this is the most useful curve in mathematics it has this unique property that creates a sort of mathematical Wormhole transforming complex problems into simple ones to see this property in action we need to look at how this curve Maps mathematical operations from one axis to another let's pick some values from our x axis and multiply them together for example 2 * 4 = 8 now let's find the corresponding y values for each x value using our curve two maps to one four maps to two and eight maps to three what's unique here is that for any multiplication problem on our x axis our y values will always form a valid sum in this case 1 + 2 = 3 if we had chosen different X values such as 1/4 * 8 = 2 we would still get a valid sum 1/4 maps to minus 2 8 maps to three and two maps to 1 giving Min -2 + 3 equals 1 so multiplication problems on our xaxis are transformed into addition problems on our y Y axis this also works for division division problems map to subtraction problems 8 / 4 = 2 maps to 3 - 2 equal 1 to see why this property is so useful we need to talk about how people did math 400 years ago Dunkirk France is 5.61 nautical miles due south and 46.6 nautical miles away from do England a navigator setting course to cross the English Channel from Dover to dunker would set the sign of their course equal to 5. 61 ided by 46.6 working out the long division gives 0.12 039 we can now look up the sign value in a book of tables that would have been available at the time such as this 1541 book by Regio om monus in the early 1600s decimal notation had not been widely adopted yet mathematicians would often work with scaled up large numbers instead of decimals to use this table our Navigator would multiply the 0.1 12039 value we computed by 6 million giving a 72234 looking up this value in the table we see that it corresponds to an angle of 6° and 55 minutes or 6.92 Dees in our familiar Decimal System so to reach Dunkirk we need to set our ship's Compass to 6.92 De south of East in 1614 the Scottish mathematician John Napier published this book which revolutionized how this and many other math problems were solved just like the Regio monus book Napier's book contains tables of signs but it also contains this column labeled logarithm naier didn't think of it this way at the time but his logarithm values are effectively the Y values for a set of points on a curve like this computed all the way to the eighth decimal place to solve our navigation problem we first need to do a quick scaling step by moving the decimal five places to the right in each of our numbers next we need to look at each of these distances in the sign column of Napier's table and get the value in the logarithmic column so our 5.61 nautical miles is converted into 28 84057 and our 46.6 nautical miles gives 7632 919 now we're literally almost done just like our curve Napier's table converts division into subtraction so instead of doing the tedious division of 5.61 by 46.6 all we have to do is subtract their logarithm values 288 0457 - 7632 919 gives 21 17113 8 now that we've completed our subtraction problem using the yv values for our curve to get our final answer we need to map back to the X values we do this by looking up our new logarithm value in Napier's table Napier cleverly combined his logarithms with sign values his book effectively contains the logarithms of signs saving an extra look up in a sign table the degrees of the angle that correspond to this logarithm are printed in the upper left of the page and the leftmost column gives the minutes of the angle so 6° and 5 5 minutes or 6.92 de the same exact answer we got by dividing our distances and looking up our quotient in the sign table Napier's method quickly caught on it's much faster and less error prone the East India Company was so impressed that it immediately began work on an English translation so its company Sailors could put it to use and it isn't just Division and multiplication as Napier lays out in his introduction his logarithms also transform Powers into multiplication Roots into Division and more naier effectively invented and spent 20 years of his life Computing a new set of numbers the Y values of the points along this curve that dramatically simplified computations Napier initially called his invention artificial numbers but ultimately decided to call these numbers logarithms meaning ratio numbers because of how they are computed within a decade of Napier's publication the mathematicians Edmund Gunter and William atred figured out that if you place tick marks on two rulers space at locations corresponding to the the Y values of the curve but labeled with the X values you could create a powerful computer these two Computing methods tables of logarithms and the slide rule were the primary methods of computation on the planet for the next 350 years until the arrival of the cheap electronic calculator in the 1970s but how is it that Napier's numbers are able to transform division into subtraction Roots into Division and so on and how did they even compute these numbers in the first place and why do logarithmically space tick marks allow the slide R to perform computations at the beginning of his book Napier starts by defining the