You may have, at some point, seen one of those bits where a series of seemingly simple math operations somehow end up proving that 1=0 or something equally clearly wrong. Most of them accomplish their joke by sneaking in a hidden division by zero. From that point on, all bets are off (see Divide by Zero).
Recently, on a YouTube channel I follow, I saw a clever example that uses a much sneakier trick. It’s harder to spot because the operation it uses is legit in two of the three possible cases.
The gag, of course, uses the third one.
I’ll go through it step-by-step. It begins with a tautology:
Nothing wrong with that! Two is certainly equal to itself.
The next step is just as correct:
One plus one equals two, probably our very first math problem. What could be more basic than that?
The next step might get into slightly more math-y territory:
But it’s entirely legit. The square root of one is one, so the equality is still clearly true.
This next step might seem a bit weird:
But it seems reasonable enough. Minus one times minus one is positive one, so it ends up being the same as the previous equation, the square root of one, which is one.
We now re-write this as:
Because that allows us to write those square roots as:
Which takes us into the complex numbers, but that’s fine. The magic number i really is the square root of minus one. So long as our calculation eventually squares away the magic, it’s perfectly legit to use in calculations.
We reduce the above to:
Which indeed allows us to square away the magic, and we’re left with:
Doing the addition we get:
Which… doesn’t seem right. It certainly isn’t what we started with!
But where did it go wrong? Did you spot where things went off the rails?
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[Jeopardy music break…]
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Everything in the above is legit except for one thing:
Which might be hard to spot, because this case is legit:
And so is this one:
But when both terms are negative, you cannot re-write them as the product of separate square roots.
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A concrete example may help to make it clear. Start with:
Which we can (legitimately!) re-write as:
Both give us the correct answer.
If just one of the terms is negative, we have:
Which we can (again, legitimately) re-write as:
If you doubt that the answer really is 6i, look at it this way:
A negative term in a square root always returns an answer that includes magical i. This ultimately boils down to having a factor of -1, the square root of which is i.
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So, the two cases above work fine, but when both terms are negative, we have:
Which cannot be re-written as the product of two square roots, because:
So, this third case — upon which the trick above depends — is not legit.
But it makes for a good joke — harder to spot than the hidden divide by zero.
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The same video mentions — but doesn’t explore — another cute trick (he has another video that does). This one starts with the tautology:
Which is legit. Anything to the power of one is itself.
This can be re-written as:
Which is legit. Four-over-four is one.
Next, we separate the fraction:
Which is also legit, because:
And:
Next, we evaluate the inner term:
It evaluates to one because:
The pattern of four repeats as the powers increase.
Now we’re left with:
![\displaystyle{i}={1}^\frac{1}{4}\left(=\sqrt[4]{1}\right)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7Bi%7D%3D%7B1%7D%5E%5Cfrac%7B1%7D%7B4%7D%5Cleft%28%3D%5Csqrt%5B4%5D%7B1%7D%5Cright%29+&bg=f9fbf9&fg=000080&s=2&c=20201002)
And since the any root of one is just one (just as any power of one is just one), that means:
Which… doesn’t seem right.
But where did it go wrong? Can you spot the trick?
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[Jeopardy music break…]
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This one is especially subtle because all the individual steps are perfectly legit taken on their own. Part of the trick lies in the accompanying text (“any root of one”), and part from a combination of steps. The latter part becomes clear when we separate the fractions the other way:
We can evaluate the inner term easily if we use the exponential notation for complex numbers:
The outer term tells us to take this to the fourth power:
So, splitting the fraction this way gives us a correct equation:
But splitting it the other way does not.
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The problem is that roots have multiple answers. The square root of four is two and negative two. A fourth root has four answers:
![\displaystyle\sqrt[4]{1}={+1},\;{+i},\;{-1},\;{-i}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Csqrt%5B4%5D%7B1%7D%3D%7B%2B1%7D%2C%5C%3B%7B%2Bi%7D%2C%5C%3B%7B-1%7D%2C%5C%3B%7B-i%7D+&bg=f9fbf9&fg=000080&s=2&c=20201002)
One of which (+1) was the “incorrect” answer we got the first time. All four answers are correct for the fourth root of one because:
In fact, the fourth root of i in the second version also has four roots, but let’s not go further down the rabbit hole. (It turns out that all four roots square to i, so, in fact, the equation is correct when split this way.)
