Where else would a purely imaginary number go? (How far East is due North?) Well, “i” sure looks like zero when we’re on the real number line: the “real part” of i, Re(i), is indeed 0. Is “0 + i”, a purely imaginary number, the same as zero? The notion of zero is biased by our expectations. A dilemma is at hand! The Solution: Zero is Relative We want the best of both: slices so thin we can’t see them (for an accurate model) and slices thick enough to create a simpler, easier-to-analyze model. We see that our model is a jagged approximation, and won’t be accurate. If the slices are tiny but measurable, the illusion vanishes. Now there’s no benefit - the ‘simple’ model is just as complex as the original! Additionally, adding up zero-width slices won’t get us anywhere.

If the slices are too small to notice (zero width), then the model appears identical to the original shape (we don’t see any rectangles!). The Paradox of Zeroīreaking a curve into rectangles has a problem: How do we get slices so thin we don’t notice them, but large enough to “exist”? Limits and infinitesimals help us create models that are simple to use, yet share the same properties as the original item (length, area, etc.). The tricky part is making a decent model.

The simpler model, built from rectangles, is easier to analyze than dealing with the complex, amorphous blob directly. The thinner the rectangles, the more accurate the model. We can break a complex idea (a wiggly curve) into simpler parts (rectangles):īut, we want an accurate model. So many math courses jump into limits, infinitesimals and Very Small Numbers (TM) without any context.

Dividing multiplying infinitesimals