motion of two points point B moving at a constant velocity and point beta which is slowing down in a very specific way if we call x0 the distance between Alpha and Omega X1 the distance between gamma and Omega and so on beta moves such that the ratio of consecutive X values is constant forming a geometric sequence for example if x0 is one and our ratio between terms is 0.9 9 then X1 would be 09 and X2 would be 9 * .9 giving 81 and so on next Napier mathematically links the motion of points B and beta together for each step forward in time point B moves a constant distance from a to c to D and so on while Point beta moves smaller and smaller distances from alpha to gamma to Delta and so on Napier now defines the position of point B as the logarithm of X of course the question remains how does mathematically connecting the motion of these two points Point magically turn multiplication into addition and simplify other operations let's construct a miniature version of Napier's table to see how at the heart of Napier's table are the sign and logarithmic columns which in our modern adaptation here would be called X and log X we'll start our logarithms at zero and have them increase by one for each step and we'll have the geometric sequence for the position of Point beta decrease with a ratio of 0 9 so our X column gives the position of Point beta and our log X column gives the position of point B now now let's say we want to multiply two numbers together for example 81 * 6561 multiplying out manually gives 531441 If instead we take the log of each number 081 maps to 2 and 6561 maps to 4 and add these logarithm values together we get six then looking up six in our table we see that it maps to an x value of 531441 which is the correct answer this works because the logarithm is keeping track of how many times we multiply 09 together the way we've set up our table the log of 081 is 2 because 081 is equal to 09 * itself 2 times and the logarithm of 6561 is equal to 4 because 6561 is equal to .9 * itself four times multiplying 081 by 6561 is equivalent to multiplying 09 times itself six times this is what the logarithm is keeping track of for us logarithm simply count the number of multiplications of some Base number required to reach a given number our curve uses the base of two the base 2 logarithm of 8 is three because we have to multiply two times itself three times to reach 8 just like Napier's table our curve converts multiplication into addition by effectively counting the number of multiplies needed of our Base number using logarithms to convert 2 * 4 into 1 + 2 is equivalent to rewriting our problem as 2 the 1st power * 2^ 2 the logarithms or Y values on our curve are just the exponents here really this is just another way of saying saying that exponents add when we multiply common bases and logarithms give us a way to rewrite problems in terms of these common bases in modern notation you most often see this fact expressed as log of a plus log of B is equal to log of a * B which is helpful for computation but hides the Simplicity of what's going on behind the notation logarithm simplify division into subtraction in a very similar way converting 8 / 4 into 3 - 2 is equivalent to rewriting 8 / 4 is 2 Cub / 2^2 this is just another way of saying that exponents subtract when dividing common bases or effectively removing two of the multiplications by two that make up eight leaving just two in modern logarithm notation log of a divid b is equal to log of a minus log of B now the problem with our examples so far is that we've cherry-picked very simple values off of our curve and miniature version of np's table for logarithms to be really useful we need to be able to look up any number to a decent number of decimal places in our tables our toy version of Napier's table uses a base of 0.9 to increase the density values on his table Napier used a ratio of 9999999 and instead of starting at one Napier started at 10 million and progressively worked down at each step multiplying by 9999999 this row of Napier's table is telling us that 9999999 multiplied by itself 109 times is 999 9892 and this row is telling us that 9999999 to the power of 63 m58 199 is 001 7453 Napier's table helped us solve our navigation problem earlier by letting us rewrite our distances as powers of 9999999 we then subtracted our exponents the logarithm values and looked up the new logarithm value in Napier's table to get our final answer now if you type the logarithm of one of Napier's values into your calculator you get a completely different answer than Napier gives this is entirely due to what the mathematician Henry Briggs did next when Briggs an influential professor at Gresham College in London received Napier's book he was absolutely astonished at what Napier had accomplished but he quickly realized that it should be completely redone the next summer of 1615 he made the long journey to Edinburgh to meet Napier and try to convince him to join forces and recreate logarithms in a simpler and more useful way way according to the astrologer William Lily when the men first met they said in silent admiration for the first 15 minutes before either set a word sounds kind of awkward finally Briggs