I have gotten mostly D’s and F’s in math, but I can’t see how I can be wrong about the following.
Zero times zero equals any non zero number. Zero divided by zero equals any non zero number. Any non zero number divided by zero equals zero.
Here is my reasoning; can anyone explain where I went wrong? Multiplication is telling how much of something you have. 0 x 0 = 0 is an absurd equation because on one side you have zero zeros, aka not zero, and on the other side you have zero. 0 x 0 must equal any number except zero because it is saying you don’t have nothing.
Non zero number divided by zero must equal zero because no amount of zeros adds up to a non zero number, and “no amount” is expressed mathematically as “zero”. The non zero number contains no, aka zero, zeros. If you have something, you don’t have any nothing.
Zero divided by zero equals any non zero number because zero times any non zero number equals zero. Zero divided by zero is asking how many zeros equal zero.
About divide by zero, I don’t understand why a math problem is declared unsolvable if it has more than one answer. Having almost infinite possible answers should make the problem really easy to solve.
Zero times zero equals zero. I think you mean ‘Any number times zero equals zero.’
The result of any number, including zero, divided by zero is undefined.
I think you are using English concepts such as double negatives to try and understand a mathematic equation. This is tripping you up. Maths is its own language and follows its own rules. In the language of maths, 0x0=0 and 0/0 is undefined. End of story.
Actually, there are good reason for accepting the convention that 0[sup]0[/sup] = 1, and you can use similar arguments to suggest that 0/0 = 1.
For the former, see sites like this: http://mathforum.org/dr.math/faq/faq.0.to.0.power.html or George B. Thomas’ Calculus and Analytic Geometry[.
(For one thing, the formulas for sums and infinite series can be consistently and concisely written if 0[sup]0[/sup] = 1 is accepted ).
For the latter, observe that 0/0 = lim (a/a) as a –> 0, and for non-zero values of a, a/a = 1.
You went wrong with ‘aka not zero’. It’s math, not English (or whatever the subject is called in other languages). If I say, ‘He didn’t not go,’ then I’ve said he went. But 0 x 0 doesn’t read like that. It’s asking, ‘How many of the numeral Zero do you have? None? Then the answer is zero.’ Or ‘If you multiply nothing zero times, then how many nothings do you have? Zero.’
It looks like your other questions are similar confusions between the mathematical concept of ‘zero’, and the linguistic use of ‘zero’.
But it’s hard to type between bites of lunch, so I’ll stop here.
EDIT: And people type faster than I eat…
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Multiplication is basically a fancy way to add things.
For example:
3+3+3+3+3 = 3 * 5 = 15
So, multiplying by zero would be akin to:
03 = 0+0+0 = 0 (30 is 3 no times so also zero)
Division is the flip of that and basically fancy subtraction.
So…
9/3 is like saying
9-3=6, 6-3=3, 3-3=0…you did three operations so 9/3=3
BUT, if you do 9/0 then you get:
9-0=9, 9-0=9, 9-0=9…forever.
That looks like infinity but infinity is not a number. If you did 1/0= infinity and 2/0=infinity then you have basically just said that 1=2 which is of course an absurd result. (Another way to write that which makes 1=2 more clear is 1/0=infinity=2/0)
So, anything divided by zero is undefined.
Your answer reminded me of the ‘on one side you have zero zeros, aka not zero, and on the other side you have zero. 0 x 0 must equal any number except zero because it is saying you don’t have nothing’ thing in the OP.
Let’s say 0 x 0 isn’t zero, since you have no nothings. Let’s say it’s 1. A zero is sometimes called a ‘goose egg’, so no nothings equals one goose egg. A starving man could feed his family just by working the equation a few times!
it’s why the population of the universe is zero. Although you might see people from time to time, they are most likely products of your imagination. Simple mathematics tells us that the population of the Universe must be zero. Why? Well given that the volume of the universe is infinite there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is as close to zero as makes no odds, therefore we can round the average population of the Universe to zero, and so the total population must be zero.
(stolen from Douglas Adams)
Love Douglas Adams but of course he is pulling a fast one on us here.
If the universe has infinite planets but only some of those are populated then you still have infinite populate planets and thus the population of the universe is infinite.
Remember, even if only one out of every Googleplex of planets are populated that is still an infinite number of populated planets (assuming there are an infinite number of planets).
This thread will go nowhere, but, for the record:
0 x 0 = 0 and indeed 0 x anything and anything x 0 = 0 by simply extending the patterns in a multiplication table, the natural patterns you observe for how things increase or decrease as you move up and down and left and right along columns and rows.
As for 0/0… We typically define x / y as the unique value which, when multiplied by y, yields x. In most cases, this works out fine. In some cases, well… There isn’t a unique value which, when multiplied by 0, yields 0. Since, as we saw above, EVERY finite whole number (and more generally than that, but let’s stick to this for now) multiplies by 0 to yield 0. So what should we take 0/0 to be? Well, one answer is just to say “It’s undefined!”, but perhaps a more informative answer with respect to the underlying phenomenon is to say “It’s indeterminate; it could be ANYTHING, and different things in different contexts”. And that is indeed what people often say, correctly.
Division by a number is a function. That is, it is an operation which applied to a given number produces one, and only one outcome. If you have multiple possible outcomes to an operation, it cannot be a function.
