What Is Smaller Than 1/2
Quick, what's one + 1? Information technology's obviously 2, right? Not so fast!
What if I was to tell y'all that I could prove that 1 + 1 is actually equal to i. And that, therefore, 2 is equal to one. Would you think I was kind of nuts? More like completely nuts? Probably. But basics or non, these are exactly the things nosotros'll be talking near today.
Of course, at that place volition be a trick involved considering 1 + ane is certainly equal to two…thank goodness! And, every bit information technology turns out, that play a trick on is related to a very interesting fact about the number null.
How does information technology all work? And what's the big ruse that the sneaky number zero is attempting to pull off? Keep on reading to notice out!
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How to "Prove" That 2 = 1
Let's brainstorm our journey into the bizarre world of plainly correct, nonetheless obviously absurd, mathematical proofs past convincing ourselves that one + 1 = 1. And therefore that 2 = 1. I know this sounds crazy, but if y'all follow the logic (and don't already know the pull a fast one on), I think you'll find that the "proof" is pretty convincing.
Hither'due south how it works:
- Assume that we have two variablesa andb, and that: a =b
- Multiply both sides bya to get: a 2 = ab
- Decreaseb ii from both sides to become: a 2 –b ii =ab –b 2
- This is the catchy part: Factor the left side (using FOIL from algebra) to get (a +b)(a –b) and factor outb from the right side to getb(a –b). If you're not sure how FOIL or factoring works, don't worry—you tin can check that this all works past multiplying everything out to see that information technology matches. The end result is that our equation has become:(a +b)(a –b) =b(a –b)
- Since (a –b) appears on both sides, nosotros can cancel it to get: a +b =b
- Sincea =b (that's the assumption we started with), we tin substituteb in fora to go: b +b =b
- Combining the two terms on the left gives us: 2b =b
- Sinceb appears on both sides, nosotros tin can split through byb to get: 2 = one
Await, what?! Everything we did there looked totally reasonable. How in the world did we end up proving that 2 = 1?
What Are Mathematical Fallacies?
The truth is we didn't actually prove that 2 = 1. Which, proficient news, means you lot can relax—we oasis't shattered all that you know and beloved about math. Somewhere buried in that "proof" is a error. Actually, "error" isn't the right word considering it wasn't an error in how nosotros did the arithmetic manipulations, it was a much more subtle kind of whoopsie-daisy known as a "mathematical fallacy."
It's never OK to divide by aught!
What was the fallacy in the famous simulated proof nosotros looked at? Similar many other mathematical fallacies, our proof relies upon the subtle trick of dividing past zippo. And I say subtle because this proof is structured in such a mode that y'all might never even find that sectionalisation by zero is happening. Where does information technology occur? Take a minute and run into if you can effigy it out…
OK, got it?
It happened when nosotros divided both sides pasta –b in the fifth step. But, you say, that's not dividing by aught—it's dividing bya –b. That'southward true, onlywe started with the assumption thata is equal tob, which ways thata –b is the same affair as zero! And while it's perfectly fine to divide both sides of an equation by the same expression, it's not fine to do that if the expression is aught. Because, as we've been taught forever, it'south never OK to divide by zero!
Why Can't You Divide By Zero?
Which might get you wondering: Why exactly is it that we tin can't carve up by zero? Nosotros've all been warned about such things since nosotros were little lads and ladies, just have y'all ever stopped to recall about why division by zero is such an offensive thing to practise? In that location are many ways to recall almost this. Nosotros'll talk about two reasons today.
The first has to practice with how sectionalization is related to multiplication. Let'due south imagine for a second that partitioning by zippo is fine and slap-up. In that case, a problem like 10 / 0 would have some value, which we'll call x. Nosotros don't know what it is, only we'll only assume thatx is some number. So 10 / 0 =x. We can likewise look at this division problem every bit a multiplication trouble asking what number,x, do we take to multiply past 0 to get 10? Of course, there'southward no answer to this question sinceevery number multiplied by zero is zero. Which means the operation of dividing past zero is what'south dubbed "undefined."
The second way to think almost the screwiness of dividing by null—and the reason we can't do it—is to imagine dividing a number like 1 past smaller and smaller numbers that become closer and closer to zero. For example:
- i / one = 1
- 1 / 0.ane = 10
- i / 0.01 = 100
- i / 0.001 = 1,000
- i / 0.0001 = x,000
- …
- 1 / 0.00000000001 = 100,000,000,000
and so on forever. In other words, every bit we dissever ane by increasingly small numbers—which are closer and closer to zero—we get a larger and larger outcome. In the limit where the denominator of this fraction actually becomes zero, the effect would exist infinitely large.
Which is notwithstanding another very good reason that nosotros can't separate by zero. And why ane + one is indeed equal to two…no matter what our screwy "proof" might say.
Wrap Up
OK, that's all the math nosotros have time for today.
Please exist sure to check out my bookThe Math Dude's Quick and Dirty Guide to Algebra. And remember to go a fan of the Math Dude on Facebook where you'll discover lots of great math posted throughout the week. If you're on Twitter, please follow me there, too.
Until adjacent fourth dimension, this is Jason Marshall withThe Math Dude'southward Quick and Dirty Tips to Make Math Easier . Thanks for reading, math fans!
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What Is Smaller Than 1/2,
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