Here’s an interesting discussion about Graham’s number, the proof in which it appeared (it was used as a bound), and some disentangling of truth and semi-truth:

Certainly we can construct larger numbers. Graham’s number is special in that it actually came up as the solution (or an upper bound) to a problem Graham was trying to solve.

Thank you π
That’s good for Graham, I guess – always satisfying to get to an answer :).
Doesn’t it depend on the problem though? Isn’t it possible to ask new questions which result in bigger answers? What was the biggest number before Graham came up with his?

Did you happen to look at the link I provided, Jesska? It’s not quite true that Graham’s number was created the way the anecdote has generally been told. The actual bound in the proof was smaller than Graham’s number. But, of course, it still IS an upper bound. π

I did….but I didn’t understand it π¦

Jesska, here’s the relevant quotation, which is non-technical:

“I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he’d made up Graham’s number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it’s bigger, so it’s still an upper bound!”

In other words, the actual number in the proof wasn’t as large as the now-famous Graham’s number, but the actual upper bound Graham DID use in that proof is harder to explain than is Graham’s number.

Now, the issue of “understanding” Graham’s number is more a matter of how many times you want to try to wrap your head around a way to build some number that is growing at rates so dizzyingly fast that you lose track of what’s happening pretty quickly (or lose concentration or motivation or all of these things).

It’s like googolplex. A googol is 1 followed by one hundred zeros. That’s a ludicrously big number, but we have scientific notation to write it briefly, namely 10^100.

A googolplex is 1 with a GOOGOL of zeros. That is, 10^googol or reverting to the previous representation of a googol, 10^(10^100). As big as a googol is, a googolplex is just ridiculously big.

Borrowing from Wikipedia, “A typical book can be printed with 106 zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 1094 such books to print all the zeros of a googolplex (that is, printing a googol of zeros). If such a book would weigh 100 grams, all of them would weigh 1093 kilograms. In comparison, Earth’s mass is 5.972 x 1024 kilograms, and the mass of the Milky Way Galaxy is estimated at 2.5 x 1042 kilograms.”

Now, Graham’s number is so much bigger than a googolplex as to make comparison pretty much meaningless. You and I would not be able to wrap our heads around it in any meaningful way, and yet if you patiently go back to how this number is CONSTRUCTED, it’s not really “hard” to follow the steps. It certainly doesn’t require any advanced mathematics, just the willingness to engage in a process that for almost anyone has no useful purpose.

Multiplication: Not repeated addition!

This is the coolest and simplest “proof” as to why this doesn’t work I’ve seen, though there are others (fractions, I think, are another good example): Let’s multiply -2 by -6.

Well, let’s use repeated addition. -6 added together two times is…minus 12.

That doesn’t work.

The best term I’ve seen used to define what multiplication actually is is “The scaling of numbers”.

Sorry, your article is great! I had just read up a little on this awhile back and I think it’s too cool a factoid not to mention.

You should start counting at 0, not -6. Add 12 to 0 and you get the correct answer: 12. Add 12 to -6 and you get your answer, 6. Multiplication is, in this case at least, repeated addition.

To clarify more, what you are doing is: -6 + (-2 x -6), not simply -2 x -6

“Multiplication Is Repeated Addition” (aka MIRA) is one of my favorite cans of worms. I blogged about it multiple times, initially thinking that Stanford’s Keith Devlin had gotten himself in over his head regarding teaching K-5 mathematics when he wrote a series of columns on this issue, then changing my mind rather thoroughly:

Your and Devlin’s assertions that the Peano axioms are “not, as are the number systems already mentioned, a descriptive axiom system that tells us how to work within it” seems to me to be completely backwards. If you handed me the complete ordered field axioms and two real numbers and asked me to multiply them, I’d have no idea how to do it, unless one of them was 1 or 0. In contrast, if you gave me the Peano axioms and two (standard) natural numbers, I would know exactly how to multiply them — by repeated addition, which is in turn defined in terms of repeated succession.

Canary, Devlin is the one who makes that assertion about the Peano postulates. I repeat it to make the point that he’s not recommending that K-5 teachers start teaching it to their students. That’s the extent of my interest in Peano regarding this particular issue: people trying to put Devlin in the category of some “New Math” folks from the ’60s, which he decidedly isn’t.

As to what an adult would do with Peano, it’s an argument in which I have no skin. There’s no instance of the phrase “descriptive axiom system” online other than in Devlin’s columns and other references to them, so it’s debatable what that “officially” means. You could, of course, write to Professor Devlin.

I think we should start with:

Consider the set N_0 such that:
0 is in N,
+ is a function from N to N\0 such that + is an injection…

And do away with all of that arbitrary one, two, three, and base ten crap we teach to kindergartners these days.

Terrific idea, Doug. You get to work on that. While you’re busy, you might (not) want to check out the work of V. V. Davydov on a primary grade mathematics curriculum that starts with measurement rather than counting. Versions of it are still in use in Russia, and there are a couple of mathematicians and mathematics educators here in the US who’ve developed materials based on Davydov’s work (e.g., Susan Addington in California, the late Jean Schmittau in New York, several folks at the University of Hawaii, Peter Moxhay in Maine, etc.).