broke the silence after some pleasantries Briggs proposed that they completely do away with the base of 0.999 9999 this value is clunky to work with at best for example if we flip our division problem from earlier this time dividing 46.6 by 5.61 we end up with a negative logarithm value after subtracting which is nowhere to be found on Napier's table in chapter 4 Napier basically tells us to add or subtract this magic number 230 25842 as many times as needed to get our logarithm back onto the table while multiplying or dividing our final answer by 10 for each operation Briggs realized that since the decimal system we're already working with is in base 10 if he recreated Napier's tables using a base of 10 instead of 9999999 he could make Napier tables much easier to work with with base 10 logarithms scaling by 10 is as easy as adding or subtracting one the log base T of 2.5 is 0. 3979 the log of 25 is 13979 and the log of 250 is 23979 so if you have a table of logarithms that only goes up to 25 but you want to find the logarithm of 250 you can just use the table to find the logarithm of 25 and add one to your final answer Napier graciously agreed to Briggs changes but they decided that since Napier was in poor health Briggs would undertake the huge task of reconstructing the tables in base 10 returning to London Briggs came to terms with the Monumental task he had signed up for while base 10 logarithms are easier to work with once you have a complete table the table itself is significantly more complex to compute Napier's base of 0.999 9999 changed very slowly when multiplied meaning that Computing successive values in Napier's table is a simple as multiplying by 9999999 sure Briggs could also compute a simple log base 10 table by just multiplying by 10 but it would of course only work for powers of 10 Briggs had to figure out how to fill in the gaps what is the log base 10 of 200 said differently how many times do we have to multiply 10 together to equal 200 10 s is 100 and 10 cubed is a th000 so the logarithm of 200 has to be between 2 and three but where do we go from here Briggs found an ingenious solution that works by zooming way in on our curve Briggs started with the fact that square root operations on our x-axis are equivalent to division by two on our y AIS in base 10 the logarithm of 10 is 1 and the logarithm of the square root of 10 is2 to zoom in on our curve Briggs repeated this operation again and again Computing the square root by hand at each step the square root of 10 is about 31622 and its base 10 logarithm is 1/2 repeating the the operation the square root of 31622 is 17782 and its logarithm is 1/4 next comes 13335 and 1/8 and then 1.1 1547 and 11/16th and so on here are Brigg's repeated square root operations published in 1624 Briggs knew that his final table needed to have a high level of precision so he computed his square roots up to the 33rd decimal place after 54 route extractions likely taking over a 100 hours of computation Briggs saw exactly what he needed if we take the very last square root value on brig's table and multiply by two it's exactly equal to the previous value on his table if we double this value it again matches the previous value this time to 32 decimal places this fact is important because it means that at this Precision in this neighborhood our curve looks exactly like a straight line mathematically this means that the logarithm of 1 + r when R is small is equal to the slope of our line time r Briggs computed the slope of the line Alpha by dividing the last logarithm value in his table by one minus the last square root value effectively taking the change in y over the change in X on our curve on this tiny interval he computed about 43429 Briggs now had an equation in hand that would allow him to compute the logarithm of any number sufficiently close to one by simply subtracting one and multiplying by Alpha next Briggs used his new formula to compute the logarithm of two two is way too far away from one for Briggs formula to work so he again relied on heavy computation to get closer to one Briggs first raised two to the 10th power and divided by 10 3 Computing 1.024 Briggs knew that these power and division operations could be easily reversed later after you found the logarithm of 1.024 1.024 is still way too far away from one for Brigg's linear formula to work at the level of precision that he needed so he again took a series of square roots this time a 47 Square s roots of 1.024 to reach a final value in his linear neighborhood 1.000000 000000001 1685 using his linear formula Briggs computed the logarithm of this number to be 000000 0000000000 00000000 07318 now that he had the logarithm value at the bottom of his table he could work backwards by multiplying by two for each of the 47 square roots that he took Computing the the logarithm of 1.024 to be about 01029 finally Briggs reversed the division by 1,00 and the 10th power to arrive at the logarithm of two about 0.301 03 Briggs had completed one value in his table thankfully not every value in brig's table required this enormous level of effort since multiplying a number by 10 is equivalent to adding one to its base 10 logarithm we automatically get the