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A Gentle Introduction to Calculus - Please Don't Panic!

Most of our presentations have been pretty high-level. But some members have asked for more detailed, more mathematics-based presentations. What can be more important to physics than Calculus? In this lecture, Larry Smith will take you, one step at a time, through basic Differential Calculus. The goal is to give you an understanding of what calculus is and how it can be applied.

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  • Mohsen K.

    Continues........

    Here we can ignore these solutions and call them undefined and nonsense or look for the possibility that there may be a domain where every number equals every other number. You can call my proposal madness. But again I am looking for possibilities. Going back to Larry’s discussion, any number divide by zero equals infinity not x as he mentions (1/0 = x ). So,
    x/0 = infinity Then,
    x = 0 * infinity Here, can we conclude that any number can arise if we couple zero by infinity. Again you can call this madness and waste of time or delve into it a bet more.

    1 · April 16, 2014

    • Mano P.

      If I may add, in Geometry the points have no size, lines have no width and planes have no thickness by definition. Mathematics is not used to fully describe the World/ Universe. A mathematical model is formulated so that it can be used to predict physical phenomena. There is a world of difference between fully describing the world and predicting some physical phenomena.

      1 · May 10, 2014

    • Victor

      wiki: "..a point is defined only by some properties, called axioms that it must satisfy. In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute." In other words the axiom defines a point to have 0 dimensions - which allows it to have an infinitely precise location in any dimension, which our digital quantum world/reality, may not be true, ... unless we live in a analog, infinitely accurate reality... and hence in a paradoxical world (i.e. Zeno's paradoxes).
      Differential equations, which are used extensively in the laws of nature, are based on real numbers, that have infinite accuracy or preciseness, in other words the axiom of a point may be turn out to be a stumbling block, not being part of the right mathematical system to describe reality.

      May 26, 2014

  • Larry S.

    Just noticed. There's a whole Wikipedia article on division by zero. http://en.wikipedia.org/wiki/Division_by_zero

    April 20, 2014

    • Victor

      Maybe call 0/0 = null.
      Null mean no information in the computer world, and had to be invented since an information placeholder can either have information (any data including 0) or no information, i.e. null.
      Use ∅ for null.

      April 22, 2014

  • Keith B.

    Round 2!

    0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways.

    April 20, 2014

    • Larry S.

      Well, 0! = 1 isn't so much for convenience as it is for consistency. But more importantly, the concept of factorials has been generalized into something called the Gamma Function (http://en.wikipedia.o...­), and in that context, 0! naturally falls out to be equal to 1.

      April 20, 2014

    • Keith B.

      I added the issue under the title Round 2 as an attempt to add a humorous touch to our extended debate. For those interested http://mathworld.wolf...­

      April 21, 2014

  • Larry S.

    Final finale: As another example, while Newton's law of gravity worked well for hundreds of years, General Relativity showed us that the world worked in terms, not of just vectors (mass "x" attracted towards mass "y", and vice versa), but of tensors (a highly sophisticated generalization of vectors). We've tightened the screws on the laws of the universe. And presumably quantum gravity will tighten the screws even further.

    One more example: there are several reasons why it's unlikely there are more than 3 generations of quarks. See http://en.wikipedia.org/wiki/Generation_(particle_physics)

    Will we ever get to the point where we can get enough experimental evidence (and a corresponding mathematical framework) where only a unique set of laws is possible? I can conceive of such. Which of course is no guarantee.

    But it would be so neat if we could!

    1 · April 20, 2014

  • Larry S.

    Finale: So perhaps Wigner's observation is due (as Max Tegmark of MIT has alluded to - see http://www.amazon.ca/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809/ref=sr_1_1?s=books&ie=UTF8&qid=1398021661&sr=1-1&keywords=max+tegmark), the universe's truly fundamental building blocks obeying mathematical laws. Of course, why they should obey those (or any other) laws is a different matter! I plead Nolo Contendere on that.

    And I mentioned deducing the existence of these "things". When a new aspect of the world is discovered (e.g. electricity), we think of all the great things it might lead to. But a different way of looking is this. We've just *restricted* the way the world works. There are now (or will be) laws (e.g. Maxwell's equations) that say "this is the way things are". Where previously we could think of electricity and magnetism as different, that's no longer valid.

