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What a fun essay! It's nice to see calculus through the eyes of a new scientist, and with the words of an experienced writer.

But the math trick isn't hard to figure out, if you know how. It isn't calculus, it's just algebra. Write it out as an algebraic equation and the trick becomes pretty obvious.

Let X be your guess. Then you write out what the steps tell you to do:

(2X + 5) * 50 + 1756) - birth year

I'm just doing the first case of having already had your birthday this year.

Now work the algebra through to get

100X + 250 + 1756 - birth year


100X + 2006 - birth year

But (2006 - birth year) is your age! Note: if you haven't had your birthday yet, the age you get is wrong... which is why it says to add 1755 if you haven't had it yet.

100X gives you your number first, plus two zeros. When you add your age to that, you get a three digit number where the first is your guess, and the second two are your age.

So you were right-- 2006 is needed; for next year the numbers will have to change.

You can also play with the numbers a lot. Anything that yields the current year after you play with it will work.

I remember always being surprised by these number games, and then not too long ago my sister asked me about one. I realized it was just algebra and worked it out. Kind of a "duh" moment in retrospect, but I think it was more fun to see how it worked than it was to be amazed at it. I feel that way about everything. It's more fun to understand!

You should teach Calculus once you're done learning it.

I think a lot of students who learn calculus learn algorithms for solving problems without really understanding deeply what it's all about. Indeed, I think that even happens with algebra -- students learn in high school how to plug and chug, but I'm not sure they have a deep understanding of what it really means to have a variable representing some unknown quantity, and why many of the various "rules" for what you can do in algebra are reasonable.

A deep understanding of what both of those are all about can only help in remembering and properly using the rules for performing the mechanics of it. But, beyond that, it's necessary if you're really going to use math to help understand science. Being able to do algebreic manipulations is one thing -- by and large, students in my non-majors intro astro course are very good at that. (I don't put too much emphasis on the equations, but there is some.) Where they have trouble is seeing that the equations aren't just a game you play to get the answer to a problem, but that they are a way of expressing the concepts that we're really working on-- and vice versa.


It's nice to read an essay about Calculus from a new scientist, to me, because when I learned Calculus a couple years ago at a community college, I learned it from a teacher who only did math and showed signs of being truly uncomfortable with the scientific mindset. On a few problems we had to work on, examples were often of course basic physics problems solved with Calculus rather than kinematic equations. Every time we would try to include the units during discussion or on the marker-board, he'd get all fussy and tell us to wait until the end and then write the units "because then you just get'em for free."

When asked why he's so persnickety about units, he went on a little rant about how math theorems can be proven absolutely so he's always been more comfortable with math rather than physics and the units bother him.

Basically he was saying that the units (And including the units in every step of a problem) reminded him of how uncertain truth in the real world is. He was actually telling us that he's uncomfortable with dealing with facts and would rather deal with the absolute rules on the chalkboard/marker-board than the approximation process that using physical units entails, like the real world.

He may be a sissy, but at least he realizes that hard truths are not found in reality and accepts it, unlike other professional mathemeticians like William Dembski, who have such an attachment to the absolutist mindsets they've attached math to that they do not understand that math models that do not reflect the flexible properties of the real world are not valid reflections of it.

If only my Calc. 1 teacher did understand that Calculus is a real answer to real questions like Zeno's Paradox. I think he could be a good phycisist.

It's fun to vicariously relive the enjoyment of getting calculus.

I want to throw in a comment on fluxions. They are equivalent to Leibniz's tools, but it's a question of where you place the focus. Leibniz implicitly began in the framework of analytic geometry, where he has coordinates, and thought in terms of changes of those coordinates along graphs of functions.

With Newton, he starts with a line, the path of an object moving through space without external influence. Then he says that if the path influenced, then there is some vector (well, line segment in his world) that gives the result of that influence, namely the vector between the undisturbed path and the disturbed path. This is the fluxion.

The two need not be perfectly equivalent. You can define a fluxion in a curved spacetime perfectly well (though Einstein told us to forget that: once you've curved the space, just throw the fluxion into the curvature as well and have done). On the other hand, the Leibniz ratios of differentials of coordinates don't always play nicely in such situations. The fluxion is defined globally since it's merely a geometric relation between abstract objects in the space, but you may have to use several coordinate systems patched together depending on where you are in the space if you actually want to do analytic geometry, and so the Leibniz equivalent of the fluxion is not a simple translation.

At this point I cannot recommend Cohen's translation of Principia too highly. I didn't feel like I understood classical mechanics of point particles until I had really internalized the first chapter of that and spent three months staring into space forcing my brain to think that way...and reexpressing all the mathematical results I knew in that visualization.

"the entire discipline is encapsulated in two fundamental ideas: (1) the derivative... and (2) the integral... Everything else involved in calculus is just variations on these two themes."

While I agree that this is what mathematicians mean by "Calculus", I would add that Calculus is a subfield of a broader topic called "Analysis", and within Analysis everything, including the derivative, the integral, and a whole lot of other things beside, are just variations on just one theme: "The Limit". The Limit is where Zeno's arrow strikes its target.