The bottom line is that, splitting fractions requires always first reducing them to their simplest form (and 4/4 is not the simplest form).
When a fractional exponent is in its simplest form, splitting it works just fine:
Because, in this case, 3/4 is the simplest possible form.
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The video that inspired this post:
And here’s one that goes into the second trick in detail:
It’s a fun channel if you enjoy math.
Stay legit, my friends! Go forth and spread beauty and light.
∇
Logos con carne 
February 15th, 2024 at 12:17 pm
Let’s go down that four-roots rabbit hole. In the first case, we have:
We evaluate the inner term as being one:
And to pull out the four roots, we re-write it as:
And evaluate it with the four values of n to get:
Which evaluate to:
Which, when raised to the fourth power, all give i:
So, putting the power of 4 inside and the power of 1/4 outside, we get four answers because there are four roots.
February 15th, 2024 at 12:43 pm
Okay, what about doing it the other way:
In this case, the four roots come from the inner term, which we evaluate first. So, we rewrite this as:
Where we’ve factored out i and pi in the exponent and expressed the 2 as 4/2 to make it compatible with the 1/2.
Now expand the inner term with the four values of n:
And evaluate each term:
Which gives us four answers for the inner term. Raising all those to the fourth power gives us:
Which we can reduce to:
So, all four roots, raised to the fourth power, evaluate to i — which was the original assertion!
Bottom line, doing it this way works, but doing it the other way doesn’t. Always reduce your fractional powers!
March 2nd, 2024 at 4:07 pm
You might like this short story by Ted Chiang:
https://fantasticmetropolis.com/i/division
March 3rd, 2024 at 9:51 am
Interesting story! It reminds me a bit of his Story of Your Life, the short story the movie Arrival is based on. Both concern a female protagonist who makes a discovery that opens a mental door that forever alters the way their minds function. (Both are relationship stories, as well.)
I think Chiang slipped in his own covert “division by zero” to make his story work the way he wanted. The introductory narration of section 6 is about Gödel’s Incompleteness theorems, and I think he subtly misstates what Gödel demonstrated. The short version is that Gödel showed that a given arithmetic system cannot possess all three properties of effective axiomatization, completeness, and consistency.
The first is the property that, given the axioms of a system of arithmetic, all possible true statements can be enumerated. Gödel demonstrated the impossibility of this using the same technique that Cantor invented to demonstrate that the real numbers cannot be enumerated. Turing used it to demonstrate the halting problem in computation.
Completeness means that any statement in a system of arithmetic can be proved as false or true. Gödel showed that this, too, is impossible (largely because all true statements cannot be enumerated).
Consistency means no statement in a formal arithmetic can be shown to be both true and false. There can be no contradictions within the system. Such as, for example, 1=2 when both 1 and 2 are well-defined as distinct.
The fastball he snuck over the plate (I think) is that, while a given arithmetic system cannot prove its own consistency, a higher system can. Of course, that higher system cannot prove its own consistency (but a higher-higher one can). We are left, though, with the notion that, ultimately, we can’t fully demonstrate mathematical consistency.
This ties back to what I wrote recently on your blog about scientism and the belief that we can ever fully understand our reality. Even our abstractions are shown to have some opacity we can never see through, even in principle. I’ve always rather liked that.
March 3rd, 2024 at 5:47 pm
Glad you liked it! Well, you know me, I never would’ve caught that mistake, just like I’ll never know what it’s like to make a mathematical discovery. The story did give me some sense of what that might be like, though.
March 3rd, 2024 at 7:21 pm
I’m not sure it’s a mistake, as such, but an intentional glossing over to enable the story. Gödel’s Incompleteness theorems are central to the story. What struck me is how he seems to be doing the same thing the 1=2 proofs do. The “mistake” there is equally intentional.
May 7th, 2024 at 8:35 pm
[…] this year, I posted about that math gag that seems to prove (very mathematically) that 2=0 (an alternate version “proves” 1=0 using the same trick: a covert division by zero, an […]