    [Ran out of room again]

    April 20, 2014

  • Larry S.

    Continued from previous post: Suffice it to say that my speculation is that at the very bottom of reality (assuming there is a bottom), we'll find (or at least, deduce the existence of) that there are "things" (I deliberately don't call them particles, forces, etc) that behave (have symmetries) that behave according to the (very simple) rules for things being a group. And as I implied, these may build up more and more complex concatenations until (maybe) we wind up with emergent properties such as, oh, I dunno, how 'bout space, time, particles, forces, big bangs, and eventually physicists!

    I know this is awfully hand-wavy. I know of no succinct way to describe group theory and what it's capable of. Maybe a Rubik's Cube, where the interchange of two opposite side "cubies" are reminiscent of 2-quark mesons and the rules governing 3 corner cubies are a bit like 3-quark hadrons.

    [Finale in next post]

    April 20, 2014

  • Larry S.

    Another Speculation About What Underlies Quantum Mechanics...

    In 1959 the Nobel Prize winning physicist, Eugene Wigner, gave a talk about "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (https://dtrinkle.matse.illinois.edu/_media/unreasonable-effectiveness-cpam1960.pdf) See also the Wikipedia article at http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

    There's a mathematical theory of symmetry called Group Theory (http://en.wikipedia.org/wiki/Group_(mathematics)). It's a bit strange in that its definition sounds almost trivial, but it has amazing, profound depth. This isn't the place to go into why and how it's so powerful (but that would make a great bi-monthly math meeting topic).

    [continued in next post]

    April 20, 2014

  • Mohsen K.

    Let us play with mathematics and look for new possibilities. We can show that any number equals another by other equations other than dividing by zero as well. For example, in algebra,
    Let a = 1 and b = 2. We can write,
    ( a – b )2 = a2 + b2 – 2ab
    (Unfortunately the web page does not reflect the exponents properly my apologies) then,
    ( b – a)2 = b2 + a2 -2ab So,
    ( a – b )2 = ( b – a)2
    +_ (a – b ) = +_ ( b – a) Then two of the solutions would be,
    a = b

    1 · April 16, 2014

    • Mohsen K.

      If Victor and Vera want to manage mathematic meetings. I will book the room for you guys.

      April 19, 2014

    • Vera B.

      as Victor suggested bi-monthly mathematics meeting think is a great idea because a) some of us are missing mathematics b) understanding math helps understanding physics and other sciences c) we can only challenge concepts well understood (and should) and d) its more fun doing math with group . I am away for the most of May , back after 25th and after that all dates are good :)

      April 20, 2014

  • Keith B.

    Oil on troubled waters!
    An article at betterexplained called "Why do we need limits and infinitessimals?" sheds some light on the issue in an interesting way.

    April 18, 2014

    • Mohsen K.

      Thanks for the input Keith. I am not sure if delving into the meaning of limits and infinitismal is the answer. For me zero is zero. Besides, if there is a thing, there should be a nothing as well. Thing without nothing cannot be defined.

      April 19, 2014

    • Larry S.

      There can be a useful distinction between zero and nothing. If I were to ask you, say, how much money you had in your shirt pocket at this moment, you might say zero. That's a specific amount. Whereas, if I didn't ask, I'd have no idea how much you had. The term "nothing" is used in some circumstances to indicate the total lack of knowledge. It's a little bit like the distinction between "speed" and "velocity". In casual parlance, they're synonyms. But in mathematical / physical terms, they're not; velocity is speed + direction. So yes, to most people zero and nothing are the same. But you can make a useful distinction between the two, if you like.

      April 19, 2014

  • Vera B.

    little madness in the right direction is not necessarily bad thing. Take for example Riemann sphere (spinning top?) very much used in physics, twistor and string theory, quantum mechanics ..
    http://en.wikipedia.org/wiki/Riemann_sphere

    Wolfram Math defines it as :
    The extended complex plane is the name gives to the complex plane with a point at infinity attached: C union {infty^~}, where infty^~ denotes complex infinity. It is also called the Riemann sphere and is various denoted C^* or C^^. it is very difficult to imagine infinity or complex infinity . the said wikipedia page states that
    "The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0 = ∞ well-behaved"

    it also talks about basic arithmetic operations of Extended Complex Numbers
    "The extended complex numbers consist of the complex numbers C together with ∞. "

    1 · April 18, 2014

    • Mohsen K.

