Symbolic Nonsense

As mentioned in the previous post, the BBC article contains video of Dr. Anderson explaining how his work allows us to evaluate the expression 00. I’ll save you the trouble of having to watch it. Here it is:

We define the number N=0/0. The number N stands for nullity, and is the new number Dr. Anderson claims to have discovered/invented. He uses a Greek letter phi to represent it, but an N will be simpler for our purposes.

He now argues:


00   =   0(1-1)   =   01 x 0 -1   =   (0/1)1 x (0/1)-1.

Anything to the first power is equal to the thing right back again, and anything to the minus one power is merely the reciprocal of the thing. Consequently, we get:


(0/1)1 x (0/1)-1   =   (0/1) x (1/0)   =   0/0   =   N.

In standard crank fashion, Dr. Anderson sums up his accomplishment immodestly as follows, “So you’ve just solved a problem that hasn’t been solved for 1200 years. It’s that easy.”

Mathematicians have an expression for this sort of thing: “Symbolic nonsense.” You can invent whatever symbols you like and manipulate them according to whatever rules you like, but don’t mistake that for actually saying something about reality.

Actually, though, this little exercise gives me a chance to present one of my favorite examples of symbolic nonsense. I’ll write sqrt(x) to denote the square root of x.

Ponder the following:


(sqrt(-1))2   =   -1.

Seems simple enough. But we also have this:


(sqrt(-1))2   =   sqrt(-1) x sqrt(-1)   =   sqrt(-1 x -1)   =   sqrt(1)   =   1.

So which is it? Looks like the complex analysis textbooks might have to be rewritten.

I’ll leave it to some clever commenter to explain precisely what went wrong.

Incidentally, does anyone out there know how to make HTML produce a proper square root sign? I found one website that said sqrt and /sqrt would do it, but that doesn’t seem to be working.

Comments

  1. #1 Chad
    December 7, 2006

    -1 = i
    Is this what you keep getting? The top part doesn’t go over the -1? I found the code to do it at http://www.azillionmonkeys.com/qed/sqroot.html
    Just view the source.

  2. #2 Jason Rosenhouse
    December 7, 2006

    Chad-

    Your code doesn’t seem to have worked much better.

  3. #3 Tyler DiPietro
    December 8, 2006

    Doesn’t the square root function in the third term violate your definition? I’d think that since you’re going with i^2 (as in the first two terms) which is -1, and i itself does not equal -1, then i*i is not equal to -1*-1, since the latter is equal to 1 while the former is equal to -1. Then it would seem that the square root of the parenthetical operation (1) in the third term would not be the square root of -1, and therefore aren’t calculating the same values. Where did I go wrong?

  4. #4 Chad
    December 8, 2006

    Yeah, better off embedding graphics.

  5. #5 Luis
    December 8, 2006

    I’m glad my secret is still safe. I was afraid this guy was onto something but he made the silly mistake of DIVIDING by zero.

    Once my math teach gave me a problem that I could not solve so I MULTIPLIED both sides of the equation by zero and that did it. Teach was not impressed. More likely, he wanted to claim the insight for his own glory. But he didn’t! I still am the Holder of the Power.

    Oh, damn, is this going everywhere? Please don’t tell anyone my secret!

  6. #6 crf
    December 8, 2006


    But that just prints a radical sign, so you must enclose the argument in brackets. That’s not much better than what you have now, except more people might recognize the radical sign.

    Complex numbers not zero or positive have two square complex roots. So -1 has two roots (i & -i). So this is not true:
    sqrt(-1) x sqrt(-1) = sqrt(-1 x -1)

  7. #7 Davis
    December 8, 2006

    00 = 0(1-1) = 01 x 0-1 = (0/1)1 x (0/1)-1.

    Here is an excellent argument in favor of leaving 00 as an undefined quantity, and dealing with it on an ad hoc basis instead.

    Complex numbers not zero or positive have two square complex roots. So -1 has two roots (i & -i). So this is not true:
    sqrt(-1) x sqrt(-1) = sqrt(-1 x -1)

    More precisely, it’s a notational issue. In order to make the square root function well-defined, we decide that the square root symbol always means the positive square root, so sqrt(4) is 2, and not -2. This has the added benefit that sqrt(a) x sqrt(b) = sqrt(ab), since you have positive x positive = positive. The positive value is the only choice that makes this rule valid, as the negative choice would give negative x negative = positive (i.e., then you would have sqrt(a) x sqrt(b) = -sqrt(ab)).

    sqrt(-1) is problematic — as with reals, there are two possible choices, i and -i. Unlike reals, we can’t decide sqrt(-1) is greater than 0, since complex numbers can’t be compared in this manner (which in particular implies there’s no notion of positive or negative complex numbers). We then bump up against a sign ambiguity problem — should sqrt(a) x sqrt(b) = sqrt(ab) or -sqrt(ab)? Jason’s example highlights this.

  8. #8 Na'Irb
    December 8, 2006

    sqrt(-1) x sqrt(-1) = (-1)^1/2 x (-1)^1/2 = (-1)^1/2+1/2 = (-1)^1 = -1

    You broke the law.

  9. #9 Jonathan Lubin
    December 8, 2006

    HTML is not TeX, nor ever will be. If you want a square-root symbol with its overbar going over a whole expression, that probably is beyond HTML as it now stands.

  10. #10 Jason Rosenhouse
    December 8, 2006

    Jonathan-

    All too true, alas.

  11. #11 mike
    December 8, 2006

    I think Euler once made the same argument you are making, Jason. The simple fact is that the imaginary unit is not manipulated ordinarily like a standard square root. You cannot merely subsume two negative numbers under a radical sign.

  12. #12 Charlie B.
    December 8, 2006

    May I suggest taking a look at MathML?