Just Lines 1mm Graph Paper: Fill & Download for Free

The easiest way is to make up or have made up some slide covers with a fine grid marked on them. A 10 micron grid will do. You can then just count the number of squares which cover the specimen. You might be able to photograph a piece of 1mm graph paper using a black and white film then diminish the image by 100 times onto another piece of film. you would then have a 10 micron grid of black lines on a clear negative.

dB (small ‘d’, big ‘B’) stands for ‘decibel’, that is, one tenth of a Bel.Bels (or decibels, which are how they are always used) are a way to express the ratio between two quantities. They’re used when the ratio can have a wide range, because it’s a logarithmic measure. It’s very practical for electronics analysis because usually circuits operate with a wide range of signal values, and the decibel allows conveniently wide-ranging graphs and measurements to be taken.On its own, the decibel is simply a ratio - it has no units. You need two quantities for it to mean anything - the thing being measured and some reference. Many standard references for various things are defined, and in those cases an additional small letter after dB tells you what the reference in use is.When you are analysing a filter circuit, the thing you are most interested in is how it behaves with respect to frequency. After all, the job of a filter is to remove certain frequencies while allowing others through. The response of the filter is usefully measured in dB, because its wide range allows you to get a really good picture of the filter’s behaviour in the limit.If we are looking at the signal voltage going in and coming out of a filter, the ratio in question is clear - it’s the ratio between input and output. The formula for dB is[math]dB = 10 log_{10}( out/in )[/math]So if the signal going in was, say 1V, and the output was 0.5 v, we get:dB = 10 x log( 0.5/1 ) = -3.01 dBSo to analyse a filter, we establish measurements of the input and output voltages using some suitable test equipment, and we sweep the input frequency through a range of interest to us. For example, if the filter is for audio, we might sweep the input frequency from 20 Hz up to 20,000 Hz.Now the frequency scale itself is quite wide ranging, with a 4:1 order of magnitude. So just like the voltage ratio, it is usually very useful to use a logarithmic scale instead of a linear one. This also corresponds to the way nature often likes to arrange things as well, so it feels ‘right’ when we do this.If we were to plot a graph of the filter’s response against frequency, using a linear scale for frequency would give us a problem in fitting it into a reasonable amount of space and still preserving essential detail. A linear scale might choose, say a 1mm grid to represent 10 Hz. 20 Hz to 200 Hz takes 18mm of space, then another 80mm to get to 1 kHz, and whopping 1.9 metres to reach our maximum of 20,000Hz! The detail at the bottom end is quite hard to see even so, with 10 Hz change fitting in only 1 mm, but at the upper end we are using reams of paper for very little of interest.The logarithmic scale solves this problem. The range 20 Hz to 20,000 Hz is divided into 4 ‘decades’, being the distance between each power of 10. So the first decade goes from 10 Hz to 100 Hz, the second from 100 Hz to 1 kHz, the third from 1 kHz to 10 kHz, and the last from 10 kHz to 100 kHz. Each decade on a graph takes the same amount of distance, but the scale is not linear - instead it compresses towards the larger values, and expands the lower values.You can buy graph paper that is already laid out like this, called log graph paper. Here’s what a piece looks like:This has a log x axis, and a linear y axis. It’s known as log-lin or ‘semi log’ paper. The log scale has 6 decades. This type of graph is ideal for plotting filter frequency responses, because it expands the details at the low end while keeping the space requirements modest at the high end. It turns out that the way electronic circuits behave really likes this sort of arrangement, because it’s literally natural.A quick description of a filter might say that it has a “cutoff” frequency of 10 kHz and roll-off or “slope” of -20 dB per decade.“Cutoff” is the point at which the output level has dropped to half of the input. In other words, when the ratio is -3 dB. It goes on to keep rolling off at a rate of -20 dB every decade. So if the cutoff is 10 kHz, at 100 kHz (one decade) the output has dropped to -20 dB, and by 1 MHz, the output is at -40 dB. This roll off rates appear as straight lines on our log-lin graph, because the dB calculation has already performed a log conversion (and hence can be plotted on a linear scale) and the ‘decade’ nomenclature implies the use of a log frequency axis. And nature being nature, the natural response of a filter is a logarithmic response, which shows as a simple straight line on our graph.

