# Why is any number multiplied by 0=0?



## Harley

So, this evening I am reading a book in which the first chapter is about 0, and namely how it is used in math and its history. While doing this, a certain thought entered my head and now I'm trying to make sense of it: *why is any number multiplied by zero equal zero?*. Because really, at first glance it all seems so simple. 5x0=0. 123x0=0. 38294034x0=0. I think that we all get blind sighted by the fact that any equation involving zero and any number is so _easy_ to solve, that we don't really care to think about why it is so.

Here's how I see it: any number added by 0 equals that same number. Makes total sense because, if I have 7 apples in a basket and I add no other apples (0) then the number will remain unchanged. Now let's look at multiplication. 7x1=7. That makes sense to me because 7 times itself is essentially just 7+0. Now what I want to know is, why can't 7x0 also be 7? In my view 7x0 is pretty much analogous to 7+0. I mean, you're not multiplying 7 by anything, so would that not be the same as not adding anything to 7?

OK, you know what, let me cross-multiply and see if I can get this right.
7x0=0 cross-multiply 0=0/7. 
Yea, still don't get it. Don't get me wrong, I understand _how_ basic arithmetic works, I'm not looking for a review in grade 3 math. I understand the logistics of 0/7=0, but for some reason looking at it the other way, just seems to lose all sense for me. 

I also don't understand why any number divided by 0 is not possible. I mean if you have 7 pizza slices and you divide it by NOTHING, would you still not have 7 pizza slices left? You virtually left the damn thing untouched.

To me, it would make more sense if 7x0=7, and cross-multiply that to be, 7=7/0. Stay with me here, because if 7 multiplied by nothing equals 7, then 7 divided by zero should then be 7, because you have not divided 7 by anything. Well that's how I see it.

Ah, fuck it, I'll create my own form of mathematics.


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## Lucretius

If you multiply a figure _zero times_, or in other words, have not even 1 _multiple _of said figure, you're going to be left with absolutely nothing.

To divide by zero is to try to divide something into "zero parts." If you divide into one part, then you are left with the original figure. In the case of 2, half of the original figure. But in the case of zero, there is just no coherent answer.


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## Harley

Alright then so, 5x0=0 simply because you start out with 0 _in the first place_. 

But if you had nothing in the first place, what would even be the _point_ of multiplying with anything anyways? I know that's unrelated to my original post, but man that is just so redundant. 



> To divide by zero is to try to divide something into "zero parts."


 I know were you're coming from yet from my POV, you yourself wrote _something into "zero parts"_ which I interpret as "I already have something (a number) and I'm going to leave it *as is*, and that is where the 0 comes from. But what n x 0 is, is essentially nothing times (or divided) by nothing. 

Ugh, I don't know the 5 (or any number) is so misleading, because I always think that there is _something_ in the equation rather than nothing. I mean, why would you want to multiply _nothing_ with _nothing_. So stupid. OK, then thanks for the clarification.


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## RedDeath9

Alright, so let's say there's a guy, and he wants to fill his hot tub. The hot tub has a volume of 100 L. In order to fill it, he's gonna take his 1 L bucket to a well, fill it up, and take it back to the hot tub, dump it in. He's obviously going to have to do this 100 times.

100 L total / 1 L per trip = 100 trips.

Now, what happens if someone, as a joke, nails planks over the top of his hot tub, effectively covering it? This would make the volume of the hot tub zero. So... How many trips does he need to make to fill the hot tub?

0 L total / 1 L per trip = 0 trips. 

Alright, so he's pissed, but he takes off the planks. But THIS TIME, that prankster cut a hole in the bottom of his bucket. Now, every time the guy goes to fill it up, the bucket is _empty_ by the time he gets back to the hot tub. So it has a capacity of 0 L. How many trips does he need to make?

100 L / 0 L per trip = ... Infinity. He'd be going back and forth forever, and forever does not exist.

So that's kind of an oversimplified explanation about why you can't divide by zero.

As for 7x1... Essentially what it means is that you have _one_ group of sevens. 7x2 -> two groups of seven. 7x0 -> no groups of seven, therefore zero.

(All credit to Mr. Hashimoto, my high school math teacher, for the hot tub analogy.)


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## Cman

Here's another thing that bothers me. Why isn't 0*(8/0) = 8?? All of my thousands of math classes have failed to give me a reason other than "it's undefined."


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## RedDeath9

Short answer: lrn2calculus

Slightly longer but still short answer:

One aspect of calculus is the concept of _limits_ - That is, values _approaching_ a certain number. Assuming you know functions, let's say f(x) = 1/x. When taking limits, we say:

The limit as x approaches 1 of f(x) is 1. (since 1/1 is 1)
The limit as x approaches 10 of f(x) is 0.1. (since 1/10 is 0.1)

Going the other way (smaller x values), we get:

The limit as x approaches 0.1 of f(x) is 10. (since 1/0.1 is 10)
The limit as x approaches 0.01 of f(x) is 100. (since 1/0.01 is 100)

So what is the limit of f(x) as x approaches _0_? As you can tell, the f(x) values get bigger and bigger as x approaches 0. That is, f(x) gets infinitely large as x gets closer to 0. Therefore, the limit as x approaches 0 of f(x) is infinity. _Approaches_ is the key word here. x never actually reaches 0, since f(x) never actually reaches infinity. But in calculus, we take "0" and "approaching 0" to mean the same thing.

