How big is a googolplex?

A googolplex is an extraordinarily large number – so large, in fact, that it is difficult for most people to truly comprehend just how massive it is. To put it in perspective, a googolplex has more zeros than there are atoms in the entire observable universe!

So what exactly is a googolplex? How did this humongous number come about? And just how mind-bogglingly huge is it really? This article will attempt to explain the origins and enormity of the googolplex in terms that our human brains can better understand.

Defining a Googol and Googolplex

A googol is the number 10^100. That is:

 10 100 zeros

Or in numeric form:

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

That’s a 1 followed by 100 zeros – a very big number indeed!

Now, a googolplex takes this to a whole new level. A googolplex is:

10^(googol)

Or in other words:

10^(10^100)

That’s a 1 followed by a googol zeros. Written out, it would look like:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, etc.

Just how many zeros is that? Well, a googol is 10^100 zeros. So a googolplex is 10 to the power of 10^100 zeros. That’s 10^10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 zeros!

Clearly an enormous number that’s essentially impossible to visualize. But let’s try to break it down in more graspable ways.

The Origins of Googol and Googolplex

The words “googol” and “googolplex” were coined by the American mathematician Edward Kasner.

In 1920, Kasner was trying to explain very large numbers to his nine-year-old nephew, Milton Sirotta. He asked Milton to come up with a name for the number 10^100. Milton suggested “googol” as a fittingly massive-sounding word.

Kasner then extended the concept by naming 10^googol as a “googolplex”. He introduced these terms in his 1940 book “Mathematics and the Imagination”.

So the fanciful-sounding names googol and googolplex originated from a child’s imagination and an uncle’s desire to explain abstract mathematical concepts. The numbers themselves are firmly rooted in math, but the colorful language captures public fascination.

How Big is a Googol?

Now that we know where googol and googolplex came from, let’s break down just how enormous these numbers really are.

First, let’s look at a googol. A googol is:

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

That’s a 1 followed by 100 zeros.

To start grasping the size of this, let’s compare it to some everyday quantities:

 1 trillion seconds 31,710 years 1 quadrillion seconds 31,710,000 years 1 quintillion seconds 31,710,000,000 years 1 googol seconds 3.17 x 10^90 years

A googol seconds is an unfathomable length of time – far longer than the age of the observable universe.

 1 light year 9.5 trillion km 1 parsec 30 trillion km Distance across our galaxy 100,000 light years 1 googol light years 9.5 x 10^98 km

A googol light years would span far beyond the edges of the observable universe.

Or let’s look at it in terms of money:

 1 million dollars \$1,000,000 1 billion dollars \$1,000,000,000 1 trillion dollars \$1,000,000,000,000 1 googol dollars \$1 x 10^100

The total value of all goods and services produced on Earth is around \$100 trillion per year. A googol dollars could fund our entire global economy for an inconceivable length of time.

So a googol is magnitudes beyond everyday human scales of time, space, and money. But believe it or not, a googolplex takes this to a totally different level.

The Mind-Boggling Scale of a Googolplex

Let’s now explore just how vast a googolplex is compared to a googol.

A googolplex is defined as 10^(10^100). That’s 10 to the power of a googol, or 1 followed by a googol zeros.

Let’s try to illustrate this scale:

 100 1 century 10,000 All of recorded human history 10^100 (a googol) Incomprehensibly long time 10^10^100 (a googolplex) Exponentially far longer than a googol

If a googol is an unfathomably long time, then a googolplex is an exponentially greater length of time. The difference is simply astronomical.

Or in spatial terms:

 100 meters Length of a soccer field 10,000 km Distance across Europe 10^100 km (a googol km) Far larger than the observable universe 10^10^100 km (a googolplex km) Exponentially greater still

Or we can consider the number of elementary particles in the universe. Estimates place this around 10^80. A googol (10^100) dwarfs this number, while a googolplex (10^10^100) is on an altogether more mind-bending scale.

The difference between a googol and googolplex is so monumentally massive that it’s practically meaningless to the human mind. We have no analogues or experiences that can provide any sense of scale for a googolplex.

Some creative analogies help highlight the monumental size:

– There are estimated to be around 10^80 atoms in the observable universe. Writing a googolplex in full would require more zeros than there are atoms in the universe.

– If every atom in the universe represented a quantum supercomputer performing 1000 operations per second since the Big Bang, the number of total operations performed would still only be around 10^108. Still absolutely miniscule compared to a googolplex.

– If you wrote a googolplex as a digit 1 on each atom in the universe, you would still be nowhere near using up all the zeros.

– If you printed a googolplex zeros on sheets of paper and piled them up, the stack would easily dwarf the size of the observable universe.

These analogies provide some glimpses into the sheer unimaginable scale. A googolplex fully dwarfs any physical scale we can meaningfully conceptualize.

Uses of Googolplex

Given how inconceivably huge they are, googol and googolplex don’t have any direct real-world applications. Their main purpose is to dazzle and amuse with their enormous scale.

Mathematicians and theoretical physicists sometimes use googolplex and even larger numbers to discuss immense quantities and scenarios. These might include:

– Possible values of physical constants in cosmology and quantum gravity models.

– Numbers of parallel universes in speculative multiverse theories.

– Bekenstein bounds – theoretical upper limits to information content in finite volumes of space.

– Complexity classes in computational theory – e.g. problems requiring googolplex time steps.

Googolplex also features in thought experiments about ethics, philosophy, and logic:

– Utility functions in utilitarian ethics – e.g weighing current suffering against googolplex future lives.

– Simulations of reality – could we be living in a simulation created by a googolplex future supercomputers?

– Counterfactuals and possible worlds – e.g. how many possible paths history could have taken.

So while not useful for practical calculations, googolplex and other enormous numbers spark the imagination in math, science, and philosophy. The vastness provokes awe and highlights the limits of human intuition and perception.

Conclusion

A googolplex is an unimaginably vast number – so huge that it defies any physical analogies or intuitions.

It originated as a colorful name to convey abstract mathematical scale, but the mental image of an exponential stack of googol zeros crystallizes the idea of “incomprehensibly big”.

While a googolplex has no practical applications, it stretches our mental faculties and provides fodder for thought experiments about ethics, consciousness, and the physical limits of the universe.

The iconic image of the googolplex highlights the boundaries between human comprehension and the endless scales of mathematics. Our minds can create abstract symbols to represent inconceivable levels of immensity, even if we can’t fully grasp them.

So in the end, assigning a cold hard number is perhaps not the point. The googolplex sparks wonder at the capabilities of human logic and open-ended imagination.