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We are told over and over of the fact that you just cannot divide by 0. Why? Yes, I know it doesn't make sense to divide by 0, but really, there has to be a mathematical reasoning of why this cannot be accomplished.
l46kok said:We are told over and over of the fact that you just cannot divide by 0. Why? Yes, I know it doesn't make sense to divide by 0, but really, there has to be a mathematical reasoning of why this cannot be accomplished.
Because division is defined to operate on nonzero denominators -- your question is like asking why we use the word "blue" to decribe light with a wavelength of 465 nm.l46kok said:We are told over and over of the fact that you just cannot divide by 0. Why? Yes, I know it doesn't make sense to divide by 0, but really, there has to be a mathematical reasoning of why this cannot be accomplished.
Hurkyl said:Because division is defined to operate on nonzero denominators -- your question is like asking why we use the word "blue" to decribe light with a wavelength of 465 nm.
There are lots of reasons why this definition is a useful one -- the other posts demonstrate some bad things that we would face if we attempted to define a new division operator that allowed division by zero.
l46kok said:We are told over and over of the fact that you just cannot divide by 0. Why? Yes, I know it doesn't make sense to divide by 0, but really, there has to be a mathematical reasoning of why this cannot be accomplished.
JonF said:Because that’s how fields are defined.
I think my answer is the best one to this question because it let's him go as deep as he wants it has the: “Because that’s how we define it” element but if he wants to know WHY we define it that way he need to study fields.Zurtex said:I know you can define a field like that, but I was always taught to define a field by these:
http://mathworld.wolfram.com/FieldAxioms.html
Where devision is defined through the multiplicative inverse, the Field Axioms then imply division by 0 isn't possible because 0 has no multiplicative inverse. Sorry I may seem a little over the top about this, I just don't feel in this case it's a very sufficient answer for someone who is trying to learn something new.
It can also nicely be shown that a field becomes inconsistent if you allow 0 to have a multiplicative inverse.
l46kok said:We are told over and over of the fact that you just cannot divide by 0. Why? Yes, I know it doesn't make sense to divide by 0, but really, there has to be a mathematical reasoning of why this cannot be accomplished.
CRGreathouse said:Let's try another way.
Then [itex]\{a\}/\{0\}=\mathbb{R}[/itex] if [itex]a=0[/itex] ..
?rbj said:this is potentially true (but i always worry about what it is in the limit, but there doesn't appear to be a limit here), but there is no way we can tell what particular real number [itex]\{a\}/\{0\}[/itex] is. it could be anything, making [itex]\{a\}/\{0\}[/itex] pretty meaningless whether a is zero or not.
morphism said:What CRGreathouse did was define operations on sets. What he wrote down isn't "potentially true", it's absolutely true. Under his definitions, {a}/{0} is not a real number, it's a set.
rs1n said:I prefer to use layman's terms in this case. Dividing is essentially a partition into equal parts. So, if you have A/B, you are essentially dividing A into equal parts of size B, and A/B represents the number of these equal parts. For a concrete example, take 16/2:
If you partition 16 into equal parts of size 2, then you would need 8 such equal partitions. However, what happens when you try to form partitions of size 0. How many would you need? An infinite number of them (and intuitively, even an infinite number of them may not be enough, which leads to indeterminate forms -- but that's for another rainy day). Unfortunately, infinity isn't a number -- it's a concept (at least as far as I am concerned; though it's considered a number in the extended reals).
Diffy said:This is not a mathematical argument, and it assumes too much. It may be easier to think about it in terms of metaphors, but that is not mathematics, it is philosophy!
Why use "laymen's terms" when we have an exact construction and contradiction in proof by CRGreathouse?
rbj said:CRGreathouse said:Let's try another way.
Then [itex]\{a\}/\{0\}=\mathbb{R}[/itex] if [itex]a=0[/itex]..
this is potentially true (but i always worry about what it is in the limit, but there doesn't appear to be a limit here), but there is no way we can tell what particular real number [itex]\{a\}/\{0\}[/itex] is. it could be anything, making [itex]\{a\}/\{0\}[/itex] pretty meaningless whether a is zero or not.
morphism said:?
What CRGreathouse did was define operations on sets. What he wrote down isn't "potentially true", it's absolutely true. Under his definitions, {a}/{0} is not a real number, it's a set.
CRGreathouse said:It took me a few readings, but I think I see what you're getting at. I'm not claiming that {a}/{0} is a real number ([itex]\{a\}/\{0\}\in\mathbb{R}[/itex]) but that {a}/{0} is the set of real numbers ([itex]\{a\}/\{0\}=\mathbb{R}[/itex]).
In fact, I showed that there's no real number to which {a}/{0} corresponds:
[tex]\not\exists e\in\mathbb{R}:\{a\}/\{0\}=\{e\}[/tex]
Hurkyl said:Because division is defined to operate on nonzero denominators .
Division by zero is the mathematical operation of dividing a number by zero. It is represented by the symbol ÷ or / and is read as "divided by".
Division by zero is undefined because it is impossible to find a number that, when multiplied by 0, will result in a given number. This means that there is no solution to the equation x ÷ 0 = a, where a is any number.
When you try to divide by zero, you will encounter a mathematical error or "undefined" result. This is because division by zero violates the fundamental rules of arithmetic and has no meaning in mathematics.
No, division by zero is never allowed in mathematics. It is considered an illegal operation and is not defined in any mathematical system.
No, there are no real-life examples where division by zero is used. In real-life situations, division by zero would lead to impossible or nonsensical outcomes, making it irrelevant and impractical to use.