If anyone is interested in a constructive conversation on these issues, I’d be happy to engage. More of the predictable sarcasm will be ignored.

It’s a poor way of looking at it, but you can still get to the correct answer using if you still want to consider it in that fashion.

So you can see the negative applies to both -6 values, and it is still straight forward addition/subtraction -(-6)-(-6)=12

Or if it helps you visualise it think in terms of adding/subtracting vectors. A deceleration twice as strong in the opposite direction is the same thing as an acceleration (x2)

I have a couple of blog post about sharing Graham’s number with kids. It was fun topic and the boys really enjoyed it. Playing around with the last digits was really neat.

I do not know the details, but there’s a citation on MathOverflow that says that Graham’s Number is not the largest number ever used in a paper. Now, maybe that’s different from “used in a proof”? Anyway, I thought it relevant to pass along. Here’s the quote from the MathOverflow page:

Though irrelevant to Tim’s question, Graham’s number is small potatoes compared to some of the numbers cooked up by Harvey Friedman, e.g., his paper Long finite sequences, JCT(A) 95 (2001), 102-144.

I question Mr. Goldenberg’s August 16th 2016 posting to Jesska about describing a Googleplex (from Wikipedia’s page). If a book contains 10^106 number of zeroes and you multiply it by 1094, isn’t that just 10^106 times (approx) 10^3? 1st Law of Exponents tells us that this is only 10^109 power. There is not enough paper or space on our planet to actually write out 10^10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

Why stop at the 64th number in the chain?

No need to stop if you don’t want to, of course.

Here’s an interesting discussion about Graham’s number, the proof in which it appeared (it was used as a bound), and some disentangling of truth and semi-truth:

http://mathoverflow.net/questions/117006/reconstructing-the-argument-that-yields-grahams-number

Certainly we can construct larger numbers. Graham’s number is special in that it actually came up as the solution (or an upper bound) to a problem Graham was trying to solve.

Thank you π

That’s good for Graham, I guess – always satisfying to get to an answer :).

Doesn’t it depend on the problem though? Isn’t it possible to ask new questions which result in bigger answers? What was the biggest number before Graham came up with his?

Did you happen to look at the link I provided, Jesska? It’s not quite true that Graham’s number was created the way the anecdote has generally been told. The actual bound in the proof was smaller than Graham’s number. But, of course, it still IS an upper bound. π

I did….but I didn’t understand it π¦

Jesska, here’s the relevant quotation, which is non-technical:

“I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he’d made up Graham’s number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it’s bigger, so it’s still an upper bound!”

In other words, the actual number in the proof wasn’t as large as the now-famous Graham’s number, but the actual upper bound Graham DID use in that proof is harder to explain than is Graham’s number.

Now, the issue of “understanding” Graham’s number is more a matter of how many times you want to try to wrap your head around a way to build some number that is growing at rates so dizzyingly fast that you lose track of what’s happening pretty quickly (or lose concentration or motivation or all of these things).

It’s like googolplex. A googol is 1 followed by one hundred zeros. That’s a ludicrously big number, but we have scientific notation to write it briefly, namely 10^100.

A googolplex is 1 with a GOOGOL of zeros. That is, 10^googol or reverting to the previous representation of a googol, 10^(10^100). As big as a googol is, a googolplex is just ridiculously big.

Borrowing from Wikipedia, “A typical book can be printed with 106 zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 1094 such books to print all the zeros of a googolplex (that is, printing a googol of zeros). If such a book would weigh 100 grams, all of them would weigh 1093 kilograms. In comparison, Earth’s mass is 5.972 x 1024 kilograms, and the mass of the Milky Way Galaxy is estimated at 2.5 x 1042 kilograms.”

Now, Graham’s number is so much bigger than a googolplex as to make comparison pretty much meaningless. You and I would not be able to wrap our heads around it in any meaningful way, and yet if you patiently go back to how this number is CONSTRUCTED, it’s not really “hard” to follow the steps. It certainly doesn’t require any advanced mathematics, just the willingness to engage in a process that for almost anyone has no useful purpose.

Multiplication: Not repeated addition!

This is the coolest and simplest “proof” as to why this doesn’t work I’ve seen, though there are others (fractions, I think, are another good example): Let’s multiply -2 by -6.

Well, let’s use repeated addition. -6 added together two times is…minus 12.

That doesn’t work.

The best term I’ve seen used to define what multiplication actually is is “The scaling of numbers”.

Sorry, your article is great! I had just read up a little on this awhile back and I think it’s too cool a factoid not to mention.

(Sorry, unthinking mistake in my comment – negative 6 added together NEGATIVE two times would be 6. Still not the right answer.)

You should start counting at 0, not -6. Add 12 to 0 and you get the correct answer: 12. Add 12 to -6 and you get your answer, 6. Multiplication is, in this case at least, repeated addition.

