number” mean?
When we talk about the “biggest number”, there are a few key aspects to consider:
- We generally mean the largest possible finite number, not infinity itself.
- The “biggest number” refers to the largest numerical value that can be expressed meaningfully and unambiguously.
- It depends on the system being used – for example, the biggest whole number or the biggest decimal number.
So we have to define some rules and a number system first. Let’s stick with whole numbers (0, 1, 2, 3…) without decimals or fractions.
Why do we need big numbers?
Working with very large numbers, even just hypothetically, turns out to be important in math and science. Here are a few reasons we care about big numbers:
- To test the limits of computational systems and number representation.
- To describe vast quantities like the number of subatomic particles in the observable universe.
- To study mathematical patterns and generalization.
- To explore the relationship between infinity and very large finite numbers.
Thinking about big numbers gets us comfortable working on scales far beyond our everyday human experience. This ability proves useful in fields like physics and cosmology.
What are some very big numbers?
Let’s start by looking at some established large numbers to get a sense of scale:
- Million – 1,000,000
- Billion – 1,000,000,000
- Trillion – 1,000,000,000,000
- Quadrillion – 1,000,000,000,000,000
- Quintillion – 1,000,000,000,000,000,000
- Googol – 10100 = 10 duotrigintillion
- Googolplex – 10googol
A googol is a 1 followed by 100 zeroes. That’s already unimaginably huge. But a googolplex takes it to another level – it’s 1 followed by a googol of zeroes. Just writing out the full value would take more space than there are atoms in the observable universe!
Some huge numbers in physics
To describe aspects of the physical universe, we need to go even bigger. Here are some examples:
- Number of subatomic particles in the observable universe – estimated around 1080
- Number of photons released by the Sun every second – around 1045
- Possible number of distinct quantum states of the early universe – around 1010,000
These begin exceeding a googol and even a googolplex. The last number here has 10,000 zeros – more than all the particles in the observable universe!
Defining bigger numbers
Let’s try systematically building up to bigger and bigger numbers by nesting operations. We’ll start with powers:
| Number | Name |
|---|---|
| 10100 | Googol |
| 1010,000 | 10-tetration |
| 1010100 | Googleplex |
| 101010,000 | 10-super-4 |
We can go another level by making the exponent itself a power, like in googolplex. The naming gets tricky here, but we’re starting to transcend some incredibly huge scales already.
Next, let’s try iterating exponentials:
| Number | Expression |
|---|---|
| 1 | f(1) |
| 10 | f(f(1)) |
| 10,000,000 | f(f(f(1))) |
| 1010,000,000 | f(f(f(f(1)))) |
Where f(x) = 10x. By iterating f multiple times, we can quickly build much larger numbers from smaller ones.
Graham’s number – a famous huge number
Let’s look at one incredibly large number that appears in a mathematical proof, known as Graham’s number. It’s famously bigger than many other well-known huge numbers.
Graham’s number is defined using up-arrow notation. For example:
- 2↑3 = 222 = 16
- 2↑↑3 = 2↑(2↑2) = 216 = 65,536
Following this pattern, Graham’s number is defined as:
g64 = 3↑↑↑↑3
With 64 up arrows! Calculating its exact value is impossible, but just 3↑↑↑3 is already larger than a googolplex. With 64 up arrows, Graham’s number dwarfs everything else we’ve discussed.
Infinite numbers
What about infinity (∞)? Isn’t that the “biggest number” of all?
Infinity is a difficult concept. It represents something without bound rather than an extremely large finite value. There are different sizes of infinity as well – for example, the infinity of real numbers is larger than the infinity of integers.
Infinity isn’t technically a number at all. But it’s a useful idea when thinking about the vastness and unboundedness of mathematics and nature.
The search for bigger numbers
The quest for ever-larger finite numbers continues. Here are some records that have been set and broken over the years:
| Year | Biggest number | Logically defined by |
|---|---|---|
| 1940 | Skewes’ number | Skewes |
| 1970 | Moser’s number | Moser |
| 1980 | Graham’s number | Graham |
Others have found even larger computable numbers since Graham’s in the 1980s. Set theorists continue pushing the boundaries of logical systems to define bigger and bigger numbers through recursive functions and other methods.
The biggest possible number
Is there a limit to how big numbers can get? What’s the biggest finite number that could possibly exist?
There are a few potential candidates:
- The busy beaver number – related to Turing machines
- Rayo’s number – developed by philosopher Agustin Rayo
- The limited principle of omniscience number
However, properties like “the biggest possible number” may simply be undefinable or only exist informally. But exploring these large numbers pushes our understanding of just how far mathematics can take us beyond ordinary human scales.
Conclusion
The question of finding the largest possible number is an endless and perhaps unanswerable one, but it leads to fascinating explorations at the limits of mathematics, computation, and philosophy. While we may never determine the biggest number in any absolute sense, imagining and working with incredibly large numbers allows us to grapple with infinity and comprehend scales far beyond everyday experience. The pursuit of huge numbers has led to important insights in many fields while capturing the human imagination and curiosity about just how far numbers can go.