The Power of Zero
The Power of Zero
Category: School, College, Greek
Samuel Thomas Poling, Blog 124, The Power of Zero
This is an essay I turned into my college math instructor last week. She has said "it's an enjoyable read," but hasn't finished it yet, nor commented further.
Samuel Thomas Poling
October 30, 2006 A.D.
The Power of Zero: The Mistake
Math is a world of logic, nothing more. Many contend that logic and math are different things, but math is just a certain area of logic to do with amounts, values, and things around that area. To navigate any mathematical problem you must rely on reasoning. Every theorem and shortcut must be provable within the world of logic. This is math. Anything else, anything else at all, can never be used to support a mathematical idea. Not desire, not how many people believe it, not by who said it to be true – only logic can say what works and what doesn't.
Another thing one must realize about mathematics is that it's simply a language. The sequence 1 + 1 = 2 appears as words in our minds. They are symbols used to represent values and processes. There are not actually any 1s floating around, nor are there any 2s. There isn't a great plus sign in the sky whose power you summon – to satisfy the deity of equals. They don't actually exist, they are just ideas used to make logical statements about amounts and so forth. With another luck we might have evolved to use the label of division in place of addition, or the word three in place of the word four. It doesn't matter; these numbers and sequences are just ways of explaining our reasoning.
One such method of explanation has been the mathematical method of powers. Exponents. You have a base number, like any other number, with another smaller number floating next to it, top right. What does this say to the human race in mathematical language? What did we decide to make that mean? The exponent, the power, the smaller number floating above, is meant to tell you how many times the base number should be multiplied by itself. That's what it means, clear and simple. It's a shorter way of writing something out. Instead of 3 * 3 * 3, you can simply write 3^3. As I type this, I couldn't find the ability to make the exponent float up next to the base on Microsoft word, but fortunately the "^" symbol works in mathematical language as well. 3^3 means 3 to the power of 3. Which means 3 multiplied by itself 3 times. If it were to the power of 4, then it would be 3 multiplied by itself 4 times, or 3^4. It's quite simple.
So following what powers have been made by our human race to represent, we can figured out what 3^3 would be. 3 * 3 * 3 = 27. Ergo, 3^3 = 27. What about 3^2? Easy, it is telling you the number 3 twice, multiplied. 3 * 3. And that is equivalent to 9. How about 3 to the power of 1? Easy again, that's three one time. Having nothing to multiply to, it is just left as three. 3^1 = 3. This is how it is contended by all.
What is a number to the power of zero equal to? Currently mathematicians hold it to be true that it is equal to one. No matter what the base number is, if the exponent is zero, then it is equal to one. However, if you follow the same reasoning of meaning we have given exponents to possess, then we see a different picture. 3^0 would be 3 zero times. Having nothing to multiply to, we are left with zero, and not with one. Following the same pattern, the same meaning, the same logic as we have with 3^2 = 9, and 3^1 = 3, we arrive at 3^0 = 0. Why? 3^2 is two 3s. 3^1 is three 3s. And 3^0 would be no 3s. "To the power" means you multiply the base by itself the number of times shown by the exponent. If you have 3^0, the exponent is telling you that you have no 3s to multiply together. This is just using common sense. To refute this, you need some pretty special and pretty solid evidence.
The math world thinks it has found the evidence. I have scoured online and off, hearing several supposed "solid" proofs for 3^0 = 1, but I have seen flaw in all of them so far. A common flaw at that. It's difficult to explain, but it's there. To show you what I mean, I'll have to first express to you the supposed proofs they have mustered together.
The world of exponents has several short cuts. Short cuts can be found in math all of the time. However, initially they are risky. You see, you have to have proof the shortcut works. You must solve the problems the long, hard, usual, certain way as well as the short cut way, and check them together to make sure they arrive at the same solution. You must not only check these short cuts in many different areas (positive, negative, fraction, decimal, zero, even, odd, ect.) to make sure the cut holds true, but you must also be able to explain why it is a short cut, and how it works the way it does. You must mathematically prove it. This is pretty much been done well with the exponent short cuts, now called exponent "rules."