      Thanks Homer for your input. I understand that claiming a = b sounds ridiculous. I see that by substituting 2a = 2b in eqn 1 we can normalize the calculations and get rid of the seemingly ridiculous proposition. But we are playing with math to ck for new possibilities.
      We are going off track now. Let us forget about a=b issue

      April 19, 2014

    • Larry S.

      There's an even easier way to see that you can't be naïve when dealing with square roots. a^2 = (-a)^2. So a = -a. 'Nuff said.

      April 19, 2014

  • Mohsen K.

    Complex system’s unit circle is normally used in theoretical physics calculations. It is a polar version of complex numbers where the value oscillates between 1 and -1. It is used for oscillatory phenomena such as wave function of objects.
    http://en.wikipedia.org/wiki/Unit_circle
    In unit circle the real value of objects periodically appears and disappears and as such you may conclude that they are discrete.

    April 17, 2014

    • Larry S.

      Polar coordinates are 2-D. They need 2 real numbers. One of them (normally called theta) if the angle it makes with the x-axis. The other coordinate, "r" is the distance from the origin. How do you represent 3 + 4i in polar coords? By the Pythagorean Theorem, r = sqrt(3*3 + 4 * 4) = 5. The angle it makes is arcsin 3/5, or approximately sin 37 degrees. So 3 + 4i in polar coordinates is (r, theta) = (5, 37). Yes, there are some circumstances where the area of interest is limited (e.g. to within the unit circle), but that's just a subset of pairs of real numbers.

      April 17, 2014

  • Mohsen K.

    Sorry Larry, I am not sure who is wrong here. Look at any text and your Wikipedia reference again. Complex numbers are combination of real and imaginary numbers. Imaginary numbers are not real numbers. http://en.wikipedia.org/wiki/Imaginary_unit
    We may use numbers to count the units of imaginary numbers such as 2i,3i,4i…….. But that doesn’t make them real numbers. Argand diagram does not have the ordinary X and Y axis.

    April 17, 2014

    • Larry S.

      Sorry, but you just don't understand. Yes, originally mathematicians thought of complex numbers as you describe. But a more sophisticated way of looking at them is as vectors. A standard real number can be considered a one-dimensional vector with its origin at (0, 0) and extending to the right (or left for -ve #s). A complex number x + iy is isomorphic to the ordered pair (x, y), a 2-D vector with its origin at (0, 0) going to (x, y). "i" is merely the vector (0, 1). In this formulation, all you have are real numbers.

      April 17, 2014

  • Avo

    Very interesting points raised by Larry and David (Hmmm. The Larry David postulate? :) (Not to be confused with Larry David of Seinfeld!)). As of now, Calculus is intensively used in QM. Will this take us from “uncountable” sets (big infinity) to “countably infinite” sets (small infinity)? Digital Physics? An interesting thought to ponder.

    April 17, 2014

  • David

    I think Larry has mentioned a very important point when dealing with physics.
    Mathematical concepts include infinity, but if physics is quantized, then this would suggest a mathematical description of reality would have to be constrained by the number of quanta that actually exist.
    For example, only so many quanta would be possible in any given length, so the math would have to be limited to that number of units within the given length.

    April 17, 2014

  • Larry S.

    My Feeling As to What's Wrong With Quantum Physics (At Least In Part)

    Standard physics is based on real numbers (or more than one real number, such as complex numbers). Real numbers are infinitely dense. Between any two points, there are an infinite number of other points. To take just one example, between points a and b, there exists the number (a + b)/2, which is larger than a but smaller than b, and so is in between a and b.

    But there's more and more and more evidence that the world is quantized. So using traditional tools (like calculus) that are based on the assumption that
    it makes sense to talk about points an arbitrarily small distance apart (e.g. a trillion orders of magnitude small than the Planck length) is probably a fundamentally flawed approach. Physicists may be using the wrong tool for the job.

