Meaning of "much greater than"? - Page 1
Forum section: electronics / beginners

author: indeterminatus
author_rank: Contributor
timestamp: October 30, 2020, 06:57:40 pm
content: Hi!In the literature (e.g. "The Art of Electronics"), there's often the talk about some quantity being "much larger/greater than" another quantity:$$x\gg y$$I have been wondering what that "much" would be, but could not find any definition thus far. Is it an order of magnitude? Two orders of magnitude? Three or more orders of magnitude? Or depending on context and how great an error term is acceptable?

author: Zero999
author_rank: Super Contributor
timestamp: October 30, 2020, 07:36:32 pm
content: It depends on the context. It normally means at least an order of magnitude.

author: ataradov
author_rank: Super Contributor
timestamp: October 30, 2020, 07:42:18 pm
content: I always assume an order of magnitude for linear values. It gets trickier for logarithmic values. So yes, depends on context.

author: Mechatrommer
author_rank: Super Contributor
timestamp: October 30, 2020, 07:57:43 pm
content: simple answer is you can usually safely assume in ratio of 1:1000000

author: coppice
author_rank: Super Contributor
timestamp: October 30, 2020, 08:10:59 pm
content: You typically meet x≫y where x == y is some kind of transition point, like the turnover point of a low pass filter. Its not a brick wall transition, so the relationship only approaches the stated behaviour as you get well beyond x == y. For example, for a simple RC low pass filter I could say that for frequencies ≫the turnover frequency, the output of the filter is shifted by about 90 degree. However high the frequency gets the shift never quite reaches 90 degree, but it gets really close. How close it needs to be to 90 degrees before you consider it effectively 90 degrees is application dependant. So, there is no hard and fast answer to how high x needs to be, but the relationship embodies a meaningful trend.

author: tggzzz
author_rank: Super Contributor
timestamp: October 30, 2020, 08:21:19 pm
content: Quote from: indeterminatus on October 30, 2020, 06:57:40 pmIt means that y is sufficiently unimportant that it can be ignored.The "sufficiently" implies "in this context", so the context has to be defined.For example consider 1mV variation on a 10V signal.If you are measuring a power supply voltage, then the 1mV is unimportant and can be ignored.If you are measuring a voltage standard then 1mV would indicate gross errors in the measurement technique or equipment, which would have to be tracked down and eliminated.

author: Nominal Animal
author_rank: Super Contributor
timestamp: October 30, 2020, 11:30:53 pm
content: Quote from: indeterminatus on October 30, 2020, 06:57:40 pmThere are a few practical definitions, depending on the context. They all boil down to a variant of$$x + y \approx x \quad \text{ iff } \quad \lvert x \rvert \gg \lvert y \rvert$$where the context is what dictates the meaning of the approximately equal sign, \$\approx\$.In general, especially in computer science and software engineering, the approximate sign is understood via (machine)epsilon, \$\epsilon\$, which represents the precision at which we are operating in:$$x + y = x \quad \text{ iff } \quad \lvert y \rvert \le \epsilon$$Simply put, \$\epsilon\$ refers to the greatest positive number that is still considered zero. (Sometimes it is defined as the smallest positive nonzero value, in which case you replace the \$\le\$ with \$\lt\$ above.) For example, if you use scalars with three decimal digits of precision, then you implicitly define \$\epsilon = 0.0005\$ (assuming you use rounding, \$0.000999\$ if you use only truncation).In many cases, theepsilonis not only dependent on the equation itself, but also the inputs. For example, consider the case where the user provides you a number of 3D points, and your program must return the number of unique points. There, you use some form ofepsilonto describe the maximum distance or maximum coordinate difference to tell whether two points are separate or not. If you use floating-point numbers, the precision of those values already give you an implicitepsilon(which depends on the terms in the expression, as both the arguments and the result are only the closest values one can represent using floating-point). If you rotate or translate those points before making the decision, you really need to consider the precision (orepsilonof) the original coordinates, as floating-point operations have rounding error – equivalent to quantization error in analog-digital circuits.(This is also why so many programming manuals talk about exact equality comparison of floating-point values being usually wrong.)In electronic circuits, we can add a few additional meanings to \$\gg\$.One is that when \$x \gg y\$, the magnitude of signal \$x\$ is so large compared to \$y\$, that the effect of \$y\$ is lost in the inherent (physical, thermal, etc.) noise. You can consider this roughly analogous to floating-point issues in computation, since real-world electronics with very large or very small or very precise signals has to consider the "non-electronics physical effects", like thermal effects from heating up the conductors and so on. Again, here the "very large" or "very small" is completely dependent on the situation.Another is that the effect of a property \$x\$ completely overwhelms property \$y\$. This is the case where \$x\$ and \$y\$ are not signals, but properties of e.g. a component, say the \$\beta\$ of a transistor. When writing out the exact description of the system, many of the variables are "neglible"; do not really affect the output at all. Depending on how you write the exact description, you can either approximate them as zeros, or say the more important variables are much greater than the variables with little to no effect within the normal operating conditions.Consider the common practical example from physics, speed addition (velocities in the same direction, opposite if different signs):$$v_{sum} = v_a + v_b \quad \text{ iff } \quad \lvert v_a \rvert, \lvert v_b \lvert \ll c$$What is the limit at which we consider velocities no longer "much smaller than the speed of light"? There is no such absolute value. For the longest time, we couldn't measure velocities at sufficient precision to even check if relativistic correction was necessary; our measurementepsilonwas greater than the difference between the relativistic velocity sum and the direct sum, so we couldn't tell the difference. If your measurements can only provide a couple of digits of precision – which isn't bad at all if we are talking about really huge velocities – anything more than a few percent slower than the speed of light is sufficiently "much slower".If you are now having a very uncomfortable feeling about the lack of precise numerical limits, or even a precise definition of such a limit, welcome to the physical world.There have never been such limits; any claims for those are just coddling the weak minds. The expectation of such limits is a limitation that we must cast away. One way I personally recommend doing that, is by habitual application ofdimensional analysis, followed by "top of the envelope estimates", i.e. like dimensional analysis, but with all numerical values replaced by the nearest power of ten, to find out the approximate magnitude of the result. It will be off, but it will immediately tell you if a model is completely bunk. This kind of criticality, as long as you are willing to consider any model or claim up to that utterly basic verification stage, will release you from the expectation of rock-solid absolute numbers, replacing it with the Real Big Guns to deal with anything thrown your way.

