Name: 為什麼你不能被零除掉？- TED-Ed (Why can't you divide by zero? - TED-Ed)
Uploaded: 2021-01-14T07:34:40.000Z
Duration: 4 min 51 s
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many strange results are possible  when we change the rules.

But there's one rule that most of us  have been warned not to break:

How can the simple combination  of an everyday number

and a basic operation  cause such problems?

Normally, dividing by smaller  and smaller numbers

So it seems like if you divide by numbers

that keep shrinking  all the way down to zero,

the answer will grow  to the largest thing possible.

Then, isn't the answer to 10  divided by zero actually infinity?

But all we really know is  that if we divide 10

And that's not the same thing as saying that 10 divided by zero

Well, let's take a closer look  at what division really means.

"How many times must  we add two together to make 10,”

Dividing by a number is essentially  the reverse of multiplying by it,

if we multiply any number  by a given number x,

we can ask if there's a new number  we can multiply by afterwards

If there is, the new number is called  the multiplicative inverse of x.

For example, if you multiply  three by two to get six,

you can then multiply  by one-half to get back to three.

So the multiplicative inverse  of two is one-half,

and the multiplicative inverse  of 10 is one-tenth.

As you might notice, the product of any number and its multiplicative inverse

we need to find  its multiplicative inverse,

This would have to be such a number that multiplying it by zero would give one.

But because anything multiplied by zero is still zero,

After all, mathematicians  have broken rules before.

there was no such thing as taking  the square root of negative numbers.

But then mathematicians defined  the square root of negative one

opening up a whole new mathematical world of complex numbers.

say, that the symbol infinity  means one over zero,

imagining we don't know  anything about infinity already.

Based on the definition  of a multiplicative inverse,

zero times infinity must be equal to one.

That means zero times infinity plus zero times infinity should equal two.

the left side of the equation  can be rearranged

And since zero plus zero  is definitely zero,

that reduces down to zero times infinity.

Unfortunately, we've already defined  this as equal to one,

while the other side of the equation  is still telling us it's equal to two.

Oddly enough,  that's not necessarily wrong;

it's just not true  in our normal world of numbers.

There's still a way it could be mathematically valid,

if one, two, and every other number  were equal to zero.

is ultimately not all that useful  to mathematicians, or anyone else.

There actually is something called  the Riemann sphere

that involves dividing by zero  by a different method,

In the meantime, dividing by zero  in the most obvious way

But that shouldn't stop us  from living dangerously

and experimenting  with breaking mathematical rules

to see if we can invent  fun, new worlds to explore.