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Maybe we can say that 1 divided by 0 is positive infinity. As we get smaller and smaller positive numbers here, we get super super large numbers right over here. But then, your friend might say, well, that worked when we divided by positive numbers close to zero but what happens when we divide by negative numbers close to zero?
- The problem with dividing zero by zero (video) | Khan Academy
One can argue that 0/0 is 0, because 0 divided by anything...
- The problem with dividing zero by zero (video) | Khan Academy
2 Δεκ 2020 · Division is defined by solving an equation that can be easily solved for all pairs x, y of numbers where y ≠ 0. You can give arguments why it is reasonable not to define division by zero, but the fact is that it is not defined. We could easily define division by zero: We could just say that x / y := x whenever y = 0.
Try Multiplying By Zero. So let us try using our new "numbers". For example, we know that zero times any number is zero: Example: 0×1 = 0, 0×2 = 0, etc. So that should also be true for 1 0: 0 × 1 0 = 0. But we could also rearrange it a little like this: 0 × 1 0 = 0 0 × 1 = 1. (Careful!
The reciprocal function y = 1 x. As x approaches zero from the right, y tends to positive infinity. As x approaches zero from the left, y tends to negative infinity. In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case.
6 Δεκ 2021 · The argument follows that as the number approaches zero, the result approaches infinity. Therefore, we should approximate the result for 1/0 as infinity.
This resolves your problem because it shows that $\frac{1}{\epsilon}$ will be positive infinity or infinite infinity depending on the sign of the original infinitesimal, while division by zero is still undefined.
One can argue that 0/0 is 0, because 0 divided by anything is 0. Another one can argue that 0/0 is 1, because anything divided by itself is 1. And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another.