author: greenpossum
author_rank: Frequent Contributor
timestamp: October 31, 2020, 12:07:23 am
content: It means something like the smaller term is TI;DB (Too Insignificant; Don't Bother).

author: indeterminatus
author_rank: Contributor
timestamp: October 31, 2020, 12:39:15 am
content: Thank you all for taking the time to help me (and hopefully other fellow beginners) understand! I think I get it now.Quote from: Nominal Animal on October 30, 2020, 11:30:53 pmYou hit the nail on the head. That's exactly what troubles me. And I think your elaborate reply was exactly what I needed to read to fill the gaps. Thank you!

author: helius
author_rank: Super Contributor
timestamp: October 31, 2020, 01:11:58 am
content: The first example of "much greater than" taught in mathematics is that\$ \sin x \approx x \$ when \$ |x| \ll 1 \$Because the slope of sin(x) around the origin is 1 [i.e., cos(0)=1]. This is a very useful "identity" seen widely in engineering. How much lesser does it need to be? Even at x = 1, they are only 18% different. By x = .1, they differ by about .2% which is often an inconsequential difference.

author: pwlps
author_rank: Frequent Contributor
timestamp: November 01, 2020, 10:51:27 pm
content: Quote from: helius on October 31, 2020, 01:11:58 amAnd sometimes the "O" notation is used to indicate how the error of the approximation behaves e.g.\$ \sin x = x + O(x^3) \$ when \$ |x| \ll 1 \$means that the eror varies like x3when x is close to zero.

author: Nominal Animal
author_rank: Super Contributor
timestamp: November 01, 2020, 11:58:50 pm
content: To add to what helius and pwlps wrote above:The way \$\ll\$ and \$\gg\$ connect to \$\approx\$ and its variants is very interesting/informative/useful.The notation varies, and since there is no agreement among mathematicians which symbols to use, I'll use the ones from Unicode:≃: Asymptotically equal to. In maths,~is most commonly used for this, but the symbol varies.≈: Almost equal to. I've seen≃used for this.≅: Approximately equal to. In maths,≈is most commonly used for this, but the symbol varies.Asymptotically equal to is typically used with limits;asymptotic analysis. Typical limits are +∞, -∞, with 0 being rarer in "asymptotically equal to" context, but the limit itself can be anything, even zero. Thus, this is not limited to "very large" or "very small" values; it is more about "near to" or "far from".Approximately equal to is what I discussed above: within some precision or error range (which can be absolute or relative, or a set of conditions), the left side can be treated as equal to the right side.Almost equal to is the hand-wavy something between the two above. You use it when you do not want to explore thequalitativeequality (asymptotic? approximate? under some conditions only?) between the left and the right side, and just want to make aquantitativestatement: for now, in this case, we can consider these two equal.While the symbols vary, and not everyone uses them "correctly", it does mean that we do have tools in Maths tospecifythe context in which each claim/axiom/formula/equation should be valid. It also means that whenever an author does NOT specify the context, the entire work is suspect. (Often, especially in narrow domains, the context is considered "obvious" and therefore left unstated; in those cases, look at diagram scales, example values, etc. across articles from few different authors.) When the context is indeed specified, for us math-appliers/programmers/whatever, it tells us in which cases the formula/equation can be applied. Sometimes they even tell us the error/difference scale.Which brings us toBig O notation. It describesthe order, the upper bound of the growth rate, of the thing it represents: the difference between an approximation and the exact value, the time or space complexity of an algorithm, or so on. Because it is the upper bound of the growth rate, constant coefficients are ignored, and only the highest powers of the variables are listed. For example, finding a specific value amongNsorted, but otherwise unknown, values has O(logN) time complexity, because depending on the exact binary search implementation, you need to examine at most ⌈3+log2N⌉ of theNsorted values to find the one (or determine it does not exist), with ⌈⌉ representing ceil, or rounding up. (The base of the logarithm isn't mentioned, because it is just a constant coefficient: logaX=ClogeX, whereC= 1 / logea.)

