Mohit Pandey Posted September 2, 2007 Report Posted September 2, 2007 Without actual calculating, find which is greater ?31^11 or 17^14Tell the method also. Quote
eric l Posted September 2, 2007 Report Posted September 2, 2007 This is just an estimation :17^14 should be the greatest of both31 is nearly the double of 17 (less actually)31^11hould be less than (17^11) * (2^11)2^10 is about 1000 or 10 (you know from kB being 1024 bytes)17^14 = (17^11) * (17^3)17^3 is greater than 10^3so 31^14 should be smaller than (17^14)*(17^3) Quote
Mohit Pandey Posted September 3, 2007 Author Report Posted September 3, 2007 Isn't there any method to calculate such large numbers without actual calculation? Anybody else? Quote
Qfwfq Posted September 3, 2007 Report Posted September 3, 2007 To be more exact than Eric's estimate, convert 31^11 into base 17. Do you know how to convert exponentials from one base to another? If so you should be able to figure out the trick. :eek_big: Quote
eric l Posted September 3, 2007 Report Posted September 3, 2007 To be more exact than Eric's estimate, convert 31^11 into base 17. Do you know how to convert exponentials from one base to another? If so you should be able to figure out the trick. :eek_big: Wouldn't that involve a lot of calculation ? I have a system for making estimations for such calculations that involves both a lot of mental calculation and knowing by heart the logaritms of the entire numbers from 1 tot 10, to 2 or 3 digits - well, knowing how to find them back is just as good, then the only one you have to remember is log 2. Of course, a number like 17 doesn't fit very well in this system, unless you say that 17 is close to (50/3). So you see it involves a lot of mental calculation, and I'm affraid most young people nowadays are not familiar with that. And remember it's still only estimation. Any good pocket calculator will do it faster and with more precission. Quote
CraigD Posted September 3, 2007 Report Posted September 3, 2007 (edited) Without actual calculating, find which is greater ?31^11 or 17^14 Tell the method also.Most of the ways that come to my mind to answer this involve finding the logarithm of 31 and 17. The question, [math]31^{11} \overset{?}{<} 17^{14}[/math] then becomes [math]11 \log 31 \overset{?}{<} 14 \log 17[/math] [math]\frac{\log 31}{\log 17} \overset{?}{<} \frac{14}{11}[/math] The trick, then, is how to estimate the logarithms of 31 and 17. One can note [imath]31 \dot = 32 = 2^5[/imath] and [imath]17 \dot = 16 = 2^4[/imath], so [imath]\log_2 31 \dot=5[/imath] and [imath]\log_2 17 \dot=4[/imath], so [imath]\frac{log 31}{log 17} \dot= \frac54[/imath]. [imath]\frac54 < \frac{14}{11}[/imath], so [imath]31^{11} < 17^{14}[/imath] Because [imath]17 > 16[/imath] and [imath]31 < 32[/imath], we also know that [imath]\frac{\log 31}{\log 17} < \frac54[/imath], so we can be completely confident in our answer. This approach fails if the problem is just a little different, however, such as [imath]31^{13} \overset{?}{<} 17^{16}[/imath], in which case we get [imath]\frac54 \not < \frac{16}{13}[/imath]. We can’t, however, confidently conclude from this that [imath]31^{13} < 17^{16}[/imath] (it doesn’t), but must find a better approximation of [imath]\frac{\log 31}{\log 17}[/imath]. For this, something like the Mercator series-derived [imath]\ln a = 2 \sum_{n=0}^\infty \frac1{2n+1} \left ( \frac{a-1}{a+1} \right )^{2n+1}[/imath] is needed, and involves more calculating than simply finding [imath]31^{13}[/imath] and [imath]17^{16}[/imath] – but a lot less than a direct approach to a variation on the question like: [math]31^{8219739813} \overset{?}{<} 17^{9962709445}[/math] Edited February 12, 2012 by CraigD Fixed many broken LaTeXs Quote
