![]() But audio and video engineers know they don’t need a perfect reproduction, just quality good enough to trick us into thinking it’s the original.Ĭalculus lets us make these technically imperfect but “accurate enough” models in math. We resist because of our artificial need for precision. This “imperfect” model is fast enough to trick our brain into seeing fluid motion. Video shows still images at 24 times per second. Can you tell a handwritten note from a high-quality printout of the same? But can you tell the difference between a high-quality mp3 and a person talking in the other room?Ĭomputer printouts are made from individual dots too small to see. We’re tricked by “imperfect but useful” models all the time:Īudio files don’t contain all the information of the original signal. We might know the model is jagged, but we can’t tell the difference - any test we do shows the model and the real item as the same. The trick to both approaches is that the simpler model was built beyond our level of accuracy. Infinitesimals build the model in another dimension, and it looks perfectly accurate in ours. ![]() Limits stay in our dimension, but with ‘just enough’ accuracy to maintain the illusion of a perfect model. To you, the rectangular shape I made at the sub-atomic level is the most perfect curve you’ve ever seen.” It’s like getting to the imaginary plane from the real one - you just can’t do it. The precision is totally beyond your reach - I’m at the sub-atomic level, and you’re a caveman who can barely walk and chew gum. Infinitesimals: “Forget accuracy: there’s an entire infinitely small dimension where I’ll make the curve. Oh, you have a millimeter ruler, do you? I’ll draw the curve in nanometers. What’s the smallest unit on your ruler? Inches? Fine, I’ll draw you a staircasey curve at the millimeter level and you’ll never know. Limits: “Give me your error margin (I know you have one, you limited, imperfect human!), and I’ll draw you a curve. ![]() Let’s see how each approach would break a curve into rectangles: These approaches bridge the gap between “zero to us” and “nonzero at a greater level of accuracy”. You see, there are two answers (so far!) to the “be zero and not zero” paradox:Īllow another dimension: Numbers measured to be zero in our dimension might actually be small but nonzero in another dimension (infinitesimal approach - a dimension infinitely smaller than the one we deal with)Īccept imperfection: Numbers measured to be zero are probably nonzero at a greater level of accuracy saying something is “zero” really means “it’s 0 +/- our measurement error” (limit approach) But an atomic measurement would show some mass change due to sweat evaporation, exhalation, etc. Here’s a different brain bender: did your weight change by zero pounds while reading this sentence? Yes, by any scale you have nearby. 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.
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