Multiplying by Zero. Need help understanding.
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 sonickrahnic
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Multiplying by Zero. Need help understanding.
OK, so we all know basic math rules that tell us that when you multiply zero by a number, the product is zero. I rationalize it like this: if you have zero items and multiply them by five, you still have zero items. But what I don't understand is this. If you have five items and you multiply them zero times, shouldn't the five items still be there, unchanged? If you have something and do not multiply it it has to be unchanged right? In my opinion anyway. I am definitely not a mathematician, but I am very intrigued by numbers and their relationships and have always wondered why it works this way. Can someone please explain this?
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 gmalivuk
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Re: Multiplying by Zero. Need help understanding.
No, 1 is the number that keeps things the same when you multiply. Think of it like area: if you've got a box 5 feet long on one side, and 1 foot long on the perpendicular side, then you've got 5 square feet. But if the width is zero, then there's no area at all.
Re: Multiplying by Zero. Need help understanding.
Different people's intuitions will differ, but what always made the most sense for me with these types of questions was just continuing the pattern.
If you believe
5 times 3 is 15
5 times 2 is 10
5 times 1 is 5
then surely 5 times 0 being 0 continues that pattern.
You're not multiplying them zero times; you're multiplying them by the number zero.
Five items multiplied by one would be like you have one set of five (which is the same as having a set of five and not doing anything to it). Five items multiplied by two would be two sets of five. Five items multiplied by zero means you have ZERO sets of five, i.e., zero items.
(Also, are you happy with two times three and three times two being the same as each other? Does it seem like the pattern "a times b equals b times a" should continue, in order to "feel right"?)
When dealing with multiplication, it's a mistake to think of the number zero as "not doing anything".
If you're adding, it makes sense to think of zero as "the number that doesn't do anything".
If you believe
5 times 3 is 15
5 times 2 is 10
5 times 1 is 5
then surely 5 times 0 being 0 continues that pattern.
sonickrahnic wrote:OK, so we all know basic math rules that tell us that when you multiply zero by a number, the product is zero. I rationalize it like this: if you have zero items and multiply them by five, you still have zero items. But what I don't understand is this. If you have five items and you multiply them zero times, shouldn't the five items still be there, unchanged?
You're not multiplying them zero times; you're multiplying them by the number zero.
Five items multiplied by one would be like you have one set of five (which is the same as having a set of five and not doing anything to it). Five items multiplied by two would be two sets of five. Five items multiplied by zero means you have ZERO sets of five, i.e., zero items.
(Also, are you happy with two times three and three times two being the same as each other? Does it seem like the pattern "a times b equals b times a" should continue, in order to "feel right"?)
sonickrahnic wrote:If you have something and do not multiply it it has to be unchanged right?
When dealing with multiplication, it's a mistake to think of the number zero as "not doing anything".
If you're adding, it makes sense to think of zero as "the number that doesn't do anything".
 sonickrahnic
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Re: Multiplying by Zero. Need help understanding.
Well, that kind of makes sense, and I know multiplying a number by one gives the original number as the product. What I dont get is that it makes perfect sense one way but not another. I can see how your example makes sense on the one hand, but on the other, if you extend the line segment five feet but do not ad the one (or other number) width, it has no square footage, but it still has linear footage, five feet.
But I can see what you are both saying and I am definitely understanding it more. I do understand that if an equation works one way it should work the same going the other way. I program a bit and know that there are multiple ways of doing the same thing within one language so obviously, programming being based on math, it should be the same in math. It has just always bothered me that if you have five items and multiply them zero times, that there should now be zero items. I still need a little clarification if thats ok.
On a side note. I have a book around my house somewhere called Zero, or something of the like, that explains zero in great depth. But I'm currently engrossed by The Drunkard's Walk by Leonard Mlodinow, which documents the intricacies of randomness and probability so once I am done that one, I will have to find the Zero one and maybe my illusions will be put to rest.
But I can see what you are both saying and I am definitely understanding it more. I do understand that if an equation works one way it should work the same going the other way. I program a bit and know that there are multiple ways of doing the same thing within one language so obviously, programming being based on math, it should be the same in math. It has just always bothered me that if you have five items and multiply them zero times, that there should now be zero items. I still need a little clarification if thats ok.
On a side note. I have a book around my house somewhere called Zero, or something of the like, that explains zero in great depth. But I'm currently engrossed by The Drunkard's Walk by Leonard Mlodinow, which documents the intricacies of randomness and probability so once I am done that one, I will have to find the Zero one and maybe my illusions will be put to rest.
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Re: Multiplying by Zero. Need help understanding.
sonickrahnic wrote:It has just always bothered me that if you have five items and multiply them zero times, that there should now be zero items. I still need a little clarification if thats ok.
Zero is NOT the number of times you're performing the action.
You are multiplying the five items by the NUMBER zero. It's like saying, "I'm going to consider this set of five items, and ask myself what would happen if I had zero groups of five, or zero sets of five."
Re: Multiplying by Zero. Need help understanding.
