Category Archives: Math

Rediscovering i

No, this is not going to be a post about me reacquainting myself with my inner child.  This is going to be a post about imaginary numbers!  Yay!

Yesterday, during a conversation with friends, we somehow got onto the topic of i.  I know what you’re thinking, how can you NOT have a conversation with friends and have the topic turn to imaginary numbers?  Am I right?  You remember i, right?  i = \sqrt{-1}?  Of course you do!  We were trying to remember what i^2 equaled.  Having worked with imaginary friends much more recently than I’ve worked with imaginary numbers, I was leaning towards i^2 = 1.  A friend, who happened to be a math major in another life, was leaning towards i^2 = -1.  Neither of us were confident in our answer, though, so I did what any smartphone wielding dork would do and looked it up.  Yeah, i^2 = -1.  Never doubt a math major.  About math.  Doubt them about everything else because they thought it was a good idea to be a math major.

This whole exercise, of course, got me thinking about one of my favorite subjects; how we teach our children.  Given the way we learn about squares and square roots, it would be perfectly natural for someone to think that i^2 = 1.  After all, we have it drilled into our heads that the square of any number results in a positive number.  I had that in mind when I was leaning towards i^2 = 1.  I remembered enough about all the stuff that we were taught were rules that turned out to be more like generalizations to be unsure that I was right, but someone who was never exposed to imaginary numbers would likely encounter a cognitive dissonance that would be difficult to overcome after years and years of believing one thing.

So why don’t we teach children about imaginary numbers immediately?  We don’t need to get into the nitty-gritty of how to figure out that i^k = i^{k\mod{4}}, but it sure would be nice to stick something into the back of those sponge-like minds that there are these things called imaginary numbers that they may run into someday that don’t fit the mold that is being taught.

The big problem, of course, is that we teach children individual things and we require them to understand that one individual thing as quickly as possible.  If you can’t add two numbers together after second grade, you are a failure.  If you can’t multiply two numbers together after third grade, you are a failure.  Our standardized tests show it to be true.  Nonsense, I say.  I cannot count the number of times I attempted to learn some math subject only to have it make complete sense after something in another subject made the pieces all fit together and I would wonder why these discrete pieces weren’t taught together as part of a whole.  We need to stop teaching children answers and start teaching them solutions.

Today, we teach kids that 123 + 321 = 444.  Add the ones and carry the remainder, then add the tens and carry the remainder, then add the hundreds and carry the remainder.  It works, but it’s not getting kids intimately familiar with numbers like they need to be if they want to succeed in more difficult subjects.  Kids need to learn that numbers can be split apart and replaced and moved around using a set of very simple rules.  Yes, 123 + 321 = 444, but it also equals 100 + 20 + 3 + 300 + 20 + 1 = x + 300 + 20 + 1 = x + y = 444.  I don’t think things like this are too difficult for the second grade mind to be able to grasp even if it takes them years more to fully grasp the implications.

Math Comedy Gold

No, really!  You need to read this.  What happens when five math experts try to split a check?  I think my favorite part is this:

Economist: I mean it! If there were no taxes, I would have ordered a second soda. But instead, the government intervened, and by increasing transaction costs, prevented an exchange that would have benefited both me and the restaurant.

Engineer: You did order a second soda.

Economist: In practice, yes. But my argument still holds in theory.

*snort*

3.14159265359

Happy Pi Day everyone!  It is a day where we all give thanks for all of our favorite circular objects that would not be possible without pi.  I am thankful for elliptical orbits without which our Earth would be thrown into the vast frozen wasteland of the universe.  What are you thankful for?

Here are 10 Youtube videos about pi to get you through your day.

How Not To Get Out Of Debt

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The above graphic is from the facebook page of some “financial guru” named Dave Ramsey.  If anyone ever gives you financial advice like this, you should back away from him as quickly as possible and never take advice from him again.

Getting out of debt IS about math.  It’s about learning the math and recognizing how much you’re saving by paying off your debts the right way.  If you know the math, you will feel good about what you’re doing no matter how long it takes you to pay off an individual debt.  Dave Ramsey is basically telling you that you’re too stupid to learn the math so you might as well feel good about yourself even though you’re likely throwing away money by doing so.