This leads me directly to the first problem with the evidence asserting a number to the zero power equaling one instead of zero. You must prove a short cut with the long, normal way of doing things, not the other way around. Although the thinking, reasoning, long-cut way of 3^0 claims that it equals zero with common sense and definition of exponent, the short cut says otherwise. It is in the shortcuts where 3^0 will equal one. And only in the short cuts. I would normally expect you to then realize that there is something incomplete with the description of the short cut! However the math world made a mistake and instead of fixing their short cuts, they used the faulty short cuts to change reality.
Let's say I say to you there is a faster way of solving a division problem. I just switch the denominator with the numerator and subtract, and then claim it was just a short cut. You do the long-cut, more difficult (yet more trustworthy) way of dividing something, and you notice we get different answers. But instead of me changing my short cut, I change the usual laws of division. What is my proof to back up my insanity? My short cut. This is, of course, rediculous backwards reasoning.
Yet this is exactly how the mathematicians are trying to prove 3^0 = 1.
They have faulty short cuts, and when it is discovered these short cuts don't work in the case of zero, which many short cuts in math do not, instead of adding it to the rules of exponents, the change the reality.
Here is the most common "proof" of 3^0 equaling 1:
There is an otherwise proven short cut exponent "rule" that says when you are multiplying two numbers of the same base, you can just add the exponents. For example, if you have (3^3) * (3^2). They have the same base, it is 3. Ergo, says the short cut, you can easily add up their exponents and just put it over the base. So (3^3) * (3^2) = 3^5, because 3 + 2 = 5. Would this trick work in the case of (3^3) * (4^3)? No, because, as you can clearly see, the bases are not the same. For this rule to work, the bases must be the same. Base 3 is not the same as base 4; the shortcut does not apply here.
Understand? Alright, so lets get right to it and see how an exponent of zero will play the part. (3^3) * (3^0) will be our problem. It has the same base, doesn't it? They both have the base of three. Ergo, the shortcut says we can go ahead and add the exponents. So we will end up with 3^3, right? Wait, if what I said is true, then we should end up with zero! (3^3) * (3^0) should equal zero, if what I said is true, because 3^0 = 0. And anything by zero will be zero. The shortcut proves me wrong. Or does it? It proves me wrong the same way my example division "short cut" proves division wrong. But let's see, if any number to the zero power does, in fact, equal one, then the short cut isn't broken, because any number, including 3^3, by one should equal itself, as it was shown above to actually happen. The short cut really does work against me well, doesn't it? No wonder so many math experts believe a number to the power of zero equals one.
But again, I assert there is an exception with zero being in one of the powers. There are several mathematical rules and short cuts with exceptions to zero, many of which are other exponent rules themselves. For example, 0^0 can't equal one, can it? There's an exception there, isn't there? The shortcut is incomplete; it needs to add an exception to zero being in one of the powers.
Although, it isn't so much incomplete. Just saying there is an exception to zero would make it easier for students to learn this. The rule, as it stands, already says that (3^3) * (3^0) = 0. You just have to look more closely. Pay attention, this is my trump card. When multiplying two numbers with exponents, you can only use the short cut if both the bases are the same, as I explained earlier. However, in the case of (3^3) * (3^0), the bases are not actually the same. It's an illusion.
Here is the part that is the most difficult to explain. Once this is a shortcut, it can't really prove one way or the other, 3^0 = 1 or 0. Circular reasoning will enter into this if you try to use it to prove one way or the other. But I'm not trying to prove 3^0 = 0 right now, I'm only trying to disprove 3^0 = 1. Show another possibility. So pay attention as I bring this to even ground. If I am right, then 3^0 = 0. In which case the base numbers are not the same and you cannot use the short cut, but the short cuts own rules! "Oh, but both base numbers are 3!" No, they're not. It just appears that way. If I am right, then 3^0 = 0, which is not at all the same base number as 3. Writing it in 3^0 to begin with was a stupid way. If I am right, then 3^0 = 4^0 and any other number you want to put in front of the zero power! (3^3) * (3^0) is really (3^3) * (X^0), X being any integer. Ergo, they do not, in reality, have the same base. And if that isn't good enough for you, then look at another mathematical process. You can write 3^3 out nicely. By the definition of powers, 3^3 is telling you 3 multiplied by itself 3 times. So 3 * 3 * 3. Looking at the definition of powers, 3^0 means 3 multiplied by itself never. So nothing. So multiplying 3 * 3 * 3 by nothing gives you nothing. Ergo, the short cut still doesn't work, in the case of the raw definitions and meanings of powers. What exponents mean testifies against 3^0 = 1, and testifies for what I say to be true.
