Math Carnival Submissions

The Math and Multimedia Carnival is coming up again soon, and the next one will be posted here on Math Concepts Explained!  It will be the 21st edition of the carnival, so hopefully it will have a great turnout by the carnival contributors.  The number 21 obviously has a lot of significance in all sorts of ways, so hopefully the 21st Math and Multimedia Carnival will have a wide variety of interesting articles to mirror the many different meanings for this special number.  I'm looking forward to hosting it and can't wait to see what great articles are going to be submitted for it!  Before that, I just want to briefly explain the math carnival submission process.

For those of my readers who are interested in submitting an article to the 21st Math and Multimedia Carnival that will be on my website, you can either submit your contribution to the carnival submission form site, or email it directly to me at mathconcepts101[at]gmail[dot]com.  Feel free to comment or ask me questions at this email address as well.

The carnival will go live on Monday, March 19, 2012.  I would appreciate if any submissions that you want considered for the carnival could please be submitted by Friday, March 16 (either through the blog carnival submission website linked above or emailed to me).

Personally, this is the first time I've ever hosted the Math and Multimedia Carnival.  Blog carnivals are new to me, so hopefully I'll be able to do a good job at it!  On that same note, I have also recently submitted my first article to a blog carnival, which can be found on the latest (20th) edition of this same math carnival.  

For those who are like me and are new to blog carnivals, Guillermo (who maintains this carnival, and is also the owner of the website Math and Multimedia) has provided several good resources.  Some of these are:

• "What is a Blog Carnival" is a very good overview that describes what they are and what can be expected.
• Index of all the previous carnivals.

So, here's hoping for another successful Math and Multimedia Carnival!  Remember to get your submissions in on time, and remember to come back to Math Concepts Explained on March 19 to see all the new articles!

Square Roots - Part II (The Irrational Number)

Picking up where I left off in my previous post (Square Roots - Part I), I'm going to briefly explain irrational numbers. (Also be sure to check out Part III - Factoring Square Roots!)

A rational number is one that can be expressed as a quotient of two integers (ie. as a fraction). So, all fractions are rational. And all whole number integers are rational (since they can be expressed as "something over one").

Conversely, irrational numbers can NOT be expressed as a fraction. The number pi is commonly given as an example of an irrational number, since it cannot be written as a fraction, and the numbers after the decimal point keep going and going. Going back to my last post, the square root of 24 is also irrational. If you punch it into a calculator, you will see that it is 4.8989794855663....... On this note, you can say that the square roots of non-perfect squares are irrational.

Now, to find the square root of a non-perfect square by hand, you just do the same trial and error method we learned when I explained square roots in my last post, only this time, as you narrow your guesses down, you can use numbers with more and more decimal points. Observe:

Find the square root of 24:
4 x 4 gives 16....
5 x 5 gives 25...
4.5 x 4.5 gives 20.25
4.9 x 4.9 gives 24.01.... needs to be a bit smaller
4.89 x 4.89 gives 23.912.... needs to be a bit bigger, but less than 4.9
4.898 x 4.898 gives 23.9904.... bit bigger still... but still less than 4.9
4.8989 x 4.8989 gives 23.999221.... even a bit bigger, but still less than 4.9

As you can see, this can go on and on, until you have as many decimal places as you want.

Alternately, and more commonly, you would leave your answer as an irrational number, rather than recording decimal places (since technically, writing to so many decimal places, unless you write FOREVER, can be expressed as a fraction.... tenths, hundredths, thousandths, ten thousandths, etc...)

However, you don't want to leave your irrational number in a form that hasn't been reduced yet, do you? Check back to my next post to find out how to factor square roots and express them in simplified radical form.

Square Roots - Part III (Factoring Square Roots)

Where I left off in my previous post about irrational numbers, we were trying to solve for the square root of 24 by trial and error.  Reducing an irrational number can be a tedious job if you do it this way!  Luckily, unless specified otherwise, you are allowed to leave your answer in the "simplified radical form," which you can get by factoring square roots.

"Simplified radical form" is exactly what it sounds like: you simplify your expression and leave it expressed as some radical.  But, you have a little bit of work to do to reduce it.

Factoring square roots is quite simple.  It has to do with factoring perfect squares.  But first, you have to determine the factors of the number under the radical sign, and then if any of those factors are perfect squares, you can pull it (the square root of the perfect square factor) out from underneath the radical sign and put it in front to multiply by it.  That sounds awfully wordy and probably isn't the most concise definition, but I think an example will go a long way to helping you understand factoring square roots, and then leaving them in simplified radical form.

