A (very) Brief History of the Complex Plane

up until elementary algebra and trigonometry most individuals are not exposed to anything but real numbers then when trying to find if an algebraic equation has rational roots one stumbles upon complex numbers though not studied nearly as intensely as real numbers within the standard curriculum these numbers are an essential part of mathematics with a whole branch of mathematical analysis being dedicated to them called complex analysis the applications are far and wide with the prime examples being in areas of physics such as dealing with electricity despite how fundamental they are it took until about the 19th century for imaginary quantities to be recognized as truly useful why did it take so long for mathematicians of the past to recognize the value of complex numbers why was it only 200 some odd years ago that the complex plane which applies a proper geometric construction of complex numbers was invented the prospect of a negative within a square root was baffling to mathematicians for many years they considered it to be an impossibility it wasn't until the 16th century that any term was given to the square root of negative one piudi meno which literally translates to more or less it was coined by raphael bombelli an italian mathematician who was the first person to really write anything remotely formal on the square root of negative 1. alas the name didn't stick and in the 17th century the square root of negative 1 gain a term that would actually stick imaginary this one was coined by rene descartes a 17th century french-born mathematician philosopher and scientist most notable for being the father of modern philosophy and making fundamental contributions to foundational geometry he introduced this term in 1637 in la geometry which was the fundamental work that gave birth to analytic geometry in the 18th century leonard euler a swiss mathematician considered by many to be the most prolific mathematician ever exists coined i for the square root of negative 1 in a 1770 paper roughly translated to on differential angular most irrational which is logarithms and circular arcs can be integrated and in the 19th century carl friedrich gauss a german mathematician whose contributions to mathematics gained in the title the greatest mathematician since antiquity coined the term complex number for imaginaries in 1831 paper there was great perplexity at and avoidance of negative numbers for a long time too as many couldn't conceive of the prospect of a quantity being less than nothing yet another reason that compounds why it took so long for people to accept the square root of negative one before we actually dive in let's give some insight into what a complex number actually is a complex number is an ordered pair of any real numbers let's say x and y represented in the form x plus iy where i equals the square root of negative 1. for ease we'll set this complex number equal to z so z equals x plus iy we call x the real part of z and y the imaginary part of z now taking a look at the complex plane we see the familiar cartesian graph but now the vertical axis represents the imaginary counterparts the real part of z and the imaginary part of z should make more sense now setting y equal to zero z rests on the real axis and setting x equal to zero z rests on the imaginary axis the rules of vector manipulation apply to z and one is able to ease more complicated calculations using polar coordinates the first sighting of a negative under a square root was in the 1st century a.d by heron of alexandria a greco-egyptian mathematician and engineer often considered the greatest experimenter of antiquity he was given a problem regarding the frustum of a pyramid with a square base the problem later being presented in a work apparently written by him called stereometria his goal in the problem was to find the height of the frustum but given the information for the calculation he ended up with a subtraction that yielded a negative number under a square root to avoid the negative heron simply swapped the quantities in the subtraction to yield a positive under the root an obvious arithmetic error showing that heron most likely didn't even want to consider the notion of a negative under a square root around 200 years later diophantus of alexandria an alexandria mathematician most notable for his contributions to elementary algebra also had an encounter with a negative under a square root he showed this in his work arithmetica a series of 13 books containing a large quantity of problems along with their answers a triangle problem in book six stated as given a right triangle with area equal to 7 and perimeter equal to 12 find its sides yielded a negative under a radical for both roots of the quadratic that came out of this information based on this diophantus stated that no equation actually existed since there existed no rational solution it is unclear whether diophantus just completely missed this or was just dumbfounded by the result [Music] over a thousand years later in the 16th century italian mathematicians began studying the depressed cubic a cubic equation missing its square term a depressed cubic always has at least one real root and one can often encounter two complex roots depending on the equation a few of the mathematicians involved in this cubic venture were del ferro an italian mathematician who first discovered a method to solve such equations nicolo fontana tartaglia an italian mathematician engineer and surveyor notable for making the first italian translations of euclid's elements girolamo cardano an italian polymath who made fundamental contributions to the foundation of probability and bombelli who we mentioned was the first to give a name to the square root of negative one the story begins with