One of the goals I wanted for this website was to provide math software for my users.  I finally took the first step to realize this goal.  I am experimenting with some programming, and have produced a math screensaver.

This first product was made with two objectives in mind.  First, I wanted something that my kids would enjoy, and, secondly, I wanted something that my users would like and would be willing to beta test for me.  Well, my kids liked it – they are able to point out and name numbers from 0 to 9.  If you wish to add it to your set of screensavers, please let me know of any bugs and the operating system you are using.  I have tested it out on several computers, and it worked fine on all of them.

I plan on generating more math programs in the near future.  I’ll make some more math screensavers, and I am planning to make some downloadable and online math games.  Anyway, I hope your kids will enjoy this math screensaver, and I hope it will help them with their math skills!

Numbers Saver Math Screensaver

Numbers Saver is a math screensaver for children.  It displays comical digits from 0 – 9.  Here is a screen capture:

Download File Name:

Numbers Saver Install.exe (672 kb)

Requirements:

Windows XP, Windows Vista, Windows 7

Installed File Size:

844 kb

Click Here to download.

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Tags: math screensaver, math software, Math Tricks

Just a quick update to let you know why there has not been much activity of late here at math tricks. We had a new addition to our family here!  We now have a nice round (prime) number of kids here . . . 3 boys!  Needless to say, things have been hectic here!  I hope to post new articles on a regular basis again very soon.

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Back in grade school, I was first introduced to the Binomial Theorem.  The title alone was quite enough to intimidate me, let alone the seemingly impossible to understand equations involved with it.

I’ll not go into the mathematics of the binomial theorem here.  Instead, I’ll introduce you to math tricks which can be used instead.  First, let me refresh your mind on why we were taught the binomial theorem.  Remember when you were asked to expand the equation:

(x + y)²

If you recall, this equation can be expanded to the equivalent equation:

x² + 2xy + y²

The binomial theorem will allow you to solve a higher order problem of the example above.  For instance, what is the expansion of the equation:

(x + y)⁵

Generally, an equation of this type can be expanded as:

where c₁, c₂, … are the binomial coefficients in the expansion.   So given any n, you can determine the expansion without the coefficients.  Expanding our example above:

(x + y)⁵ = c₁x⁵y⁰ + c₂x⁴y¹ + c₃x³y² + c₄x²y³ + c₅x¹y⁴ + c₆x⁰y⁵

So how do you determine the binomial coefficients?  You can determine the binomial coefficients individually using the equation:

for k=0 to k=n.  This works fine, but is a little bit cumbersome – especially for large values of n!  So what is the math trick to solve this quickly?

Before I can answer this, I have to introduce to you Pascal’s Triangle.  Pascal’s triangle is a mathematical progression which is determined by constructing a triangle with numbers using a very simple algorithm.  First, take a look at this example of Pascal’s triangle:

At the very top is row 0, which is simply a 1.  In row 1, there are two numbers, both 1s.  In row 2, there are three numbers: 1, 2, and 1.  Notice that the 2 in row two is the sum of the two numbers above it; this is how you determine the numbers in the triangle – simply add two side-by-side numbers to get the result below and between the numbers:

Construction of Pascal’s Triangle¹

So how can you use Pascal’s triangle to find the binomial coefficients when you expand the equation (x + y)⁵?  First, notice that the equation is raised to the 5^th power.  So now simply go to the 5^th row of Pascal’s triangle (remember, the top row is row 0), and those numbers are the required coefficients:

1 5 10 10 5 1

And so,

(x + y)⁵ = x⁵y⁰ + 5x⁴y¹ + 10x³y² + 10x²y³ + 5x¹y⁴ + x⁰y⁵

Quite a time saver!

¹File by Hersfold, en.wikipedia.org/wiki/User:Hersfold

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Tags: binomial expansion, binomial theorem, Math Patterns, Math Tricks, Pascal's Triangle

Many of you may have seen this one before – it is very old (I first saw this in grade school, which makes this example of math tricks ancient!).

Yes, it is the infamous 2=1 “proof”.  Here I present it to you as a graphic an also as a video set to Beethoven’s 5th (yea – it was raining yesterday so I had a lot of time on my hands!).

First, here is the graphic of the proof:

And now for the video proof:

So that is the proof.  Can you spot where the error is?

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Tags: 2=1, math proof, Math Tricks