    2 · April 14, 2014

    • Larry S.

      And where did you get the idea that real numbers are ever discrete? That's just a wrong concept on your part.

      April 16, 2014

    • Mohsen K.

      Please see my response in the website

      April 17, 2014

  • Betty

    A fine and comprehensive presentation. Thank you Larry, you never disappoint! If you have any more topics you'd like to present on, do let us know!

    April 15, 2014

  • Victor

    At first I had my doubts about this topic being delivered successfully, but amazingly it was well done by Larry.

    April 12, 2014

    • Mano P.

      Mohsen, I see your point. To explore reality is the pursuit of science. Mathematics is but a tool. To tackle new problems new mathematics is made (evolves?) with new rules etc. For example from Euclidean geometry to Riemannian geometry. Einstein made this point at a speech in Berlin in 1921 - As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

      1 · April 14, 2014

    • Victor

      The game in mathematics is to have non-contradictory statements, consistency, and non-paradoxes in following the smallest number of axioms possible. Coincidentally reality also "tries" to never have any "miracles" or paradoxes or inconsistencies in its causality, so it should be not a surprise that mathematics and science, in having the same "goals", sometime crisscross each other - but are not the same! For example Euler's 1700's beta function was later used by string theorist Gabriele Veneziano. Dirac had the same viewpoint regarding math sometime appearing in nature's "secret" rules or game.
      If they were the same, then a bunch of math equations scribbled on a blackboard could initiate a big bang. Math does not.

      April 14, 2014

  • Larry S.

    Why you can't divide by zero - two reasons...

    (1) a/b = c means a = b * c. For example, 10 / 2 = 5 means 10 = 2 * 5

    Consider 1/0 and 2/0.

    If 1/0 = x, then 1 = 0 * x = 0
    There 1 = 0. This is nonsense.

    And if 2/0 = y, then 2 = 0 * y = 0.
    So 1 = 0 and 2 = 0, so 1 = 2. Yet more nonsense.

    (2) Another way of looking at a/b = c means that you can subtract b from a, c times.
    For example, for 10/2, you can subtract 2 from 10, 5 times.
    But if b = 0, then you can subtract forever and never get an answer. So that's
    another reason why dividing by zero is undefined. And please, don't waste
    everyone's time by throwing in the magic word "infinity".

    2 · April 14, 2014

  • Vera B.

    Great job Larry. This lecture reminded me of the favorite math teacher

    April 12, 2014

  • Daniel W.

    A very good presentation Larry.

    April 12, 2014

  • Gayatri

    Thanks to Larry for a well thought of and a good introduction to Calculus. I especially liked how all the examples fit together and the link to Snell's Law was a nice touch. The get together afterwards at Milestone's was wonderful. I prefer this place to Baton Rouge.

    1 · April 11, 2014

  • A former member
    A former member

    I just now noticed there's no room number. Is it the same as the biology group?

    April 11, 2014

  • Barbara

    As requested by Betty I reserved a table at Milestones (5095 Yonge Street,[masked]) for 20 people for 9:15pm.

    1 · April 1, 2014

    • Gayatri

      I agree with Dan. Thanks Barbara for making the après plans for the group. See you all in a few days and have a good week.

      April 7, 2014

    • Barbara

      I am feeling sad and won't join tonight. I called Milestones to reconfirm and they'll be happy to meet you at 9:15pm as arranged.

      1 · April 11, 2014

  • Larry S.

    I've uploaded my slides for my Calculus presentation this Friday. Anybody who wants a running start at this long talk is welcome to look them over. They're at http://www.meetup.com/northyorkphysics/files/

    April 9, 2014

    • Maris

      That looks very nice. Great inclusion of Snell's law and Fermat!

      April 10, 2014

  • Allistair

    This should be interesting. It's well known that mathematics is the language of Physics.

    1 · March 21, 2014

    • Victor

      Generally differential equations is the language of physics laws so far (lagrangian?) ... but non-linear equations (chaos .. see lorentz ) is the next step to refining the laws, with the help of computer models or algorithmic equations.

      April 7, 2014

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