Hyperoperations and even bigger numbers | Data structures in Mathematics Math Foundations 179

A powerful approach to exploring big number arithmetic is to extend the notion of arithmetical operation. By considering hyperoperations starting with +,x,^ and then triangle, square etc we can ramp up arithmetic considerably. We can in fact inductively define an operation *_k for any natural number k.

But then an interesting new possibility emerges, which opens up a whole new world of arithmetic--a rather lofty and hard to comprehend world, to be sure! We can consider a diagonal limit operation.

This video is an introduction to the remarkable successor limit hierarchy which allows us to go far beyond our usual arithmetic! But don't worry, we don't actually get to infinity. That's because there is no such place.

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[FareSkwareGamesFSG]

I was so curious about this topic, I'm glad I found this video! Appreciate the effort and organized explanation. Thank you!

Also, it's unbelievable how large these numbers actually are, yet still finite. 🤯

[juliusbourodimos]

Wow it gets so huge so quick. I can’t calculate past the *4, even if I use smaller numbers.

For example, Let’s say I want to solve m *5 n, where m=2, n=3, and *5 is the 5th operation. You get to the point where you have to raise 2 to the power of 2, a whopping 256 times, which is pretty much impossible to calculate with my personal calculator. The magnitude truly is incomprehensible!

Great explanation too, this helped me a lot I’m only a Junior in huh school and i just started calculus, but your explanations were still straightforward and easy to follow.

Also, one thing I noticed was the behavior of the operations when m and n are both 2. If m=2 and n=2, then all the operations (no matter how far you go) will have an answer of 4. For example, 2+2=4, 2x2=4, 2^2=4, 2△2=4, and it continues on and on!