There are two things you kinda need to know. They are:

0.00... 01 = 0
and
0.99... 9 = 1

There are proofs for these... Look them up if you don't believe me.

So, going back to f(x) = 1/x. As x _approaches_ 0, f(x) _approaches_ infinity. Similarly, as x approaches infinity, f(x) approaches 0.

*Now, going back to your question^*

You actually have a point. Let's say you had a function

f(x) = 8x/x.

As x approaches 0, we get:

8(0)/(0) = 0/0, which makes no goddamn sense whatsoever.

So in calculus, what you're allowed to do is _cancel_ the x's, which represent values approaching zero. Now, in regular math, you wouldn't be allowed to cancel zeroes - It fucks shit up - but this is calculus, and we're dealing with a value that isn't actually ZERO, it's APPROACHING zero. So you cancel the x's, and you get:

lim as x approaches 0 of f(x) = 8. So you're kinda right.

But what you've done isn't a function, or a limit. Also, you've put brackets around 8/0, which means THAT has to be calculated first. As explained above, 8/0 equals infinity. Now we've got 0*&#8734;. Whether in simple math or calculus, this makes no sense. Below, I've explained why:

So we know any real number divided by zero = &#8734;, right? So let's say:

c = any real number

c/&#8734; = 0 -> equation 1

We want to know what 0*&#8734; is. So we set 0*&#8734; = y -> equation 2

(and although it probably doesn't matter to you, let's say both of the infinities in equation 1 and 2 are the same)

So, we said that 0 = c/&#8734;. So we can replace the zero in equation 2 to get:

c/&#8734; * &#8734; = y.

We can change this expression into a function, and say:

y = cx/x

x can take the place of both infinities, since we said them to be equal.

Now, we take the limit of y as cx/x as x approaches infinity. Without canceling, we get &#8734;/&#8734;... Which equals 1, since we said both infinities were equal. But we could also do this by canceling the x's, since we're dealing with limits. By canceling, we get 

y = c.

y was our answer to 0*&#8734;, and c was defined as _any real number_. That's why 0*&#8734; is indeterminate - It could be any real number.


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## Kirsa

Consider the 7 piece pizza's point of view. It doesn't want to be divided at all. It will miss it parts, and so when zero comes along to snatch its piece, the zero will take none of the pizza in the real number system because it has so many choices of pieces from which to choose. The indecisive zero can't decide which one and therefore opts for none, but in a complex number system of an alternate reality, that dastardly zero can take any number of pieces it wants amidst the screams of torment that pizza feels as it's ripped apart into infinity and the pizza will still be 7 in prime's universe.


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## dalsgaard

Because multiplication is just a factor, it tells you how many of something else you have. If you put 0*5 that means that you don't have any fives:

0*5=0 zero fives
1*5=5 one five
2*5=5+5 two fives
3*5=5+5+5 three fives
4*5=5+5+5+5 four fives
.
.
.


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## carson

The other possbility is that the 7 pieces of pizza change in a way that cannot be measured by mathematics. The act of applying mathematics to an object eliminates the mathematics, but changes the status of the object. 7 x 0 may equal 0, but what happened to the x? Where did the act multiplication go?

You had 7 pce pizza, it was there and yummy, smelling great ready to be eaten...mozzerella, tomato, garlic, basil...maybe even anchovies. You reach out, but just before you touch it, mathematics turns up and says...this is not a pizza, it is seven objects! 

Now instead of seeing a tasty treat, you see seven pces of lifeless inanimate object. An object that will never mold or decay throughout time. None of your senses can find the pizza. Now it is only comprehended through numerical values. It is an Italian tradgedy of the darkess kind. Mario slaved over hot coals, pouring his love and experience into your meal and now you have betrayed his love with numbers, stolen, discarded it, and in the process, deadened your senses. 

How may you rejoin your pizza in it's now still beauty? If only you had eaten it sooner, oh the loss, the pain. You caress it's tender crust. Goodbye my love...goodbye... the memory of your tasty filling will live with me always... and throw yourself from the ramparts.

Mario finds your lifeless body outside the doorway of his shop. (He works just below the ramparts). Tears fall from his eyes and he rages...why do people count my slices! They do not ask to be counted. Is not the love of magherita enough?!Damn you mathematics! 

The curtain closes and you vow never to count another pizza slice again, knowing full well that mathematics will always lose, even as it snatches the taste of love from your mouth. You see a pizza seller as you walk home and he sees you, once, and then twice, he knows you know and you know he knows you know.


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## Nenad

I think you're looking at it the wrong way. It's not 7 pizza slices multiplied by 0 times (7x0). It's more like 0 slices multiplied by 7 times (0x7). If you have 0 slices and multiply them by any number you still get zero in the end.


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