Multiplying zero times would be 5^0. 5*0 is like adding 5 zero times, which reasonably leaves you with nothing.
 sonickrahnic
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Re: Multiplying by Zero. Need help understanding.
skullturf wrote:You are multiplying the five items by the NUMBER zero. It's like saying, "I'm going to consider this set of five items, and ask myself what would happen if I had zero groups of five, or zero sets of five."
OK, that makes way more sense. I learn through language so if I don't have something presented in a certain way, I may not pick up the whole meaning. But your explanation makes a lot of sense.
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Re: Multiplying by Zero. Need help understanding.
sonickrahnic wrote:
OK, that makes way more sense. I learn through language so if I don't have something presented in a certain way, I may not pick up the whole meaning. But your explanation makes a lot of sense.
Cool, I'm glad.
And you're certainly not alone in your confusion. I remember times when one of my students would solve an equation, and then at the end get 0 as their answer. In some cases, they might say something like "So... there are no solutions?"
That remark is like a confusion of levels, in a way. The 0 doesn't mean "there aren't any solutions"  on the contrary, there IS a solution, but it happens to be the number 0.
Re: Multiplying by Zero. Need help understanding.
Here's how I think about it:
multiplication works by taking the number 0 and adding x sets of y.
x*y = 0 + x*y
in your example if we multiply 0*5 we add 0 sets of five to 0 and adding 0 sets means adding nothing.
multiplication works by taking the number 0 and adding x sets of y.
x*y = 0 + x*y
in your example if we multiply 0*5 we add 0 sets of five to 0 and adding 0 sets means adding nothing.
I am imaginary, prove me wrong!
Re: Multiplying by Zero. Need help understanding.
sonickrahnic wrote: If you have five items and you multiply them zero times, shouldn't the five items still be there, unchanged? If you have something and do not multiply it it has to be unchanged right?
For this example, it's more accurate to assume you are drawing the items out of a bag.
Thus "zero times five" would translate to drawing zero items out of the bag five times, while "five times zero" would be drawing five items out of the bag zero times. This should highlight that in your query, you're assuming you've already drawn once, and THEN drawing zero times. It's a bit like a woman saying "I can't be pregnant! I haven't had sex in months!"
I can see though, how translating "zero times five" into a real life situation is easier to visualize than "five times zero."
For example, someone might say "I've been fishing five times, and every time I come home empty!" To calculate the number of fish caught, you would think: zero fish each time, went fishing five times, so 0*5=0.
However, if someone says "Every time I fish, I bring home five," it's natural to assume the person has gone fishing at least twice, and thus have caught at least ten fish. You simply don't expect a person to say "I catch five fish every time I go fishing, but I've never gone fishing."
This brings up the concept of something being vacuously true. That is, the concept that if you assume something false as true, then logically, everything is true.
So to use the previous example, the blank in this sentence can be filled with literally any true/false statement:
"Every time I go fishing, ______________________. However, I've never been fishing."
The name "vacuously true" is meant as an analogy to an item in a vacuum. Since there are (by definition) no items in a vacuum, if you assume there ARE items in the vacuum, you can give them any property you want.
So, in a vacuum, every cat is a dog. In a vacuum, every cat is not a dog. That Saint Patrick drove the snakes from Ireland is vacuously true, as long as you take it literally.
 GenericPseudonym
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Re: Multiplying by Zero. Need help understanding.
skine wrote:[...] However, if someone says "Every time I fish, I bring home five," it's natural to assume the person has gone fishing at least twice, and thus have caught at least ten fish. You simply don't expect a person to say "I catch five fish every time I go fishing, but I've never gone fishing." [...]
Perhaps a less (to use your own words) vacuous analogy would be to day "every time I go fishing I catch five fish, but I haven't gone fishing today" therefore having caught [imath]5 x 0 = 0[/imath] fish today, but possibly more over the course of one's life.
Re: Multiplying by Zero. Need help understanding.
In addition to all those arguments using intuition an common sense, there is also a mathematical proof.
Hee it comes:
Given the following field axioms,
(A) 0 + a = 0
(B) 1 * a = a
(C) a + (a) = 0
(D) a + (b + c) = (a + b) + c
(E) a * (b + c) = a * b + a * c
one can proof that 0 * x = 0 , where x is an arbitrary number, as follows:
(I put the applied axiom beside the equality sign, I'm to lazy to do it properly in TeX)
0 * x
(A)= 0 * x + 0
(C)= 0 * x + (x + (x))
(B)= 0 * x + (1 * x + (x))
(D)= (0 * x + 1 * x) + (x)
(E)= (0 + 1) * x + (x)
(A)= 1 * x + (x)
(B)= x + (x)
(C)= 0
Hee it comes:
Given the following field axioms,
(A) 0 + a = 0
(B) 1 * a = a
(C) a + (a) = 0
(D) a + (b + c) = (a + b) + c
(E) a * (b + c) = a * b + a * c
one can proof that 0 * x = 0 , where x is an arbitrary number, as follows:
(I put the applied axiom beside the equality sign, I'm to lazy to do it properly in TeX)
0 * x
(A)= 0 * x + 0
(C)= 0 * x + (x + (x))
(B)= 0 * x + (1 * x + (x))
(D)= (0 * x + 1 * x) + (x)
(E)= (0 + 1) * x + (x)
(A)= 1 * x + (x)
(B)= x + (x)
(C)= 0
 agelessdrifter
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Re: Multiplying by Zero. Need help understanding.