Note that I am interpreting the question as "Why is math harder to master than language in general?". Ask a bunch of English math majors whether they find it more difficult to learn mathematics or to learn Chinese. Secondly "mastery" of math is a fuzzy concept.Source of the below text: http://worrydream.com/KillMath/It's an opinionated piece, but I agree with it. Part one is more relevant to the question, but they deserve to be read together.Language and Visceral InterpretationThe ability to understand and predict the quantities of the world is a source of great power. Currently, that power is restricted to the tiny subset of people comfortable with manipulating abstract symbols.By comparison, consider literacy. The ability to receive thoughts from a person who is not at the same place or time is a similarly great power. The dramatic social consequences of the rise of literacy are well known.Linguistic literacy has enjoyed much more popular success than mathematical literacy. Almost all "educated" people can read; most can write at some level of competence. But most educated people have no useful mathematical skill beyond arithmetic.Writing and math are both symbol-based systems. But I speculate that written language is less artificial because its symbols map directly to words or phonemes, for which humans are hard-wired. I would guess that reading ties into the same mental machinery as hearing speech or seeing sign language.I don't believe we have the same innate ability for processing mathematical symbols.* Instead, we tend to reply on implicit physical metaphors, both for the mechanics of symbol manipulation (e.g., "moving" a term to the other side of the equation, "canceling out" two terms, etc.) and for the semantic interpretation of the symbols (e.g., exponential "blow-up", or the "smallness" of a neglibible term). To a certain extent, a person's mathematical skill is tied to their ability to "feel" the symbols through these physical metaphors, and thereby make the abstract more concrete.I believe that both of these forms of mental contortion are artifacts of pencil-and-paper technology. A person should not be manually shuffling symbols. That should be done, at best, entirely by software, and at least, by interactively guiding the software, like playing a sliding puzzle game. And, more contentiously, I believe that a person should not have to imagine the interpretation of abstract symbols. Instead, dynamic graphs, diagrams, visual models, and visual effects should provide visceral representations. Relationships between values, exponential blow-ups and negligible terms, should be plainly seen, not imagined.* Papert might disagree, and claim that a child raised in "Mathland", an immersive interactive mathematical environment that "is to math what France is to French", would become as fluent in symbolic math as in language. With regard to symbolic math, I might respond that a child raised in Antarctica would be quite tolerant of the cold, but maybe people shouldn't need that sort of tolerance.Language and Visceral Interpretation (2)Humans are built for language -- we're symbol-processing machines -- so I can't say "symbols bad". But I feel that there are things that we need to see or experience in order to truly understand. And there are things that are easy to draw or build, but impossible to describe (without years of practice in arcane specialized languages).I think that quantity and measure fall into that category. Reading "1m" and "1mm", versus actually observing those two measures -- one is just numbers on a page, the other hits you viscerally. Do you think most people understood, really felt, the difference between a $1B and a $1T bailout? Three orders of magnitude hidden inside a symbol.The point is that you need that visceral sense, that gut feel, to reason about a problem by intuition. Good circuit designers can "feel" how a circuit behaves. They look at a schematic and in their mind's eye, they see the voltage going down over here and pushing the voltage up over there, as if they were looking at a see-saw or water pump. It requires years of practice to develop this sense, this ability to look at symbols (in some domain) and feel what they represent.Likewise, people used to think that reading and making sense of huge tables of numbers was an essential skill for working with data. But then William Playfair came along and invented line graphs, and suddenly everyone could feel data through their eyes. Their plain old monkey-eyes!Complex numbers provide a similar example. Being able to work with complex numbers (as abstract values) is seen as an essential skill in many scientific fields. Then David Hestenes came along and said, "Hey, you know all your complex numbers and quaternions and Pauli matrices and other abstract funny stuff? If you were working in the right Clifford algebra, all of that would have a concrete geometric interpretation, and you could see it and feel it and taste it." Taste it with your monkey-mouth! Nobody actually believed him, but I do, and I love it.It's the responsibility of our tools to adapt inaccessible things to our human limitations, to translate into forms we can feel. Microscopes adapt tiny things so they can be seen with our plain old eyes. Tweezers adapt tiny things so they can be manipulated with our plain old fingers. Calculators adapt huge numbers so they can be manipulated with our plain old brain. And I'm imagining a tool that adapts complex situations so they can be seen, experienced, and reasoned about with our plain old brain.