Everything Everywhere Daily

Episode Summary: "Insanely Ridiculously Absurdly Large Numbers"

Host: Gary Arndt
Date: January 16, 2026

Episode Overview

In this episode, Gary Arndt explores the concept of extremely large—but still finite—numbers, discussing how humans and mathematicians approach, express, and attempt to comprehend numbers that far exceed anything encountered in everyday life or even the observable universe. The episode covers the evolution of large number notations, naming conventions, and the unimaginable size of certain mathematical numbers, culminating in Gary's own thought experiment about the number of possible photographs with a digital camera.

Key Discussion Points & Insights

The Ever-Growing Need for Larger Numbers

• Human Understanding of Numbers:

  • Early societies had limited need for large numbers. “Some small-scale or historically isolated societies didn't develop words for numbers beyond 2 or 3 because their daily lives didn't require exact counting.” (04:15)
  • With civilization’s advancement, the need for expressing and counting large quantities grew.

• Modern Examples:

  • Modern finance and science forced us to regularize huge numbers: companies valued in trillions, US national debt approaching $40 trillion, etc. (05:30)
  • The scale of physics (atoms) and astronomy (stars, universe) requires numbers magnitudes beyond everyday experience.

How We Write and Name Large Numbers

• Exponential Notation:

  • Scientists compactly express large numbers with exponents.
    • Example: “A million is 10 to the power of 6. A billion is 10 to the 9, a trillion is 10 to the 12th, and so on.” (07:30)
    • Each increase by one exponent is an “order of magnitude.”
    • Scientific notation offers further clarity: 2,300,000 as 2.3 × 10^6.

• Naming System for Large Numbers:

  • English (short scale): million, billion, trillion, quadrillion, etc. Each new "-illion" increases by three orders of magnitude.
  • Prefixes drawn from Latin/Greek roots for numbers beyond billion.
  • Fun Fact: "Some terms, like centillion, are defined more by convention than mathematical precision, and the pattern becomes unwieldy at high values.” (09:50)

• SI (International System) Prefixes:

  • Used to standardize powers of ten for units: kilo (10^3), mega (10^6), giga (10^9), tera (10^12), and up to yotta (10^24).

Relating Large Numbers to the Universe

• Tangible Examples:

  • Age of the universe: ~13.8 billion years, or “roughly 4 × 10^17 seconds.” (12:00)
  • Grains of sand on Earth: estimated at 10^18 to 10^20.
  • Stars in the observable universe: between 10^22 to 10^24.
  • Atoms in the observable universe: around 10^80, known as the Eddington number.

• Key Insight: “The Eddington number of 10 to the power of 80 is about as big as we can go as far as counting actual things. However, mathematically, we're just getting started.” (13:30)

Stepping Into the Realm of Absurdly Large Numbers

The Googol and Googolplex

• “If you can express large numbers as exponents, we can create even bigger numbers… One that most of you are familiar with, even if you don't know it: 10 to the power of 100, a googol.” (15:01)
  • Named by a nine-year-old nephew of mathematician Edward Kasner.
  • “The company Google was named after the number, although they're spelled differently.” (15:45)
• Googolplex: 10 raised to the power of a googol.
  • “A googolplex is so large that even if every particle in the universe were one bit in a massive computer, you couldn't even express the number in binary form.” (16:05)
  • Writing it out: more books than atoms in the universe.

Power Towers and Arrow Notation

• Iterated Exponentiation:
  • Exponents on exponents lead quickly to incomprehensible numbers.
  • Tetration: Repeated exponentiation (e.g., 10^^3 means 10^(10^10)).
• Knuth’s Arrow Notation:
  • Single arrow: exponentiation (a^b)
  • Double arrow: tetration (a^^b)
  • “2 double up 4 is 2 raised to the power of 2 raised to the power of 2 raised to the power of 2, which equals 65,536.” (18:44)

The True Monsters: Graham’s Number and TREE(3)

• Graham’s Number: Emerges from Ramsey theory; immense, defined recursively, incomprehensible with standard notation.
  • “Its last few digits are the only part that can actually be meaningfully discussed in decimal form. Yet it's still finite and precisely defined.” (21:20)
• TREE(3):
  • “Vastly larger than Graham’s number… the resulting value grows faster than almost any function commonly encountered in all of mathematics.” (22:04)

Gary’s Own Insanely Large Number Thought Experiment

• How Many Possible Photos With a Digital Camera?

  • Hypothetical: 50 megapixel camera, 64,000 color options per pixel.
  • Calculation: 64,000^(50,000,000)
  • Or in exponential notation: 10^240,309,000
  • “Even if every atom in the universe stored a unique image, and the universe were recreated again and again trillions of times, you still wouldn't come close to exhausting the number of possible images.” (24:08)
  • Most images would appear random—just noise. Only an infinitesimal number would look like recognizable photos.

• Key Conclusion:

  • “The total number of possible photos isn’t infinite. It’s finite. Stupidly, overwhelmingly large, but finite nonetheless.” (25:08)
  • Gary jokes: “If any mathematicians out there want to use this in a paper, please give me co-author credit so I can earn a Paul Erdős number to go along with my Kevin Bacon number.” (25:52)

Notable Quotes & Memorable Moments

• "Let me start by saying that this episode is not about infinity... This episode will be about finite numbers. Absurdly large finite numbers, but finite numbers nonetheless.” (03:55)
• "The Eddington number of 10 to the power of 80 is about as big as we can go as far as counting actual things. However, mathematically, we're just getting started.” (13:30)
• “A googolplex is so large that even if every particle in the universe were one bit in a massive computer, you couldn't even express the number in binary form.” (16:10)
• “Insanely, ridiculously, absurdly large numbers can be difficult to wrap your head around, and that's okay, because our brains truly cannot grasp their size.” (26:13)
• “Despite their massive sizes, they are not infinite, and that many things we may think of as being infinite are actually just really, really, really, really big.” (26:22)

Timestamps for Key Segments

• [03:55] – Distinction between infinity and very large finite numbers
• [07:30] – Exponential notation and scientific naming conventions
• [12:00] – Real-world massive numbers and the Eddington number
• [15:01] – The googol, googolplex, and their origins
• [18:44] – Iterated exponentiation and Knuth’s up arrow notation
• [21:20] – Graham’s number and TREE(3)
• [24:08] – Gary’s digital camera photo thought experiment
• [25:08] – Key takeaway about the finiteness of gigantic numbers

Final Takeaway

Gary closes by emphasizing the importance—and the limits—of extremely large numbers in mathematics, science, and even creativity. While their size is beyond human comprehension, the key lesson is the difference between truly infinite and simply enormous quantities. Even the most massive numbers discussed are finite and precisely defined, illustrating the imagination and rigor of human mathematical thought.