XCV wrote:(A) 0 + a = 0
typo?
Re: Multiplying by Zero. Need help understanding.
agelessdrifter wrote:XCV wrote:(A) 0 + a = 0
typo?
No. It's the newest, coolest axiom! You haven't heard of it yet?
double epsilon = .0000001;

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Re: Multiplying by Zero. Need help understanding.
What I like to do is think of things and not numbers.
As long as it makes sense. For example, it can be bunches of melons.
Say you can take any number of bunches, and all the bunches contain equally many melons.
Then 5*0 can be interpreted in two ways.
1) 5 bunches of 0 melons or 2) 0 bunches of 5 melons.
Either way you end up having 0 melons.
1) means you take 5 things that contain nothing at all. Now, no matter how many (though finitely many) you grab, they are empty, and you end with nothing.
2) simply means you take nothing, so it doesn't matter how many melons the bunches contain  you don't get any.
Cause you're right, imagining numbers is hard. Imagining bunches of melons, on the other hand, is easy.
As long as it makes sense. For example, it can be bunches of melons.
Say you can take any number of bunches, and all the bunches contain equally many melons.
Then 5*0 can be interpreted in two ways.
1) 5 bunches of 0 melons or 2) 0 bunches of 5 melons.
Either way you end up having 0 melons.
1) means you take 5 things that contain nothing at all. Now, no matter how many (though finitely many) you grab, they are empty, and you end with nothing.
2) simply means you take nothing, so it doesn't matter how many melons the bunches contain  you don't get any.
Cause you're right, imagining numbers is hard. Imagining bunches of melons, on the other hand, is easy.
Re: Multiplying by Zero. Need help understanding.
There's another algebraic proof that I think is a little bit easier than XCV's proof.
First we take the definition of the zero element and one of the the field axioms [math]0+a=a[/math] [math]a(b+c) = ab+ac[/math]
Before using the axiom, we can evaluate the a multiplying a quantity a different way if c=0 because it is the additive identity [math]a(b+0)=a(b)=ab[/math]
Now applying the axiom [math]a(b+0)=ab+a*0[/math]
Since we already proved that [imath]a(b+0)= ab[/imath] that means a*0 must equal 0.
Edit for clarity's sake.
First we take the definition of the zero element and one of the the field axioms [math]0+a=a[/math] [math]a(b+c) = ab+ac[/math]
Before using the axiom, we can evaluate the a multiplying a quantity a different way if c=0 because it is the additive identity [math]a(b+0)=a(b)=ab[/math]
Now applying the axiom [math]a(b+0)=ab+a*0[/math]
Since we already proved that [imath]a(b+0)= ab[/imath] that means a*0 must equal 0.
Edit for clarity's sake.
Last edited by B.Good on Fri Nov 19, 2010 9:33 pm UTC, edited 3 times in total.
 agelessdrifter
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Re: Multiplying by Zero. Need help understanding.
B.Good wrote:Now applying the axiom a bit more thoroughly [math]a(b+0)=ab+a*0[/math]
doesn't this beg the question?
Re: Multiplying by Zero. Need help understanding.
agelessdrifter wrote:B.Good wrote:Now applying the axiom a bit more thoroughly [math]a(b+0)=ab+a*0[/math]
doesn't this beg the question?
How so? This follows the axiom for a field and doesn't assume that a*0=0. There may just be something I am not getting.
 agelessdrifter
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Re: Multiplying by Zero. Need help understanding.
I'm sorry, I meant to quote the step above that:
BUT, I see it now. You mentioned the additive identity property of zero. I thought you were distributing the a to zero and just taking a*0=0 without showing the step.
B.Good wrote:Now we take c to be zero [math]a(b+0)=a(b)=ab[/math]
BUT, I see it now. You mentioned the additive identity property of zero. I thought you were distributing the a to zero and just taking a*0=0 without showing the step.
Re: Multiplying by Zero. Need help understanding.
agelessdrifter wrote:I'm sorry, I meant to quote the step above that:B.Good wrote:Now we take c to be zero [math]a(b+0)=a(b)=ab[/math]
BUT, I see it now. You mentioned the additive identity property of zero. I thought you were distributing the a to zero and just taking a*0=0 without showing the step.
I admit that I didn't point out the definition of the zero element as well as the rest of my post. Maybe I'll put it in the math tags with the rest of the post so it will stand out more.
Re: Multiplying by Zero. Need help understanding.
The way I proved it in maths was:
a*0 = a*0
a*0 = a*(0+0)
a*0 = a*0 + a*0
a*0  a*0 = a*0 + a*0 a*0
0 = a*0
Where 0 is defined as the additive identity (i.e. a + 0 = a)
a*0 = a*0
a*0 = a*(0+0)
a*0 = a*0 + a*0
a*0  a*0 = a*0 + a*0 a*0
0 = a*0
Where 0 is defined as the additive identity (i.e. a + 0 = a)
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