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I need a real-life example of division by zero to understand why a division by zero would not be undefined.

Exactly, reality becomes inconvenient at some point for science... If this happens to simple algebra then what about advance physics? If 1 × 0 has an result so should 1 ÷ 0. But not quite, the issue is mixing apples with pears, zero is not a number, zero = the undefined itself.