What’s the right way to pay off your debts?  It’s simple.  Which of your debts has the largest interest rate?  Pay that one off first.  Which of your debts has the second largest interest rate?  Pay that one off next.  And so on.  With very few exceptions, this is the fastest way to pay off your debts while making sure as much of your money stays yours.

Why do it this way?  Because it’s the interest that kills you.  If you buy a house for $250,000 with a 30 year mortgage at 5%, you will be paying close to $500,000 for that house if you pay off the minimum amount every year.  Yes, you pay double for the house because of interest costs.  If your loan was at 7%, you’d be paying nearly $600,000 for the house.  Interest adds up quickly.

That is an extreme example, though.  Most people who are trying to get out of debt aren’t terribly worried about their mortgage and it is likely that the interest rate on the mortgage is the lowest of their debts if they have one.  When we’re talking about debts, we’re almost assuredly talking about credit cards.  Credit cards commonly have interest rates as high as 30%.

Let’s take a simple example.  Say you have two debts.  One, a credit card with a $10,000 balance and a 30% interest rate and the other a credit card with a $5,000 balance and a 15% interest rate.  Say the minimum payment for each is $50 to avoid paying penalties.  Also say that you have $500 a month dedicated to paying off your debts.

Dave Ramsey’s advice is to pay off the $5,000 one first.  Let’s see how that goes.  We always want to avoid paying penalties (Something Dave Ramsey completely glosses over with his graphic) so we will be paying $450 a month to the 15% card and $50 a month to the 30% card.  It will take you 13 months to pay off the 15% card.  At that time, the balance on the 30% card will then be $12,200.  It will then take another 39 months to pay that off.  That’s a total of 52 months.

Now, let’s do things the correct way and pay off the 30% one first.  Again, we always want to avoid paying penalties so we will be paying $450 a month to the 30% card and $50 a month to the 15% card.  It will take you 33 months to pay off the 30% card.  At that time, the balance on the 15% card will be $5,400.  It will then take another 13 months to pay that off.  That’s a total of 46 months.  6 months sooner than Dave Ramsey’s way.  Congratulations!  You just saved $3,000 by paying off your debts the right way!

Frequentists vs. Baysians

If you don’t laugh out loud at this XKCD comic, you need to take a statistics class.  Or you have a life.  Definitely one of the two.

 

A note about statistics

There is a lot of brouhaha about pollsters (most notably Nate Silver) predicting an Obama win in the coming election.  As of this post, Nate has the chance of Obama winning at 83%.  Various news personalities are complaining that Nate must be in the bag for Obama to dare predict that Obama will win and that a Romney victory will prove that Nate Silver is a lying liar who lies.  A Romney victory would prove nothing of the sort, though.

Why?  Nate Silver has been eerily accurate at predicting electoral outcomes.  He has a model that he follows and he is mostly public about his methods.  His methods haven’t changed and they predict an over 80% chance of Obama winning.  Does that mean Obama will win?  No.  Romney still has a 20% chance of winning.  If you were to bet on the outcome, though, you’d be wise to put your money on Obama.

Look at it this way.  Take a 10-sided die.  What?  You don’t have a 10-sided die?  Ok, go back in time and play Dungeons & Dragons as a kid.  Poof, you now have a 10-sided die.  Pretend that an Obama win is represented by 1-8 and Romney is represented by 9-10.  Roll the die.  There’s an 80% chance you’d roll an Obama.  Roll it a couple of times.  Most of the time, you’ll roll a Romney in only a few rolls.  If there could be two elections, there is a 36% chance of rolling a Romney one of those times:  2/10 + (2/10 * 8/10).  A third election would give you a 49% chance of rolling at least one Romney: 2/10 + (2/10 * 8/10) + (2/10 * 64/100).  Generally, you have a 1 – (8/10)^n chance of rolling a Romney in n rolls.

Of course, this is all just a useless exercise in math and probability because there is only one election and Obama currently has an 83% chance of winning according to the polls.  But don’t go attacking Nate Silver or math if the die rolls a 9 or a 10.