As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?
So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?
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We can, and it is called [projectively extended reals](https://en.wikipedia.org/wiki/Projectively_extended_real_line). As others have said, we have to consider the consequences. Every time we add something to a number system, we lose some nice properties.
Going from real numbers to complex numbers isn’t “free” either. The biggest one (I’m aware of) is we lose total order. Namely, for any real numbers a and b, either a > b, a < b or a = b. This does not hold for all complex numbers.
Each of the following holds for all real numbers a, but does not hold for all projectively extended real numbers a.
– a × 0 = 0 × a = 0
– a – a = 0
Also, you know how when doing algebra in real numbers, you have to be careful with division by zero? For example, when given x^(2)-5x+4 = x-4, you can’t just divide both sides by (x-4). If you want to divide you have to break it into two cases where x=4 and x≠4 to arrive at (x^(2)-5x+4)/(x-4) = 1 OR x = 4
When dealing with projectively extended reals, this spreads to addition, subtraction and multiplication. In reals, only division can produce undefined results. In projectively extended reals, all four basic arithmetic operations can. For example, given x+a = x+b, we **cannot** subtract x from both sides and conclude a = b. Like the example above, we have to consider the two cases where x=∞ and x≠∞ to arrive at a=b OR x=∞.
Is defining 1/0 = ∞ useful? Maybe. Is it worth the cost of losing some nice arithmetic properties as discussed above? That is for you to decide.
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