Math Does Not Equal Calculating

I have one question for anyone who has taken, or is taking calculus:  Can you explain what a Limit is?

I have asked this question literally dozens of times to engineering upperclassmen and graduate students.  Every single time its easy to see from their tortured attempts at an explanation that they have no real idea what a limit is in calculus.  They can integrate problems left and right, but they have virtually no idea why they are integrating something or what its purpose is.  

This is a fundamental failure of math education in this country.  We teach people how to calculate.  We don't teach people math.  

The fact is, calculations are a thing of the past.  Virtually all calculations performed in real world problems are automated with computers.  If memorization is the lowest form of learning, calculating is the lowest form of math.  It teaches nothing of importance other than how to follow a strict process to solve an over simplified problem no one will ever see in the real world.  And after 12-16 years of teaching math, no one has an idea WHY they are doing any of the calculations in the first place.

So I'll let you in on a little secret that most engineers eventually learn after college:  your math classes, and most of your math professors are a waste of time.  Most engineers don't really "understand" math until they start using it in the real world.  They quickly realize their schooling left them totally unprepared for what really matters:  learning how to correctly APPLY math (which is only a tool) to real world problems.

So let me answer that first question for you.  What is a limit?  Well, what is the smallest number of sides any shape can have?  One side only makes a line, not a shape.  Two sides will either make one or two distinct lines, again, not a shape.  A triangle, with three sides, is the shape with the least number of possible sides.  Four sides?  A rectangle.  eight sides?  An octagon...and so forth.

So what happens to a triangle, if we keep adding sides to it...over and over and over again.  Well, there is a limit that you will reach where your are no longer changing the shape of the object in any meaningful way.  Any idea what shape you will be left with?

A circle!  So, as you start to add sides to a triangle, from originally just 3 sides, to an infinite number of sides, the solution you will be left with is a almost perfect approximation of a circle. 

THATS IT!  That is the meaning and reason behind a limit.  There is no magic.  No black box.  Its simply a math trick that will allow you to come as close as possible (the limit) to the true shape of a circle, starting with something completely known...a triangle.

(Of course, limits can be applied to all kinds of things other than circles and triangles...things that exist in real life...like imperfect curves or surfaces...)

Saying Goodbye to Humanity

My apologies for the dirth of updates over the last few months.  An infant at home and an unusually busy semester load will do that to you...

I felt compelled to break my silence by an update on Voyager 1 that I read over at Bad Astronomy recently.  Since I have been wanting to post about the Voyager spacecraft for a while, now is the perfect time.

Voyager 1 was a NASA probe launched in 1977 to explore Jupiter and Saturn at a distance never before possible.  After providing some of the most fantastic science ever about these two monster planets, Voyager 1 slowly became something altogether different:  the ambassador of the entire history of our planet.

Recently, Voyager 1 reached a zone astronomers refer to as the "Heliosheath".  The picture below helps illustrate where this area is.

Basically, this is an area of space around our solar system where the solar wind from our sun no longer blows.  Voyager 1 has now officially said goodbye to our sun forever.  In a few years, she will leave the Helioshealth altogether and become the first object from planet earth to ever enter deep, interstellar space.

Realize that with time, all things will end on our planet.  At some point, our sun will engulf our world, destroying everything mankind has ever accomplished.  All knowledge or memory of our existence will be destroyed and it will be as if we never were.  All the timeless monuments to our brightest and best,  Egyptian pyramids that have survived 4000 years,  the Great Wall of China...they will all be gone forever.   Hopefully we will have long left our tiny planet for greener pastures by this time, but at this point, we can only hope for such an outcome.  

No matter what our future holds, Voyager 1 and 2 will always go forward, with evidence and descriptions of our existence.  On board each is a gold record with detailed instructions.  Any intelligent species who encounters these probes could then hear greetings in 55 different Earth languages, listen to the sounds of our whales, dolphins and weather, even listen to our music.  Due to this spacecraft, we are guaranteed that for as long as the universe exists, our place within it will always be known.

Soon, we will have to say goodbye to our stellar ambassador forever.  One of Voyager's last messages home before leaving our solar system was a picture.  One of the most famous NASA pictures of all time.  If you have never taken the time to stop and look at this picture, you cannot yet possibly know what it is to live and breathe.  That pale blue dot is us.

No one has ever immortalized the meaning of this picture better than the late, great Carl Sagan.  After seeing this picture, not too long before his death, Sagan urged us all to look at this picture in awe.

"From this distant vantage point, the Earth might not seem of particular interest. But for us, it's different. Look again at that dot. That's here, that's home, that's us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies, and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every "superstar," every "supreme leader," every saint and sinner in the history of our species lived there – on a mote of dust suspended in a sunbeam."  - Carl Sagan

Nothing else ever needs to be said...