logarithms of 20 200 2,000 and so on the logarithm of 200 we were wondering about earlier is 2 plus the logarithm of 2 or about 23103 Briggs computed the logarithm of 5 by exploiting the fact that 2 * 5 is 10 since logarithms convert multiplication into addition this means that the logarithm of 5 is just equal to 1 minus the logarithm of 2 using these and many other clever computational approaches Briggs computed an enormous table covering the logarithms of 1 to 20,000 and 990,000 to 100,000 to 14 decimal places after 7 years of calculating Briggs ran out of steam and the Dutch publisher Adrien V jumped in and filled in the values from 20,000 to 990,000 publishing in 1628 over the next two centuries as logarithms became an indispensable tool for computation dozens of other authors published tables of logarithms and literally all of them just copied the values from Briggs and black in the early 1800s the mathematician Charles babage compared 22 different books of tables published from 1633 to 18 1826 and found the same errors in almost all of them aside from small Corrections the tables of Briggs and black were largely unchanged until the 20th century these 100,000 numbers along our curve computed by two mathematicians in the early 1600s became the backbone of human calculation for the next three centuries Isaac Newton Albert Einstein Robert Oppenheimer all use versions of the logarithm tables of Briggs and black they also use slide rules which make direct use of these logarithm values to perform precise but very fast calculations to multiply two numbers together on a slide rule for example 1.52 * 2.25 we move the one on the C scale to 1.52 on the D scale and move the cursor to 2.25 on the C scale and then read the answer off the D scale 3.42 done division works in Reverse to divide 4.15 by 2.5 set the cursor to 4.15 on the D scale Slide the C scale so 2.5 is under the cursor and read off where one on the C scale lines up with the D scale 1.66 what's the square root of 10 place the cursor on the 10 on the a scale and just read the answer off the D scale 3.16 slide rules are able to rapidly solve all these problems and many more because the tick mark spacing precisely follows the values on our curve an easy way to see this is to look at where powers of two land on our A and B scales the A and B scales are identical to each other and cover the values from 1 to 100 using the B scale to measure notice that the intervals between 1 and two line up between our A and B scales sliding our B scale over we can see that the distance between 2 and 4 is the same as the distance between 1 and two moving along we see the same spacing from 4 to 8 8 to 16 16 to 32 and so on this spacing exactly follows our curve each multiplication by two along our x- axis is equivalent to an addition of one on our Y axis so powers of Two end up at equally spaced intervals just as they do on our slide rule this property is precisely what allows slide rules to perform multiplication following the same approach we did earlier let's multiply 4 * 8 we'll move the one on our B scale to the four on our a scale and look for where the eight on our B scale intersects our a scale giving the correct answer 32 the sliding action of the slide rule is really about adding the number of steps it takes two steps to get to four on our a scale and we're adding three more steps to get to eight on our B scale for a total of five steps on our a scale scale so the slide rule is just adding the number of steps for us however because of the spacing of our tick marks each step is measured as a multiplication by two our five steps are measured on the scale of our slide rule as 2 5th equal 32 so the addition along the physical length of our slide rule is converted into a multiplication by the logarithmic spacing of our tick marks multiplication on the slide rule is the physical embodiment of log of a * B is equal to log of a plus log of B converting multiplication of A and B into the addition of of log of a and log of B division on the slide row works in the same way except instead of adding steps we subtract this corresponds to the logarithmic identity log of a / B is equal to log of a minus log of B in 1972 HP released the hp35 the electronic calculator that killed the slide rule its instruction manual reads our object in developing the hp35 was to give you a high Precision portable electronic slide rule remarkably to implement the logarithm functions HP Engineers used a version of Brigg's zooming in method to linearize logarithms for values close to one Brigg 1624 publication arithmetic logarithm is literally cited as prior art by HP Engineers the hp35 did everything a slid rule could do and more within a few years slide R manufacturers began shutting down production and tables of logarithms and slide rules disappeared from daily use as quickly as they had appeared in 1614 for the 358 years from Napier's publication of the first logarithm table to the release of the hp35 the points on this curve printed in tables are measured on the surface of a slide rule were the most powerful and widely used Computing instruments on the planet making this the most useful curve 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