I understand that Leibniz and Newton are credited with the "invention" of calculus, for example. But I don't understand how people can actually invent new math. How do they do it, and what are the limits to inventing future maths?
207913
Well there is still a list of unsolved mathematical problems called Hilbert's 23 challenges.  Most math develops form the need to correlate new equations to streamline comparisons or find new ways to predict observable behavior.  the former being how calculus of founded.\n\nhttp://en.wikipedia.org/wiki/Hilbert's_problems
160142
Firstly look at all the different engineering feats that men has ever engaged in - for example your desk, the home you live in, lights, cars, highways, computers, TV, Internet, commerical planes, ipod etc - everything is related to mathematics!\n\nWhile eating your meal think about this, how would a chef be able to cook that delicious dish without the use of mathematical principles of proportion.\n\nThe designer clothes that you are currently wearing, how would the designer be able to create such clothes without the help of mathematical measuring instruments.\n\nThe back engine of any business enterprise is mathematics! Without a sound mathematical business strategy, business can go bankrupt overnight!\n\nAnd the list goes on! \n\nSorry to say, but i think you had a very lousy Mathematics Teacher!
616354
Math is used all the time in the engineering sciences.  Many physical phenomena, including fluid flow, heat transport, and diffusion can be described by appropriate partial differential equations.  In order to solve these equations and make predictions, you need to be good at calculus.\n\nIf you limit yourself to addition, subtraction, multiplication, and division, you would be stuck with linear models for everything.  This means that any more complicated relationship between two variables could not be modeled continuously, and you would never be able to extrapolate with any confidence.  Thus development of new technology would be strictly trial and error (and you wouldn't even have a good trial and error system without statistics).\n\nHang in there!  Get through that math.  You don't have to like it, but you will inevitably appreciate the developments that math has enabled.
299980
if your talking about the origin of the numbers we use:\n\nthey are the Arabic Numerals that were used way back during the time of early civilizations and trade routes.\n\nthorough the trade routes, their numbers spread, and were widely used\n\nthey also used the zero, which was created by either the romans or the Greeks\n\nWe dont have any idea as to who invented them, though. Our best idea is scholars or educators.
65407
And I assume that you know that after college you will never ever again use that worthless knowledge.  What scientist is going to be doing long hand algebra when they have a computer?  Ask your teacher who uses it on a daily basis and what percentage of the population that is.  Would you even want to as a job choice?  It's really a shame that schools require such nonsense.
44404
That's untrue.  Quite in fact, the absence of the zero stifled the advancement of mathematics in the west for a long time.  Zero is a number of fundamental importance.\n\nI don't know where you're getting your info.  Perhaps someone got an award from some group of idiots that thought he was being clever, but zero is far, far, far, far, far from being nonsense.\n\nSomeone is jerking your chain, bro.  Give up onit already.  You need to recognize that you were duped and move on.
653247
Yes!!!!!!!!! Email me, and I'll try to shed more light on this. Parabolas are where my genius comes out :)
89888
to get an extra credit, maybe?\n\n---\n\nseriously, I cannot figure out how you can study calculus without knowing limits: even if you only deal with numerical procedures, every iterative formula, or better, EVERY FORMULA (such as those used to find the roots of a non-linear equation, approximated integrals, derivatives and so on) needs a preventive study for convergence conditions: this involves limits!\n\nOne more step back: DERIVATIVE IS A LIMIT ! How can you apply it without knowing what it is?
107094
These links explain the story behind the calculus and the influence of Huygens and Leibniz.
575158
Aryabhatta.... he invented "0"... without which we would not know maths or physics as it is today... and oh yeah... the computers would not be there without the 0s and 1s.
57708
That's an interesting question. But the answer may seem a bit weird at first. I suggest you read the answer carefully.\n\nThe world has invented the number system. \nIn olden days, the arabians and egyptians used numbers from 1 to 9. But the theory of negative numbers came only after a brilliant Indian Mathematician ARYABHATTA devised the number zero as the king of all numbers. Zero is undoutedly the most important and the backbone of maths and the number system.\nThe number system was extended on both sides of zero around the 1800s to 1900s to form nearly more than 100 numbers on each side. With time, the number system has advanced and now we know that its limitless.\nEverything that is made around us has many helping hands. Always remember that my friend. Keeping asking such provoking questions. They will shape you into a good character.
