Division by zero can be thought of in terms of dividing by ever smaller numbers. Divide by 4 and get 4 parts; 2, 2 parts; 1, 1 part.

Divide by 0.5 and you get double what you had. By 0.25 and you get 4 times. Keep making your denominator smaller and smaller towards zero, and your result approaches infinity. So if you divide by zero, is the result always infinity?

No, and as is revealed by calculus, the result of approaching a zero denominator can actually be a real number. So in general, division by zero is undefined until proven otherwise.