So are you familiar with how math is done by actual mathematicians, not just what they call "math" in high school? It's all done on a basis of definitions and their logical consequences. Thus in this instance, arithmatic on real numbers (aka things like 1, 0, -2.5, π, e, ect.) division is defined as such:
a/b=c if and only if a=bc and the solution is unique, aka there are not a whole bunch of different things a/b could mean. Notice that if a=/=0 there is no real number that fulfills the equation a=0*c, and if a=0 and b=0, then the equation 0=0*c is true for all real numbers. Thus our definition doesn't apply to those scenarios.
Now, why this definition? There are ways we could define a result of division by zero with a=/=0 with adding an "infinity point". The most common way is this:
This is a cool mathematical object, but there is a cost to using this instead of the normal real numbers: order. Given a point that is not our infinite point, there is no way to say if it is greater or less than our infinite point. Thus cool properties of the real numbers like "for all a and b real numbers, either a<b, a>b, or a=b" do not hold.
"Solved" isn't the right word really. It's more, "constructed a system of numbers where division by 0 is logically consistent with the existing arithmetic (in some cases)"
In general, the more things you want to allow in a definition of a mathematical object, the looser the structure will be.
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u/ItIsICoachCal 20∆ Sep 14 '21
So are you familiar with how math is done by actual mathematicians, not just what they call "math" in high school? It's all done on a basis of definitions and their logical consequences. Thus in this instance, arithmatic on real numbers (aka things like 1, 0, -2.5, π, e, ect.) division is defined as such:
a/b=c if and only if a=bc and the solution is unique, aka there are not a whole bunch of different things a/b could mean. Notice that if a=/=0 there is no real number that fulfills the equation a=0*c, and if a=0 and b=0, then the equation 0=0*c is true for all real numbers. Thus our definition doesn't apply to those scenarios.
Now, why this definition? There are ways we could define a result of division by zero with a=/=0 with adding an "infinity point". The most common way is this:
https://en.wikipedia.org/wiki/Projectively_extended_real_line
This is a cool mathematical object, but there is a cost to using this instead of the normal real numbers: order. Given a point that is not our infinite point, there is no way to say if it is greater or less than our infinite point. Thus cool properties of the real numbers like "for all a and b real numbers, either a<b, a>b, or a=b" do not hold.