CraigD Posted September 3, 2007 Report Posted September 3, 2007 (edited) To be more exact than Eric's estimate, convert 31^11 into base 17. Do you know how to convert exponentials from one base to another? If so you should be able to figure out the trick. :)Please do tell, Qfwfq, what trick you have in mind. :confused: As far as I can tell, all the trick of representing 31 in base 17 get you is (writing everything in base 10 for ease of viewing): [math](17 +14)^{11} \overset{?}{<} 17^{14}[/math], which expands (laboriously, even with the aid of Pascal's triangle) to: [math]17^{11} +11 \cdot 17^{10} \cdot 14 +55 \cdot 17^9 \cdot 14^2 \cdots +14^{11} \overset{?}{<} 17^{14}[/math] from which you algebra equally unhelpful inequalities like: [math]11 \cdot 17^{10} \cdot 14 +55 \cdot 17^9 \cdot 14^2 \cdots +14^{11} \overset{?}{<} 17^{11}(17^3 -1)[/math] prompting me to suspect I’m failing to grasp your hint, and ask, as eric has,Wouldn't that involve a lot of calculation ? I have a system for making estimations for such calculations that involves both a lot of mental calculation and knowing by heart the logaritms of the entire numbers from 1 tot 10, to 2 or 3 digits - well, knowing how to find them back is just as good, then the only one you have to remember is log 2.I, too, have this intimate knowledge of common (base 10) logarithms from 1 to 10 to a couple of significant digits precision – though in my case, it comes not from any noteworthy memory or metal calculation skills, but a trusty old relic of my computing youth, a slide rule ;)Of course, a number like 17 doesn't fit very well in this system, unless you say that 17 is close to (50/3).I suspect a key intuition you’re intended (by the question’s author) to have about this question involves 17 and 31 being close to powers of 2, making their logarithms in base 2 easy to roughly estimate.So you see it involves a lot of mental calculation, and I'm affraid most young people nowadays are not familiar with that.I suspect eric is right - the ubiquity of electronic calculators accurate to ten or so significant digits renders mental arithmetic less impressive than it was in the days when slide rules ruled the Earth (or at least the part of it within the math/science department areas of schools). Still, a competent math student needs to have a good grasp of estimation and decision strategies, and understand that the process has less to do with being able to calculate than with knowing where and how intensely to calculate. In many ways, the low-precision tools of the 17th-20th centuries were more helpful than modern, high-precision digital ones, because they were physical and intuitive, and tended to keep you aware of the limits of your precision.And remember it's still only estimation.But the answer arrived at by estimating – in this example, [math]31^{11} < 17^{14}[/math] and the complete confidence that can be had in it, are not estimatesAny good pocket calculator will do it faster and with more precission.But very few can, without some thought and skill from its user, answer variations like this one from my previous post: [math]31^{8219739813} \overset{?}{<} 17^{9962709445}[/math] or this even more sillily large one: [math]31^{3289479823425651347473136858070760265556133} \overset{?}{<} 17^{3987003536725214823364107943702062812949028}[/math] PS: Anyone caring to show off, please feel free to answer these, or make up your own :) Edited February 12, 2012 by CraigD Fixed many broken LaTeXs Quote
Mohit Pandey Posted September 4, 2007 Author Report Posted September 4, 2007 To be more exact than Eric's estimate, convert 31^11 into base 17. Do you know how to convert exponentials from one base to another? If so you should be able to figure out the trick. :confused:No, sir. Can you tell me? :) Quote
Qfwfq Posted September 4, 2007 Report Posted September 4, 2007 Now I thought Mohit meant without actually calculating those two numbers, I'm not sure he meant literally no calculation at all. Wouldn't that involve a lot of calculation ?How much is a lot? A few keystrokes on a scientific calculator? Please do tell, Qfwfq, what trick you have in mind. :confused:It boils down to what you did, or even a bit simpler, except that I left Mohit the chance of thinking it over, or asking if he doesn't know enough to figure it out on. ;) Of course it works for any base in which the logarithms are known, ln, log, lg, log_n for any n.... With the Windoze calculator in sci mode, approximations don't show any difference (and likely it gets one of the two kinds of logarithm by conversion from the other): [math]\frac{\ln31}{\ln17}=1,2120468130962642130694242732922[/math] [math]\frac{\log31}{\log17}=1,2120468130962642130694242732922[/math] 11 times this ratio is less than 14. Quote