Oh, and another thing. Calculus doesn't go "to infinity and beyond", although it does approach infinity sometimes in The Limit. If you actually want to go to infinity and beyond you need to investigate the theory of transfinite arithmetic - a whole 'nother topic. Ask your supervisor about Cantor.

Good luck with your exploration.

Especially helpful comments here -- thanks. I appreciate all the input, and it'll be nice to come back to them as I progress.

Daran, I think we're covering the notion of the Limit in a later lecture. But, FYI: the post title "to infinity and beyond" was a tongue-in-cheek pop culture reference (TOY STORY), not a literal reference to calculus...

Great essay! Just a few comments: I presume you mean Christiaan Huygens? He had a (crude) wave theory of light competing with Newton's particle theory of light.

To show how awful Newton was, I remember two anecdotes: Robert Hooke was a very short man, so when Newton said "If I saw further than others, it is because I stood on the shoulders of giants", this is a direct dig at Hooke. Second, the Royal Society sided with Newton, because he was the president of the Royal Society at the time!

Hi Jen, enjoyed this here post, and, glad to hear you haven't lost a chunk from your knee in your previous post.
So just for fun concorde flying London to New York could expect to arrive there before it left, and expect to be travelling against time on the return journey. The actual average speed (time) would be determined by the prevailing currents and winds.
Does an arrow, all things being equal (strength of bow and prevailing wind) travel the same distance whether fired East or westward, the earth's spin (and time) having no measurable(?) or practical(?) impact on distance travel. Equally so if the arrow is fired North or Southward.
With more powerful and longer distance ballistics, do the measurements become much more 'critical'?

Aerik is right! It's a lot of fun to watch someone blogging about learning calculus. Keep it up!

And Aerik's teacher is wrong! You need the units to keep yourself from making mistakes. If your final answer is supposed to be in meters and you're getting kilogram seconds-squared/degree Kelvin, you've done something seriously wrong.

Jen, once you've got Calculus under your belt (oh, and Vector Calculus, too, you don't want to miss that), where to next? Differential Equations? Group theory? Linear Algebra? Complex numbers?

I feel like a kid watching another kid in a candy store.

For those who haven't encountered Zeno before, he was a Greek philosopher living in the 5th century B.C., who thought a great deal about motion. Some might argue he thought a bit too hard about motion; the guy was always playing devil's advocate, even with his own arguments, which is how he arrived at his eponymous paradox.

Philosophical note: Actually, Zeno had seveal paradoxes, three of which are famous. His purpose wasn't to play devil's advocate, but to argue for the unreality of time and space, and the phenomenal world in general. THis was a popular position with the ancient Greeks -- see Plato's cave analogy -- but has some adherents even today. For example, the hardheaded 19th century philosopher McTaggart argued for the unreality of time using updated Zenoesque arguments.

What you present as a "corollary" isn't a corollary to "zeno's paradox", but a separate paradox of Zeno's known as the arrow paradox.

The meme that "Calculus solves Zeno's (first) paradox" is one of those items of conventional wisdom that goes unquestioned despite having such a shaky foundation. Calculus may resolve Zeno's paradox, but that's certainly not uncontroversially or straightforwardly true.

Oh, and by the way, speaking of chocolate math, you can also determine the speed of light using chocolate.

(Is there anything chocolate can't do?)

Daran, while you're correct that the standard treatment of calculus shuns actual infinities in favour of limits, there's also a more modern technique using non-standard analysis (not the same as transfinite arithmetic) that goes straight for infinite quantities first. In this approach a derivative really is the slope of an infinitesimal line segment and an integral really is an infinite sum of infinitesimal quantities.

I'm a little late to this party, but I have to say that I learned Calc originally from a guy with a Ph.D. in Chem. Eng. and 35 years in dustry before he went off to become a college prof, and he was the best instructor I've ever had at any level of mathematics (the only one who comes close was my 7th grade algebra instructor who was an aeronatical engineer...this would seem to suggest something).

Pure mathematicians get too wrapped up in the abstractness of it all, and I've yet to meet one who understood that talking to us like we already knew what they were teaching us was pointless, since if we did, we wouldn't be wasting our time in their class.

And textbooks...written by math Ph.D.s, for math Ph.D.s, and to hell with anyone who doesn't understand.

Don't get me wrong, I love mathematics. I just can't stand most of the people who've ever attempted to teach it to me.

I'm a little late, but for completeness's sake I thought I should post a link to a textbook on this "non-standard analysis" of which Jeremy Hinty spoke.

My apologies --- "Henty", not "Hinty". Stupid fallible fingers. :-/

thanks for the bloigs!! got so many ideas aquired!

I too developed an interest in mathematics (number theory, what Algebra really means, etc.) late in life (50'). I've since realized that most of the math "teachers" I had in school were either incompetent to begin with, or knew the subject but NOT how to teach! It's a national disgrace that we have allowed this to happen in the US for so many decades, especially in the public schools.

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