"But 3^0 doesn't equal 0, it equals 1!" Shouts the mathematician, falling into circular reasoning. They put in (3^3) * (3^0) and get 1 due to the short cut. They take apart the numbers as 3 * 3 * 3 * 1, again getting the solution of 3^3, meaning 3^0 equaling one works! They're using circular reasoning. They are using what they are trying to prove as part of the proof for that thing. They are assuming 3^0 = 1 being true as they are trying to prove it. When you ask them to prove to you 3^0 = 1, and they, in the proof, use 3^0 equaling one, then they are guilty of circulus in demonstrando. Circular reasoning, circular argument. Until they have proven it to you, assert and reassert that as of now 3^0 = ?. And they have to deal with that until they've proven otherwise. They'll slap 3^0 = 1 somewhere in their "proof" but you just snap their attention back to the fact that right now, until they've proven it, 3^0 = ?, not yet 1. Not until they've proven it.
But isn't circular reasoning what I was doing above? I said, "If I'm right," a few times up there, that's true. But as I said, I wasn't trying to prove 3^0 = 0, I was only trying to show that there is not yet any proof that 3^0 = 1. If they can, in their proof, slap in 3^0 = 1, then I can, in my proof, slap in 3^0 = 0. If they attack me on that, I can correctly accuse them of special pleading and hypocrisy.
Where, then, do the tables turn? In the common sense. Where I did in fact prove 3^0 = 0 was much earlier on, with the very simple description of what powers are created to mean. How many of the base numbers are being multiplied together. If there is none, then there is none. There is zero. Following the same reasoning and definitions of all the other exponents, a number to the zero power will be equivalent to zero. This is where the tables turn.
The others "proofs" I've looked at for 3^0 = 1 are other variations of the same one I just destroyed, as common sense can clearly see. Instead of putting in 3^0 = 1 in any of their proofs, go ahead and try 3^0 power, apply my same logic, and you'll see it could work that way to. There is that exception to zero, there is that fact that it isn't actually the same base, and so on. If they say that any number over another number is equal to one, and then put 3^0 over 3^0 on a fraction bar, then just write zero over zero next to that, and ask them if that equals one. (3^0) / (3^0) = 0 / 0, which does not equal 1. Hey, they could assume 3^0 = 1 in their proof, why can't I assume 3^0 = 0 in mine? I also have the bonus of the common sense and the actual meaning of exponents on my side. I'm actually more justified.
There may be evidence that a number to the power of zero is truly one, but my main point here today is that I have not seen it. I heard dozens of experts, viewed several sites, heard my math instructor, herself, explain it to me. However, their evidence is not conclusive. If they are right, they haven't proven it yet. If it does not equal zero, which I very well may not, then that doesn't mean this essay was wrong. I'm pointing out the flaws currently being made. And also pointing out that, as of now, my side is more justified. I also understand that this would change everything greatly, so close mindedness to my entire argument here by mathematicians is unfortunately likely. But their complaining would change anything until they have some conclusive proof.
So, in closing, there is a big mistake. A mistake of circular reasoning, using what you are trying to prove as part of the proof for that thing. A mistake of backwards reasoning, using short cuts to change reality, instead of reality to fix short cuts. There is no proof for 3^0 = 1, that isn't fallacious. My math instructor once told me to take 3^0 = 1 on faith. Only error needs the assistance of faith. If, in math, it's true, it can be proven logically, and, as it is so far, it has not yet been proven to me. Much we can learn from our teachers, more from our colleagues, but the most from our students.
Samuel Thomas Poling, Blog 124, The Power of Zero