To clarify more, what you are doing is: -6 + (-2 x -6), not simply -2 x -6

“Multiplication Is Repeated Addition” (aka MIRA) is one of my favorite cans of worms. I blogged about it multiple times, initially thinking that Stanford’s Keith Devlin had gotten himself in over his head regarding teaching K-5 mathematics when he wrote a series of columns on this issue, then changing my mind rather thoroughly:

http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html

Your and Devlin’s assertions that the Peano axioms are “not, as are the number systems already mentioned, a descriptive axiom system that tells us how to work within it” seems to me to be completely backwards. If you handed me the complete ordered field axioms and two real numbers and asked me to multiply them, I’d have no idea how to do it, unless one of them was 1 or 0. In contrast, if you gave me the Peano axioms and two (standard) natural numbers, I would know exactly how to multiply them — by repeated addition, which is in turn defined in terms of repeated succession.

Canary, Devlin is the one who makes that assertion about the Peano postulates. I repeat it to make the point that he’s not recommending that K-5 teachers start teaching it to their students. That’s the extent of my interest in Peano regarding this particular issue: people trying to put Devlin in the category of some “New Math” folks from the ’60s, which he decidedly isn’t.

As to what an adult would do with Peano, it’s an argument in which I have no skin. There’s no instance of the phrase “descriptive axiom system” online other than in Devlin’s columns and other references to them, so it’s debatable what that “officially” means. You could, of course, write to Professor Devlin.

I think we should start with:

Consider the set N_0 such that:

0 is in N,

+ is a function from N to N\0 such that + is an injection…

And do away with all of that arbitrary one, two, three, and base ten crap we teach to kindergartners these days.

Terrific idea, Doug. You get to work on that. While you’re busy, you might (not) want to check out the work of V. V. Davydov on a primary grade mathematics curriculum that starts with measurement rather than counting. Versions of it are still in use in Russia, and there are a couple of mathematicians and mathematics educators here in the US who’ve developed materials based on Davydov’s work (e.g., Susan Addington in California, the late Jean Schmittau in New York, several folks at the University of Hawaii, Peter Moxhay in Maine, etc.).

If anyone is interested in a constructive conversation on these issues, I’d be happy to engage. More of the predictable sarcasm will be ignored.

It’s a poor way of looking at it, but you can still get to the correct answer using if you still want to consider it in that fashion.

to calrify

2×6 = 0 + (+6) + (+6) = 12

2x-6= 0 + (-6) + (-6) = -12

-2x-2= 0 – (-6) – (-6) = 12

So you can see the negative applies to both -6 values, and it is still straight forward addition/subtraction -(-6)-(-6)=12

Or if it helps you visualise it think in terms of adding/subtracting vectors. A deceleration twice as strong in the opposite direction is the same thing as an acceleration (x2)

I’ve bumped into Graham’s number before. A good thing to know is that Ron Graham is a very skilled juggler. He’d have to be, don’t you think? π

Bahahahaha. I read the whole post and still ?????? I knew better since being mathematically challenged is hereditary. Its a family problem.

I wouldn’t worry overly much. There is virtually no chance that you’ll ever need to understand Graham’s number.

Rightπ

I have a couple of blog post about sharing Graham’s number with kids. It was fun topic and the boys really enjoyed it. Playing around with the last digits was really neat.

https://mikesmathpage.wordpress.com/2014/04/12/an-attempt-to-explain-grahams-number-to-kids/

https://mikesmathpage.wordpress.com/2014/11/28/the-last-4-digits-of-grahams-number/

https://mikesmathpage.wordpress.com/2015/10/23/grahams-number-and-skewes-number/

Wow that number’s HUGE! Nice post π Very interesting…..

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I guess it kind of depends on what you include as a “number,” but I like the Church-Kleene ordinal. https://en.wikipedia.org/wiki/Large_countable_ordinal

Here’s an interesting even bigger – waaaay bigger – number: https://en.m.wikipedia.org/wiki/Kruskal%27s_tree_theorem

Love your blog!

I want more! I want more! XD

This is an awesome blog! I will let my daughter know about it!

Have you thought of doing some with the distance formula using square roots?

This math concept reminded me of God!

Can you recommend any good sites that teaches algebra and precalculus?

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I do not know the details, but there’s a citation on MathOverflow that says that Graham’s Number is not the largest number ever used in a paper. Now, maybe that’s different from “used in a proof”? Anyway, I thought it relevant to pass along. Here’s the quote from the MathOverflow page:

Though irrelevant to Tim’s question, Graham’s number is small potatoes compared to some of the numbers cooked up by Harvey Friedman, e.g., his paper Long finite sequences, JCT(A) 95 (2001), 102-144.

I question Mr. Goldenberg’s August 16th 2016 posting to Jesska about describing a Googleplex (from Wikipedia’s page). If a book contains 10^106 number of zeroes and you multiply it by 1094, isn’t that just 10^106 times (approx) 10^3? 1st Law of Exponents tells us that this is only 10^109 power. There is not enough paper or space on our planet to actually write out 10^10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.