author: wilfred
author_rank: Super Contributor
timestamp: November 02, 2020, 12:49:44 am
content: Quote from: indeterminatus on October 30, 2020, 06:57:40 pmIt is a simple imprecise English phrase and has no meaning other than it means "more than a little bit greater than". The context usually provides a clue such as one value is swamped and can be ignored or considered negligible because another value is far more influential.

author: indeterminatus
author_rank: Contributor
timestamp: November 02, 2020, 12:23:34 pm
content: Quote from: Nominal Animal on November 01, 2020, 11:58:50 pmI guess that is it. The books of the "Art of Electronics" series are far from questionable, I consider them very solid and they take it to great lengths to make the math and the concepts work out. (I know it's a dangerous statement, but at least I get the feeling that I understand what's going on, for most part). It might very well be that I missed something (like a foot-note); I've been browsing the book and re-reading passages that make use of "much greater than", but could not really determine a pattern that applies to them all. It was not clear to me if that was context-sensitive or if there was some hidden, "obvious" knowledge in the electronics/physics world that I lack. Sometimes, the authors would state in the text that an error of 2% would be good enough for the specific circumstances, which relativizes the "much greater than" inequalities; with that, I can work. I can also work with fuzzy systems where single components themselves are not very precise, e.g. that it makes no practical sense to solve anything in a circuit to a high level of precision when say a resistor introduces a variation of 5% for its resistive value (assuming that this resistor is significant, because its value is "much greater than" the other resistive values).To sum it up: My current understanding (thanks to all of your contributions/remarks) is that there is no universal rule when the (boundary) conditions are not explicitly stated. There either must be a definition, or it is impossible/infeasible to provide a definition (e.g. inherent traits of materials/...), or it depends on the context (and the knowledge there-of comes with deeper knowledge of the domain).Unfortunately for me, most of the times that I had a hard time to figure out the limits, feel like they belong to the latter. I will have to sort them out individually (which is by far outside the scope of this general question) -- the realization that there is no universal ratio/quantification/rule already helps a lot, so I will no longer try to make it work for all the aspects.

author: Nominal Animal
author_rank: Super Contributor
timestamp: November 03, 2020, 02:00:59 am
content: Quote from: indeterminatus on November 02, 2020, 12:23:34 pmFully agreed.Quote from: indeterminatus on November 02, 2020, 12:23:34 pmContext-sensitive, most definitely.Usually the reason is physical; sometimes things like manufacturing costs not permitting perfect components, so that while there is a range (of precision, noise, leakage current, or whatever) one can buy off the shelf, there is always a tradeoff due to which it is not possible to state a quantitative limit.For example, consider ordinary diodes. In the forward direction, they have a voltage drop that depends on the current. In the reverse direction, they have a breakdown voltage but also a leakage current. The exact junction type in a diode determines its capacitance and speed. There is no perfect diode – in fact, the closest we can get to a perfect diode at low voltages probably involves MOSFETs; see ideal diode –, but within the diode family, we can say certain things do not matter compared to other things. Where the "limit" is, depends on the exact diode type, possibly even the manufacturer.Similarly for transistors, operational amplifiers, and so on.Quote from: indeterminatus on November 02, 2020, 12:23:34 pmYes. The last one is one of the reasons why real-world experience is so useful.If the context were to be fully explored in a textbook like the Art of Electronics, it would be a large series of tomes – if at all possible to write: sometimes the context is known to apply to everything we currently have, and we don't yet knowwhereit stops applying!Writing a good textbook is not justincludingthe most important stuff, it is also aboutleaving outthe context that can be expected to be already known in order to really benefit from the contents. In my experience, in both physics and electronics, having real world experience – even at the interested hobbyist level – makes a huge difference, because then things "slot" together better; even a vague understanding of the context helps build better understanding.