delfero who was the first to find a general solution to the depressed cubic equation he made restrictions in the solution that only yielded a real root and he mostly kept the solution to himself the only people he told about the solution were a few close friends and when laying on his deathbed also told one of his students antonio fiore fior wasn't a very good mathematician but given this rather exclusive formula this allowed fyodor to go on and win prize money from solving problems in computational contests fiora's head getting rather inflated from his winds challenged tartaglia to a computational match fiore had discovered tartaglia after tartaglia announced he had found the solution to solving another form of a cubic one missing its linear term instead of its squared term suspecting fiora got the depressive cubic equation from delfero tartaglia dove headfirst into finding the solution for himself apparently he discovered this the day before he was to face off with fjor and put it bluntly absolutely destroyed fior [Music] tartaglia's initial intention was to keep his solutions private hoping to eventually publish them though he never got around to it cordano was very intrigued by these solutions and beg tartaglia to tell him tertoglia ultimately gave him the solution but not the derivation and cardano swore to secrecy intending to keep it that way not knowing del ferro had solved this problem prior to tartaglia once cardano saw delfero's surviving papers on this he broke his vow of silence and published the formulas in his 1545 work arsemagna giving both del ferro and tartaglia credit though this didn't stop tartaglia from feeling betrayed and accusing of plagiarism [Music] within the work cardano introduces the problem of dividing 10 into two parts whose product is 40. this led to a quadratic equation x squared minus 10x plus 40 which yields two complex roots expressing the problem as manifestly impossible nevertheless he embarked on arithmetic manipulation of the roots adding them together and multiplying them yielding a real number in each case he apparently said of this putting aside the mental tortures involved so progresses arithmetic subtlety the end of which is as refined as it is useless clearly showing how uncomfortable the negative under a square root made him he was especially uncomfortable when the square roots ended up under cube roots when dealing with the depressed cubic equations calling such an instance irreducible in 1572 bombelli wrote the work algebra and addressed this occurrence using x to the third equal to 15x plus 4 as his example this equation yields all real roots x equals four and x equals negative two plus or minus the square root of three when using cardano's formula one ends up with the sum of complex conjugates which ultimately sum to four bombelli's brilliant insight led him to deal with the system of equations setting each cube root equal to a plus ib and a minus ib respectively where a and b are real numbers from this deduction that each imaginary part would cancel out and won't be left with only x equals 2a upon summing this allowed him to focus on just the cube of a plus ib ultimately yielding a equals 2 and thus x equals 4. this work was revolutionary as it made it clear that one could toy with the square root of negative one with the usual rules of arithmetic published posthumously in 1615 a result of french mathematician friend suave yeti produced all the results of an irreducible cubic in terms of cosine and inverse cosine avoiding complex numbers in 1637 descartes published la geometry within this work he describes the geometric construction of square roots the construction being for the square roots of any given line segment of positive length he left out much of the work commenting that he intentionally omitted so as to leave the others the pleasure of discovery as mentioned earlier even negative quantities themselves still seem to throw people off well into the 17th century descartes was no exception to this looking at geometric constructions of negatives as making no physical sense and even ignoring them as roots to an equation he went as far as to impose restrictions on certain equations so he would only yield positive roots otherwise he would just ignore thus the square root of a negative quantity wasn't going to make sense either which of course led to him calling such quantities imaginary john wallace was essentially the first to embark on a proper geometric representation of the square root of negative one his first dealings began in 1675 when analyzing a triangle with size one and two and a base of four but he only analyzed it with formal algebra as the triangle was actually impossible yielding real solutions which seemed paradoxical he didn't pursue the problem much more than that but clearly his imaginary wheels were turning flash forward 10 years he published algebra in 1685 stating in the introduction that negative numbers were indeed able to be physically represented by going to the left of zero in the cartesian plane with no issues at all and later in the work producing his attempt at geometrically representing the square root of negative one he attempted two constructions one using the mean proportional which he was not satisfied with and the other using two triangles that have two sides given and an angle given that is not shared between the sides this second attempt hinted at imaginary numbers acting vertically in a plane though wallace said nothing about this it wasn't until kaspar vessel that we were finally able to get the complex plane oddly enough vessel wasn't a professional mathematician he was actually a surveyor working hard to make the most efficient maps he could just so happened that a lot of the geometry he was working with was running into the imaginaries so