Division as a function is the inverse of multiplication. That is, if a/b = c, then cb = a, and vice versa. Thus, we know that 6/2 = 3 because we know that 32 = 6. We’d be really hamstrung doing most math if we couldn’t know exactly what the outcome of 6/2 was. Imagine if you go to split 6 objects up evenly among 3 people, and they have to argue which answer should be used!
This tells us why it is important to leave division by 0 undefined. If we try to divide a number, say 6, by 0, the answer must be that number which, multiplied by 0 will equal 6. So let the answer be x If 6/0 = x, then *x**0 = 6. But there is NO SUCH NUMBER, because anything multiplied by 0 is equal to 0.
Now let us take that to the 0/0 situation. If 0/0 = x, then *x**0 = 0. Yes, that means x could be 0. But it could also be ANY real number, because all such real numbers, multiplied by 0, give us 0. That means that dividing by 0 here would not be a function, and we want division to be a function.
As for your absurd attempt to make 00 != 0, I must simply shake my head. We can easily represent multiplication using a simple array of items. That is, we can take a multiplication problem and represent it with objects placed in rows and columns. Thus, 32 = 6 is easily represented by creating 3 rows of two items each (3 rows and 2 columns), then counting the result. So do that with 0*0. We now have an array with 0 rows, each with 0 items in them. I think you will be hard-pressed to find that your array has anything other than 0 items in it. 
ETA: ugh, or what he said just above me. 
As for Douglas Adams, my great love, he pulls a double fast one there. First, he says “But not all of them are populated; therefore only a finite number are”, ignoring the possibility, as Whack-a-Mole notes, that infinitely many are populated even though not all are. For example, not all integers are even, but infinitely many are.
Secondly, he pulls the particularly manifest nonsense (deliberately, jokingly, of course) of reasoning that if only finitely many planets are populated, out of infinitely many planets total, then no planets are populated (because the percentage of planets populated is infinitesimal, and may even be considered zero). That this reasoning is fallacious is, well, obvious. Only finitely many integers are equal to 3 (only one is, in fact!), out of the infinitely many integers total, but that doesn’t mean none of them are equal to 3!
Yes, this is precisely the sort of situation in which one might say “The percentage of objects with property so-and-so is zero; nonetheless, such objects exist”. This is the same way one might say the area of the Equator is zero (as an infinitesimally thin strip) but nonetheless, there are points on the Equator, or such things. (Cf. discussions of how the standardized formal language of “probability equal to zero” is not the same as actually being impossible)
Since the original question was immediately answered, we may as well discuss this tangent 
The Douglas Adams quote seems to be positing an infinite number of worlds, out of which a finite number are populated. In that case, it makes perfect sense to say that the average population density (appropriately defined!) is zero (and not the reciprocal of a large number as in your example). The joke is that there are still populated worlds, as Adams was well aware.
If the density were non-zero, then you would be correct in inferring the existence of an infinite number of populated planets, but that’s not what Adams wrote.
ETA ninja’d and better elucidated by Indistinguishable…
But by your reasoning, 0/0 = lim (ka/a) as a -> 0 for any fixed k and for any nonzero a ka/a = k so obviously 0/0 = k for whatever value of k you want or indeed all values of k
then of course there’s the case that if we knew of only one world which was unpopulated, what is ∞-1?
If there are an infinite number of worlds then you can never say only a finite number of them are populated. You can say that from the bit you can see only X% are populated but you cannot say that is the sum total of populated planets.
It may be that you are right that those are the only ones but far more likely that an infinite number of planets are populated not matter how rare they are (assuming an infinite number of planets in the universe).
Is Adams correct in saying the average population of any world in the universe is near, but not quite, 0? Then only his last line concluding that the rounded average of populations of all worlds in the universe times the infinite number of worlds will define the population of the universe is incorrect.
I am not saying anything, just “explaining” Adams’s double joke 
I agree that, in real life, even through a magic ideal telescope you could only observe a finite number of planets.
2 x 5 = 10 because if you have 2 bags with 5 apples in each bag, you have 10 apples.
5 x 2 = 10 because if you have 5 bags with 2 apples in each bag, you have 10 apples.
2 x 0 = 0 because if you have 2 bags with 0 apples in each bag, you have 0 apples. You’d still have 0 apples no matter how many empty bags you had, so anything x 0 = 0. This is still true for 0 x 0 = 0.
0 x 5 = 0 because if you have 0 bags of 5 apples, you have 0 apples. No matter how many apples each bag holds, if you have 0 of those bags, you have 0 apples. So 0 x anything = 0, and this still works if the anything is 0.
Wrong; zero doesn’t mean “no amount.” It’s true that there is no such number that you can multiply by zero and get a nonzero number—no number of zeroes you can add together and get a nonzero number. This means that #/0 is not a number. It’s not equal to zero, zero is a number.
Here you’re a lot closer to being correct. But a mathematician wouldn’t say that 0/0 is “unsolvable”; he’d say it’s “indeterminate.”
A mathematician wouldn’t talk about an expression or operation, like 2+3 or 2x5 or 12/4, or 0x5 or 0/5 or 0/0, as having a solution; he’d talk about it having a value. 2+5 has the value of 5. 0x0 has the value of 0. And, as you correctly note, 0/0 doesn’t have any one specific value. There are infinitely many different values it could possibly have.
Well, you might similarly say that you can never know that there are an infinite number of worlds. In the real universe there’s no way to know such a thing. But if one assumes that somehow you know that there are an infinite number of worlds, then by whatever magical means you acquired that knowledge, you may also have learned that a finite number of them are populated.