Qfwfq Posted September 4, 2007 Report Posted September 4, 2007 No, sir. Can you tell me? :)Consider [math]b^e[/math] and suppose you know that [math]b = a^l[/math]. How can be [math]b^e[/math] written? [math]b^e = (a^l)^e[/math] But raising an exponential to a power means multiplying the exponent by the new one, so: [math]b^e = a^{le}[/math] Therefore, for bases a and b, l is the ratio of the two exponents. Quote
eric l Posted September 4, 2007 Report Posted September 4, 2007 Hello GraigD How about starting a "Brotherhood of the slide rule" ? I still own two of them, but I admit to not having them used for over 10 years. And I still am better than most for mental calculation - I still like to beat shop assistants when they have to use a pocket calculator for calculating the discount. Quote
Qfwfq Posted September 4, 2007 Report Posted September 4, 2007 I've been too exhausted for that kind of thing, lately. I used to catch the cashiers that got their additions and subtractions wrong despite using a calculator. Quote
Mohit Pandey Posted September 4, 2007 Author Report Posted September 4, 2007 Now I thought Mohit meant without actually calculating those two numbers, I'm not sure he meant literally no calculation at all.Yes, I meant no calculation. calculation should be done only in mind.eric l 's solution seems to be best to me. No comlicated calculation, only estimation and logic:thumbs_up .Is there another method like this? Quote
eric l Posted September 4, 2007 Report Posted September 4, 2007 An other method like this ? In this case there is : thinking binary. 31 is almost equal to (but lower than) 32 which is 2^517 is almost equal to (but greater than) 16 which is 2^4 32^11 = (2^5)^11 = 2^(5*11) = 2^5516^14 = (2^4)^14 = 2^(4*14) = 2^56 2^56 is greater than 2^55, so 17^14 (being greater than 2^56) must be greater than 31^11 (which is lower than 32^11). In this case, working with base 2 is faster than working with base 10 or any other. But being an old slide rule guy, I simply feel more at home with decimals than with binaries. Quote
Mohit Pandey Posted September 5, 2007 Author Report Posted September 5, 2007 An other method like this ? In this case there is : thinking binary. 31 is almost equal to (but lower than) 32 which is 2^517 is almost equal to (but greater than) 16 which is 2^4 32^11 = (2^5)^11 = 2^(5*11) = 2^5516^14 = (2^4)^14 = 2^(4*14) = 2^56 2^56 is greater than 2^55, so 17^14 (being greater than 2^56) must be greater than 31^11 (which is lower than 32^11). fantastic! :lol:. I was looking for these types of answers.;) Quote
Qfwfq Posted September 5, 2007 Report Posted September 5, 2007 But Craig had already said that! And of course, if you meant without using a calculator, it's certainly the best way. :lol: Quote
CraigD Posted September 5, 2007 Report Posted September 5, 2007 But Craig had already said that!We seem to all have similar ideas on how to approach this thread’s initial question. Eric, IMHO, has done best at writing an easily understood explanation. :thumbs_up I’d be curious to see how people could do (using no calculators) with more difficult variations of the problem, where the exponents are slightly or much higher and the “31/17 is a little less than 2” estimate is no longer adequate. I’ve offered a few above, but could crank out random ones in any number, if anyone would like to try. :) All of this talk of logarithms and binary spurred me to an exercise in … base 2 logarithms between [math]1[/math] and [math]10_2[/math] written in binary. The result, stripping the spaces and dots, looks like a good template for a floor tile pattern, or, picking a couple of large primes for the number of rows and columns, a SETI message. :) Anyway, as a tribute to the abundance of file space on the internet, here’re the logarithms between between [math]1[/math] and [math]10_2[/math] in steps of [math]2^{-8}[/math] to [math]2^8[/math] bits precision: 1.00000000 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 1.00000001 0.0000000101110000100111000100011011010111101010101100011101110100110010011001001101000010000110111111010101000100101001001100110110001011001011001100000111111001101000110011000111100111001100010011010101101000111000110010010111011000100000001101001110010011 1.00000010 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