Can a number be bigger than anything you can count?

It’s Talking Math with Your Kids week here on OMT.

We’ll get started with a favorite topic: large numbers.

My son Griffin was thinking about large numbers in the car the other day. He was trying to figure out what good it is to have a number (here, googolplex, which for the record is 10^{10^{100}}) that is larger than anything you can count.

Griffin: If you put all the things [in the world] together, would that make googolplex?

Me: No.

G: Even if it’s nanoinches?

Me: Nope. Still not googolplex.

G: Even if it’s half-nanoinches?

G: Even if it’s all of the seconds of the world being alive?

Me: Nope.

G: Even if all the seconds of the universe existing?

Me: No.

I love the developing proportional reasoning embedded in Griffin’s questions.

For each example, he scales it up when his first try doesn’t do it.

If nanoinches don’t work, surely half-nanoninches will! Plenty still to learn about orders of magnitude, I’m afraid.


2 responses to “Can a number be bigger than anything you can count?

  1. Even though Griffin might have plenty more to learn about orders of magnitude, this could easily be a discussion I have with my high school kids, and Griffin would be holding is own. For starters, he realizes that dividing by 1/2 is the same as multiplication by 2, which is nontrivial, even in 11th grade. Second, he realizes that doubling things does make them grow quickly, so it’s not a bad strategy. In fact, it won’t take him any more then about 3 doublings to bump up an order of magnitude (since \frac{1}{\log(2)} \approx 3.32). He can hardly be blamed for having a tough time grasping how many doublings would be involved in the case of a googolplex. Third, this question of “What good is it to have a number that…?” is rampant in my curriculum as we move through, e.g., irrational and imaginary/complex numbers. So yeah, Griffin’s a stud. I want to teach your kids some day.

    Here’s something that might foster further discussion.

  2. I have this kind of discussion all the time. I like the current estimation that all the particles in the universe add up to ~10^80 and so even just googol is not very useful from a counting perspective.

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