Squares and Square Roots
Remember roots? Not the tree things that you trip over all the time, the excellent television miniseries, or the Canadian apparel company. Roots are the solutions to polynomials that are set equal to zero.
For example, in the equation
(x – 4)(x + 11) = 0,
the roots are 4 and -11, our two favorite numbers. Square roots, meanwhile, are solutions to an equation of the form
x2 – c = 0.
There's a square (x2) hiding out in there, so that makes sense. More specifically, the square root is the positive solution to the polynomial equation x2 = c.
Remember that a solution looks like .
We use the notation of the square root to show both the positive and negative solutions to the equation. That's right, count 'em: two. The concept seems a little bizarre, but it's certainly no weirder than Catdog, so let's move on.
We show both answers by defining the square roots as the two possible values of . In other words, . Instead of turning to fractional powers, which we must only use for good and never for evil, we will only be using the radical sign √ in this section. We think they're fun, in a checkmark-that-went-haywire kind of way.
How do square roots appear? Who invited them in the first place? If we take any positive number s and draw a square with side length s, the area of that square is s × s = s2. This should be graffitied on the inside of your brain by now, so take a moment to memorize it if you haven't already.
We call s2 the square of s or s squared, and we say that s is a square root of s2. Since (-s)(-s) = s2, -s is also a square root of s2. A square root of a number is anything that gives that number when multiplied by itself. By now, you've heard us say the word "square" so many times that it's probably dancing around inside your head. Sorry for all the square-dancing. Wait, we're never sorry about square-dancing. Swing your partner 'round and 'round!
The square below has side length 2 and area 4. The square of 2 is 4, and 2 is a square root of 4.
Since (-2)(-2) = 4, -2 is also a square root of 4. We can't draw a square with a side length of -2, though, or at least without the assistance of one of those newfangled "negative space" pencils.
A perfect square is any number that is the square of an integer. We hesitate to call them that, because it will give them such big heads. Like their egos aren't already off the charts. It's a good idea to know the first fifteen or twenty perfect squares by heart. If you forget them you can always figure them out by multiplication, but you never know when it might be useful to recognize that 169 is the square of 13. That might the bit of knowledge that saves your life someday.
We can't imagine a scenario where that would be the case, but better safe than sorry.
For a positive number t that isn't a perfect square, we could still have a square with area t. We just don't want to keep rubbing in the fact that it's imperfect. However, the side length of the square wouldn't be an integer, and it might not even be a rational number. Now it's imperfect and irrational. Sounds like our favorite PE teacher from junior high.
Any positive number has one positive square root and one negative square root. The positive square root is called the principal square root, since that's the one we usually want for real-life problems. Sorry, negatives, it comes with the territory of being invisible. In fact, the negative square root is called the negative square root. It's so inapplicable in real situations that it doesn't even warrant a fancy new name.
Since mathematicians love abbreviating things, instead of writing
"the principal square root of 9 is 3''
For one thing, it's shorter. It also includes a symbol, which mathematicians love. (Other things that mathematicians love: long walks on the beach in the approximate shape of the golden ratio and candlelit dinners spent discussing the Millennium Prize Problems. Who said romance was dead?)
The symbol is called a radical sign and is an abbreviation for "the principal square root of." Radical, right? Gnarly, one might even say. If you're not sure what those words mean, ask your parents.
For the negative square root, we write , as in .
The thing under the radical sign (in the case, the 9) is called the radicand. Don't say radican't; say radicand!
In the statement , the radicand is 16.
We can square any number we like, rational or irrational, although it's easier to write down examples using rational numbers. However, when have you ever known us to do things the easy way? (Uh, all the time. Then we tell you how.)
The square of an integer is an integer since the product of two integers is always an integer, and the square of a rational is rational since the product of two rational numbers is always a rational number. The square of an irrational number may be irrational, such as π2, or rational, such as . Are you starting to see now how they got their name? They're completely unpredictable and never follow the normal rules. Sounds irrational to us.
We can take the square root of any non-negative number we like, rational or irrational.
The square root of an integer may be an integer, such as , or not, such as . Similarly, the square root of a rational number may or may not be rational. The square root of an irrational number, however, will definitely be irrational. Once again, irrational numbers refuse to subscribe to the pack mentality. What rebels.
When finding the square root of a number, ask yourself "what can I multiply by itself to find this number?" Then ask yourself if you remembered to brush your teeth this morning, because from where we're standing, we're not so sure. Oral hygiene is very important.
A Few Square Roots
- The square root of is , since .
- . The square root of 25 is 5, and 2500 has two extra zeros, so each factor (square root) needs one of the extra zeros. Geez, if we had known there were extra zeros being handed out, we would have gotten here earlier.
- . We had a feeling the number 2 would factor into the square root of 0.04 somewhere. Since the square root needs to have one decimal place, squaring it gives us two decimal places.
It can be helpful to factor when finding square roots of numbers that aren't so obvious. Always keep that factoring trick handy, like a Swiss Army knife in your back pocket. (Except at school or when going through airports, of course.) Be careful when sitting down, though. There are few things less comfortable to sit on than a corkscrew.
For numbers that aren't perfect squares, we need to use a calculator or a table of square roots. Most people use calculators instead of tables these days, though since tables are so much harder to carry around in your pocket. Especially with the legs still screwed in.
Be careful: Just because we're using a calculator doesn't mean we're allowed to stop thinking. That's what weekends are for. Seriously, though, it's good to keep a ballpark estimate in mind so we can tell right off the bat if our answer is horribly, horribly wrong. You know things are getting real when we bust out the baseball metaphors.
Even More Square Roots
- needs to be somewhere between 2 and 3, since 7 is between the squares of those two numbers (4 and 9). A calculator will tell you that = 2.64575131... Guess this calculator character knows what it's doing.
- should be a little bit above 5, since 26 is fairly close to the square of 5 (25). The calculator says , which is indeed a little above 5. Maybe our calculator should take the test, except that brings us uncomfortably close to Skynet territory.
- Since 81 < 92 < 100 or, in other words, (9)2 < 92 < (10)2, we know that .