Check out this website! www.homeschoolbytes.com

Just a quick post here to mention a great site worth a visit. Homeschool Bytes has put out a recent post called Math At Play Blog Carnival #7 - Onomatopoeia. They even mention my site! It's a great resource for math, homeschooling, and other educational topics. Pay them a visit if you get a chance and explore their site. It's really informative and well done. :)

Square Roots - Part III (in progress)

Reducing an irrational number!

What is the most simplified way of writing the square root of 24?

To do this, you want to know the numbers that can multiply up to 24 (factors).

Square Roots - Part II

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 Square Roots - Part III!)

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 in the 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 more.

Square Roots - Part I

Make sure you check out my Square Roots - Part II and Part III posts!

The concept of square roots often gives students trouble. (In fact, it is the topic most searched for on my site!) However, the initial uncertainty and hesitation with this topic is quite unnecessary. Square roots sound daunting, but they're really a simple concept.

The SQUARE ROOT of a number "x" is some number "y", such that when "y"is multiplied by itself, its product is "x". Sounds confusing... but you will see that it's not. The square root sign looks like this: √25 (usually with a line over the top of the number.)

Example:
Find the square root of 25.

So, going along with the definition I gave above, let's say x = 25. So then, we want to know y... that is, what number, when it is multiplied by itself, will equal 25. In this example, most people will be able to say immediately "5 times 5 equals 25!" And they will be right. The square root of 25 is 5, because when 5 is multiplied by itself, it gives 25.

Now, I'm sure that most of you will be saying something like "That's easy! But what about when the numbers are big... or weird... like 529?" While most calculators can tell you the square root by the touch of a single button, figuring it out by hand can take a little more guesswork. To find the square root of, say, 529 (by hand), you have to just keep trying to multiply numbers by themselves to reach it. Watch:

10 x 10 = 100..... not big enough
15 x 15 = 225..... still not big enough
20 x 20 = 400..... getting closer
25 x 25 = 625..... too big! so we've narrowed it down to between 20 and 25
22 x 22 = 484
24 x 24 = 576
23 x 23 = 529 BINGO!

It takes a bit of work, but you see how it can be done. Now, having seen this, there are a few things to note.

1) While it can be said that the SQUARE ROOT of a number "x" is some number "y" that, when multiplied by itself, gives "x", the SQUARE of a number is the result you get when you multiply a number by itself. So, the square root of 25 is 5, whereas the square of 5 is 25. It is important to understand these definitions and not to mix them up. Pay attention to what the question is asking!

2) The examples and method I described use PERFECT SQUARES. A perfect square is a number whose square root is an integer (whole number). So, the square root of 25 is 5, but the square root of 24 is.... less than 5.... but more than 4 (since the square of 4 is 16). Therefore, 25 is a perfect square, but 24 is not. We would call the square root of 24 an IRRATIONAL NUMBER.

I'll have a bit more to say about squares and square roots in following posts. Stay tuned!