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)](http://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}](http://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.
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