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• In the world of math,

• 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:

• don't divide by zero.

• How can the simple combination of an everyday number

• and a basic operation cause such problems?

• Normally, dividing by smaller and smaller numbers

• gives you bigger and bigger answers.

• Ten divided by two is five,

• by one is ten,

• by one-millionth is 10 million,

• and so on.

• 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?

• That may sound plausible.

• But all we really know is that if we divide 10

• by a number that tends towards zero,

• the answer tends towards infinity.

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

• is equal to infinity.

• Why not?

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

• Ten divided by two could mean,

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

• or, “two times what equals 10?”

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

• in the following way:

• if we multiply any number by a given number x,

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

• to get back to where we started.

• 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

• is always one.

• If we want to divide by zero,

• we need to find its multiplicative inverse,

• which should be one over zero.

• 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,

• such a number is impossible,

• so zero has no multiplicative inverse.

• Does that really settle things, though?

• After all, mathematicians have broken rules before.

• For example, for a long time,

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

• But then mathematicians defined the square root of negative one

• as a new number called i,

• opening up a whole new mathematical world of complex numbers.

• So if they can do that,

• couldn't we just make up a new rule,

• say, that the symbol infinity means one over zero,

• and see what happens?

• Let's try it,

• 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.

• Now, by the distributive property,

• the left side of the equation can be rearranged

• to zero plus zero times infinity.

• 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.

• So, one equals 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.

• But having infinity 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,

• but that's a story for another day.

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

• doesn't work out so great.

• 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.

In the world of math,

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# なぜ0で割ることができないのか？- TED-Ed (Why can't you divide by zero? - TED-Ed)

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Evangeline に公開 2021 年 01 月 14 日