he needed some way to geometrically represent them in 1797 vessel wrote a paper with the help of the then president of the science wing at the royal danish academy of sciences which discussed vessel's research with the new plane he had created the title of the paper was on the analytic representation of direction and attempt vessel had a beautiful understanding of geometry completely defining the complex plane as we know it today yet he got no real acknowledgement for it in his time as it was written only in danish credit was finally given when the paper was rediscovered in 1895. vessel began his paper by discussing vector manipulation instead of calling what he was manipulating a vector vessel simply called them line segments first describing addition and then multiplication where multiplication was vessel's original contribution since wallace had expressed similar ideas regarding addition of line segments a century earlier not long after vessel's paper was written the complex plane popped up again this time in france because of a paper written by another mathematician jean-robert argand who is also notable for providing the first rigorous proof of the fundamental theorem of algebra in 1806 argonne wrote his paper on the complex plane in the form of a pamphlet under the title essay on the geometrical interpretation of imaginary quantities it wasn't until 1813 however that argon got recognition for this writing which only came after a plea from the mathematician jacques francais notable for his research mechanics land thrust in particular he wrote his plea in anal du matamatique asking that the author of the essay come forward evidently the mathematician adrian marie lujon notable for lujon polynomials and lujon transformation found argon's pamphlet and wrote a letter to jack's older brother francois francais a mathematician who worked extensively in differential calculus but francois died shortly after receiving the letter while going through francoise belongings after his death zech found the letter discussing argon's pamphlet but lujon had failed to mention argon's name in the letter thus the play neither of the men knew a vessel's paper on the complex plane written years before and surprisingly vessel who was still alive in 1813 never heard about argonne's pamphlet in the same year argonne got recognition for his pamphlet a gustan koshi a french mathematician notable for his contributions to the foundations of mathematical analysis developed his contribution to the theory of complex numbers koshi submitted a memoir in complex integration to the french academy of sciences and within his paper introduced the contour integral which is integration within the complex plane and the idea of residue which is to explain why the value of an iterated integral might be different depending on the order in which the individual integrations are performed koshi's work wasn't published until 1825 and the members of the academy did not realize at the time that koshi was about to launch an entirely new branch of mathematics that of complex function theory for the rest of his life koshi spent a great amount of his time developing the theory of the complex numbers making it far more concrete so concrete that by 1850 complex function theory was highly advanced in 1828 yet another geometric representation of complex numbers arose after years of silence though it was largely unnoticed it was written by reverend john warren a fellow of jesus college at cambridge university titled a treatise on the geometrical interpretation of the square roots of negative quantities and his book was similar to the descriptions of our gand in 1831 gauss made his contribution to the theory of complex numbers which included his geometric representation finally making the geometry of complex numbers popular as early as 1796 gauss had already made a complex plane of his own but felt the world was not ready for it to be published thus waiting for it to ripen [Music] in 1835 william rowan hamilton an irish mathematician notable for his important contributions to optics classical mechanics and algebra produced his own work regarding complex numbers that focused only on the algebra it was titled theory of conjugate functions or algebraic couples with a preliminary essay on algebra as a science of pure time hamilton had read warren's book a year after its publication and was in disagreement with warren's presentation he felt that complex numbers should stay purely algebraic that algebra with its square root of negative one was separate and distinct from geometry in his paper ordered pairs of real numbers were introduced as couples for the complex plane he also defined addition and multiplication of these couples making the numbers less confusing and more believable as people were still having trouble accepting the idea of imaginaries for the years up until his death hamilton just like koshi contributed a great portion of his time to the algebra of complex numbers as one can see the conception of the complex plane was no easy feat what we are so accustomed to today was quite a foreign idea to many until just recently beginning with heron of alexandria on the first century ad the lack of acceptance for square root of negative one continued even through the 17th century but it is from there that the magic happened and the path of the complex plane unfolded rapidly developing into the concrete fundamental tool we use today if you enjoyed the video please click that like button and subscribe and if you generally just enjoy the content of this channel please consider supporting on patreon as always thank you for watching and i'll catch you next time

Channel: moderndaymath

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Length: 18min 10sec (1090 seconds)

Published: Thu Dec 31 2020