366743
18. Use induction.  If n=0, then it's just (f*g)(a) = f(a)g(a).\n\nAssume that Leibniz' formula holds for some n>=0, and show that it holds for n+1.\n\n(f*g)^(n)(x) = sum k=0 to n (n choose k) f^(k)(x) g^(n-k)(x)\n\nDifferentiate this with respect to x:\n\n(f*g)^(n+1)(x) = sum k=0 to n (n choose k) [ f^(k+1)(x) g^(n-k)(x) + f^(k) g^(n-k+1)(x) ]\n= sum k=0 to n (n choose k) f^(k)(x) g^(n+1-k)(x) +\nsum k=1 to n+1 (n choose k-1) f^(k)(x) g^(n+1-k)(x)\n= f^(0)(x) g^(n+1)(x) +\nsum k=1 to n [(n choose k)+(n choose k-1)] f^(k)(x) g^(n+1-k)(x)\n+ f^(n+1)(x) g^(0)(x)\n\n(n+1 choose 0) = (n+1 choose n+1) = 1, and\n(n choose k) + (n choose k-1) = n!/[k!(n-k)!] + n!/[(k-1)!(n-k+1)!]\n= [(n-k+1)n! + kn!]/[k!(n+1-k)!] = (n+1)!/[k!(n+1-k)!] =\n(n+1 choose k)\n\nputting all the terms together gives the result.\n\n22a.Suppose a is a double root of f.  Then f(x)=(x-a)^2*g(x), so a is a root of f.  By the product rule, f'(x)=g(x)*2(x-a) + (x-a)^2*g'(x), so f'(a)=0.  Suppose f(a)=0 and f'(a)=0.  Then f(x) = (x-a)r(x) for some polynomial r(x).  f'(x) = r(x) + (x-a)r'(x), and since f'(a)=0, it follows that r(a)=0, so r(x)=(x-a)g(x), meaning f(x)=(x-a)^2*g(x).\nb. f'(x) = 2ax + b, so its root is -b/2a.  For this to be a root of f(x) as well, must have a(-b/2a)^2 + b(-b/2a) + c = 0, or b^2 - 4ac = 0.  Geometrically, the graph is tangent to the x-axis at x=-b/2a.\n\n27. Let L = f'(0), and define g(x) = f(x)/x for nonzero x, and g(0) = L.  Then g(0) = L = f'(0) = lim (x->0) f(x)/x = lim (x->0) g(x), so by the definition of continuity, g is continuous at 0.
183915
Yes, modern day inventions are fascinating. Imagine the stuff that companies have developed but have not shown us. Just the other day, I was reading about quantum teleportation and about creating force fields with plasma. It is amazing how far we have come and where our imagination can take us.
573704
In the history of math, it took certain conditions to kindle an interest in various branches of maths and related interests (astronomy, geometry, algebra). Accounting was necessary only where personal consumption varied strongly from personal production, i.e. in cities where communal irrigation or similar communal activities overrode the pure agricultural (or otherwise primary production) activity. Keeping a calendar became another agricultural necessity tied in to maths (or at least counting and accounting).\n\nMore recently, under the Nazi regime there was a school of German physicists which aimed at teaching physics not from mathematical formulary but in a purely textual approach, to counter the presumed "jewish" way we still practise at university.\n\nWhen I started school, another teaching paradigm was to start maths education with set theory.\n\nBoth of these approaches were ideologically based rather than culturally, but if a few generations of mathematicians are taught that way, it might change their outlook at maths to make a cultural difference.
167750
I'm sure you know more about calculus than I do, but I had to mention that Neil Stephenson wrote a pretty good trilogy called the Baroque Cycle or something along those lines.  First book is called Quicksilver.  It's a historical fiction and it uses Newton and Leibniz as character as they work out the basics of calculus.  You might find it interesting.\n\nhttp://www.amazon.com/exec/obidos/tg/stores/series/-/90841/ref=pd_sr_ec_ser_b/103-2231829-6875059\n\nAs for the original question, Newton was first but he didn't publish first, so he didn't get the full credit.\n\nEdit: Yes, my grammar is very poor as I never did well in English class.  I'll try and fix things a little just for you. :)  I don't really think so.  It's pretty much 90% off topic by now.
306569
I'm assuming you mean Einstein, but I would have to go with Newton. He invented calculus and was absolutely revolutionary in developing elementary physics. Now, obviously Einstein was brilliant too, but at least he had a long background of physicists to build on. He stood on the shoulders of giants, one of which is newton.
43474
If you want to have a career cleaning toilets, you probably won't ever use advanced math or calculus. If you want to be a prison guard, you probably will not ever use advanced math or calculus. If you want to be a professional tap dancer or a comic, you probably won't ever use advanced math or calculus. If you want to drive a taxi, you probably will never use advanced math or calculus.\n \nHowever, if you want to be an architect, or a chemist, or a statistician, or an engineer or a molecular biologist, or a hedge fund manager, or an economist, or a weather forecaster or an oceanographer or design airplanes or work with electronics or any one of thousands of interesting professions, you will probably need advanced math or calculus. \n\nHere are some problems:\n\n1. Design a technique for producing better computer speech synthesizers for interacting with users, or reading written material to blind people. \n\n2. Use an MRI machine to make pictures of the inside of the human body. ALL of the data is analyzed extensively using fairly advanced mathematics to make the picture. \n\n3. Finding oil is all done with a large number of equations. \n\n4. Designing airplanes is all advanced math and material well past elementary calculus\n\n5. Power systems are all designed with sophisticated mathematics. \n\n\nI could give thousands and thousands of examples. When you use the internet, it is all mathematics. Your computer software is all mathematics. The weather forecast is all mathematics. The odds at the race track and casino games are all mathematics. \n\nAnd not just simple mathematics. Calculus is just the beginning. After all, Calculus was developed 300+ years ago. You could easily imagine that we might have made some advances in over 300 years, couldn't you?