Let's continue with the example from my last post, looking for the square root of 24.  In this case, let's just reduce it to its simplified radical form, and not bother wasting time trying to find the exact decimal answer.

So, the first step is to ask yourself what the factors of 24 are.  (I'm going to ignore the √ for a minute)  For this, you can determine this quite easily by trial and error.  You can find that the factors of 24 are 1,2,3,4,6,8,12,24.  From these now, you want to see if any are perfect squares.  Again, you should be able to easily say that 4 is the perfect square.  So, 24 can be expressed as 4 x 6. That's what we want.  Now, let's look at what we have done so far:

√24 = √(4x6)

Now, as I said before, we can take the perfect square that is under the radical sign, and bring it outside the sign.  This is because of the properties of square roots (property #2), which in this case allows us to write:

√(4x6) = √4 x √6

With that property in mind, it should be a bit easier to see why we are interested in the perfect squares, because now, by factoring perfect squares, we can just rewrite the square root of 4 to 2!  So, we can finally write our expression in simplified radical form as:

√24 = 2√6

I hope this demonstration has explained to you the basics of factoring square roots, and leaving you answers in the simplified radical form.  Let me know in the comments if you'd like another example, and I'll do one for you!  Remember to +1 me if this helped you!  :)

A Valentine Graph Equation

This is a fun little math trick that will give you a romantic Valentine graph!  (OK, maybe romantic isn't the right word...) Please remember to click the Like button if you like this!

Several months ago, Google added the ability to graph equations directly from their search line.  All you have to do is type in the equation instead of performing a search query, and Google will return to you a graph of your equation!  So, instead of having to pay money for a fancy math graphing calculator (such as the TI83), or doing it the long and slow way of graphing out by hand on graph paper, now you have a very easy to use graphing calculator for free whenever you are on the Internet!

Just in time for Valentine's Day, @Huckberry on Twitter has noted that if you type in the following equation into the Google search line, this is actually a Valentine graph equation and Google will return a result that shows you a nicely graphed out heart!  Try searching for this Valentine graph equation on Google and see what happens!

sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), -sqrt(6-x^2) from -4.5 to 4.5

This graph is composed only of standard trigonometry functions (ie. cosine), square roots, absolute values, exponents... all familiar concepts introduced in your math classes fairly early.  However, you've probably never seen them put together quite like this!  I can point out a few things about this equation though.  The absolute value is involved in creating the mirror image, and the 300x part of the cosine function actually is the part that determines the frequency of the plotted line.  If you reduce the 300 to something smaller (try 20), you can actually see that the interior of the heart (the "solid" colour) is actually just a line bouncing up and down along the domain of the equation.  The 300 makes the frequency so tight that it appears solid.  (Thanks to Samantha Murphy for pointing this out in her post!)

A Valentine graph is just one of many impressive graphs that can be created by plotting mathematical expressions.  As many of my readers will undoubtedly remember, a few months ago I posted an image of the Batman equation and Batman graph.  A lot of people have since searched for that and have really appreciated what a mathematical equation can actually represent.  Hopefully this Valentine graph equation also helps to demonstrate that math isn't just all about problem sets and tests... math can be pretty fun and cool too!

I'm going to send an email to my wife with a link to that search result page.  How's that for romantic?  :)

Announcing... the Mathematics and Multimedia Carnival 21!

I am pleased to announce that the 21st edition of the Mathematics and Multimedia Carnival will be hosted here on Math Concepts Explained in March!

For those who are new to blog carnivals (like myself!), Guillermo has put together a nice summary on his website that explains what is a blog carnival.  Quite simply, it is just a collection of related blogs or websites that are submitted by readers and then have been summarized and listed into a single blog entry by the carnival host.  A carnival may be compared to a magazine, in that various writers submit their articles to be published, and then the magazine editor reviews all the entries and puts them together into a final, polished magazine.

As I mentioned, I have never hosted a carnival, nor have I participated in one, so I admit to being very unfamiliar with the process surrounding this.  However, I am looking forward to it as an opportunity to make some new friends and learn some new things.  So, please bear with me as I figure this out as I go!

With that, look for an official call for entries in the coming days.  I will explain the submission process at that time.  :)

In the meantime, I am happy to direct you to Guillermo's website GeoGebra Applet Central, where he is hosting the 20th edition of the Mathematics and Multimedia Carnival later in February.