 # large numbers of 2

I disagree with the answer listed here:

I propose the following alternative answer:

So when, “2” is the set {2.0 <= R < 3.0}, then for large values of “2” it’s double is equal to the set “5”.

That is to say, there exists a subset of {2.0 <= R < 3.0} that, when doubled, is equal to the set {5.0 <= R < 6.0}.

Okay… and the double of “2” is also “5” when “2” is 2.5.

No, as an interpretation that fails completely. The saying is not “2 × 2 = 5 for a sufficiently large value of 2”, but “2 + 2 = 5 for sufficiently large values of 2” – not to mention that you are apparently treating one of your two 2s as approximate and the other as exact, with no evident justification, seeing that (2.5-ε) × (2.5-ε) = 6.25-ε, which is a good deal more than 5.

Your formulation also fails in its (note, in passing, how “its” is properly spelt) spectacularly incorrect reference to ‘the set “5”’.

Um, x + x = 2x

But technically, the large values of 2 do not have to equal each other, just both be large values of 2. 2.4 and 2.3 both round to 2, but the sum of 2.4 + 2.3 = 4.7, which rounds to 5. Which is the original explanation in the column.

Multiplying by 2 assumes you are using the same large value of 2, which is not the original statement. Ergo, multiplying by 2 is out, you must sum the two potentially different values.

I would also point out that if “2” is the set {2.0 <= R < 3.0}, then for large values of “2” you actually can get 6. 2.9 + 2.8 = 5.7, rounds up. So that definition fails the original proposal. Or should I say extends beyond the original proposal.

I hate that column, no offense to Dex.

2.3 is not a value of 2. 2.7 is not a value of 2. The joke in the first place is not about rounding. The joke, or better, observation, lies in the way that people have a tendency to put new interpretations on values of any kind, even if they are supposedly fixed numbers, to support the claims they want to make. It’s about psychology, not math, and putting actual numbers into the joke misrepresents it at a minimum. More realistically, it totally misses the point.

It isn’t in pure math, but it is in what we used to call “apple math”. Furthermore, “2”, “2.0”, “2.00”, “2.000”, etc., are all different values, and 1/0 is illegal, or ∞, or ±∞, depending on what is most useful.

Applied math is as different from pure math as topology is from geometry.

I agree that it misses the point. Basically, it’s not a math joke, it’s a semiotics joke. Or possibly a FORTRAN joke.

I agree with what you’ve said here, but not with what I think you meant. Numerical analysis is certainly a branch of mathematics, just as topology is a branch of geometry.

Exapno Mapcase, I agree there is more to the joke than just the rounding. The point of the joke is, as you said, about fudging interpretations to support the answer you want.

But the joke wouldn’t make any sense without understanding the math and the rounding that is applied. In other words, the psychological angle of the joke requires one to first understand the math element. If you cannot grasp how 2 can be a “large value of 2”, then you are in the same position of Adam, i.e. lost.

Hmmm…

IIRC this was a joke too from my computational math class.
Basically, when working in variables (x, y, a, b, you know…) you can approximate the results.
As x approaches infinity, consider:
(x)(x+1) = x^2+x

The squared term becomes so much larger (“dominates”) that it really doesn’t matter what the other terms are; if I offered you x^2 +x pennies versus x^2 how badly would you care about the difference?
x=1 penny - 2 versus 1
x=10 - \$1.00 vs. 1.10
x=100 -\$100.00 vs \$101.00
x=1000 \$10,000.00 vs \$10,010.00 - at this point, who wants to wait while they count out the last ten dollars?
x=1 million; One trillion pennies, or \$10B versus \$10B plus \$10,000.00

So you can see, “for sufficiently large values of X, we can say that X^2 is approximately equal to X^2+X”. This sort of approximation becomes a cliche in that field of math.

Why is it useful? The highest term (largest exponent) dominates. This sets the shape of the graph of the function f(x). When you scale really large, that’s the number you need to know about.

For example, searching the google database for a keyword; if you have X entries, and look at them one at a time, your time will be X. If the list is in order, and you can hop back and forth by half the remaining list each time (binary sort) your search time is limited by log(X)

or you are searching a face database for facial recognition; or a fingerprint database, etc. The classic exercise for orders of magnitude in processing is teh sort algorithm. Anything from x^2 to xlog(length(x))

For a computer that does millions of calculations a second, that does not sound like much. But when there are billions or trillions of items to search, it CAN take a while. This is where the dominant term is important.

So the cliche was “x=x+1 for sufficiently large values of x”. The joke that “2+2=5 for sufficiently large values of 2” is just geek humor at work. It’s not pretty.

I understand what you’re saying but I still have to disagree. Your interpretation still makes it about rounding. My interpretation insists that 2 is 2, but that people will simply discard this absolutism to make their results fit. It’s more a way of acknowledging that people want their versions of reality to be correct no matter what the underlying inconvenient facts happen to say. If that’s true then it’s crucial to the joke that the numbers can’t be right under any set of mitigating circumstances. Nor do the numbers have to be numbers. The joke can be applied to any of a million threads here at any given time whether the subject is numerical or not. Look at how often people use a definition of a word to mean something different from the way everybody else in the world defines it to argue a point that nobody else agrees with. Same principle.

This level of analysis pounds the humor out of a joke and there wasn’t much there to begin with. Let’s agree to disagree.

Expano: a very interesting interpretation, that the joke is about fudging data. When I researched the phrase to write that Staff Report (long ago), I didn’t find that interpretation anywhere. When people used the statement, the expression “for very large values of 2” was expressed matter-of-factly (as if there are such things) rather than sarcastically (where “for very large values of 2” means “when I say it does.”) I also think the context really pushes for the rounding interpretation; otherwise, the expression would be “2 + 2 = 5 if it needs to” or “2 + 2 = 5 for sufficiently sliced data points” or something of the sort.

So, I think your interp is very interesting, but i don’t think I’ll revise the Staff Report. I stand by my approximations. Silliness!

2

The average male has less than two testicles.

Inerestingly enough, so does the average female. We’re more alike than we thought!

Maybe it’s YOUR absolutism and self-righteous acceptance of “elegant” data points at face value that this joke critiques. It points out that a person of subtle thinking will understand the non-absolutism of even a plain value.

(I say this not in direct response to your post, but having encountered many of your posts for some time.)

Exapno Mapcase said:

The guy who asked the question that Dex responded to. Did you read the column? Since you say directly that you’re not responding to my post, that makes this a definition of an ad hominem. Are ad hominem attacks allowed in CSR? Let’s find out.

Irishman: Sure I read the column. But I didn’t memorize it. Good grief. You guys are arguing in circles. Read my previous post!

The “for sufficiently large values of…” is a standard phrase in math theory, to allow you to approximate, by dropping all terms except the one with the largest exponent. For sufficiently large values of a variable (when we get into the billions, say) all other terms are so small as to be irelevant.

“X^2 = X^2+1 for sufficiently large values of X”

i.e. If your computer only does 32 bits of precision in calculations, any difference of less than 1 in 2^32 is irrelevant.