If division can be thought of as simply another way to do subtraction, why does dividing by zero not result in infinity?

Question

First off, I won't claim to be a math expert: it was debatably my least favorite subject in school, and Pre-Calculus was where I reached the limit of my mathematical capabilities. However, I recently saw Alan Becker's new video Animation vs. Math which helped me visualize some of the concepts I struggled with in school. One novel way the video helped me conceptualize division was as simply another means of doing subtraction. You take the dividend and the divisor, then subtract the divisor from the dividend repeatedly until you reach zero. The number of times you subtracted the divisor is the quotient. So, if we follow this logic, why does dividing by zero not result in an infinitely large number, since you would subtract zero infinitely many times?

Answer 1

Writing $\frac{1}{0} = \infty$ does make a certain kind of sense, and there are contexts in mathematics where we adopt this convention. Still, this definition can be problematic.

For example, $0 = 1-1$, so $\frac{0}{0} = \frac{1}{0}-\frac{1}{0}$ using ordinary rules of arithmetic. Based on your intuition for how division works, I'm guessing you'll say that $\frac{0}{0} = 0$, so we end up concluding that $0 = \infty - \infty$. But now, let's consider $1 = 2-1$. Diving again by zero, we get $\frac{1}{0} = \frac{2}{0}-\frac{1}{0}$, or $\infty = \infty - \infty$. Now we're forced to conclude that $0 = \infty - \infty = \infty$, so $0$ and $\infty$ are the same. This is clearly absurd.

Another thing you can think about is how to define $\frac{1}{0} \times 0$. Do the zeros cancel out and we get $1$? What about $\frac{2}{0} \times 0$? Is this $2$? Does this mean that $1 = \infty \times 0 = 2$?

These examples pretty much give the intuition as to why dividing by $0$ isn't straightforward. Whatever convention you adopt, combined with the simple rules of arithmetic, forces you into all kinds of absurdities.