nth Roots

This brief post is going to explain to you what nth roots are.  Following posts will show you how to work with them in your equations.

Going back to elementary algebra lessons, you were taught the concepts of square roots.  For example, the square root of 25 is 5.  This notion was explained, but likely never generalized much past saying "what times what gives you your number?"  However, like many things in the mathematics world, there is far more to roots than this.

Enter the concepts of "nth roots".

Everyone knows the symbol for square root, and that it means "what times what gives the number underneath this symbol".  Some of you may have seen a tiny 3 written on the top left just above the root sign.  What this means is a touch more complicated: "what times what times what gives the number underneath this symbol?"  It is called the cube root.

It's always simpler to go forwards before going backwards, so let me show you one:

2 x 2 x 2 = 8

You can say 2³ (2 to the power of 3) equals 8.  Straight-forward, right?  Now, go backwards.  Find the cube root of 8.  Well, in this case, "what times what times what" is "2 x 2 x 2", so the cube root of 8 is 2.

So then, we now understand square roots, and cube roots.  Now, naturally, we can expand on those.  In general terms, nth roots can be defined as this:

a^n = b     (a to the power of n equals b)

and therefore a is the nth root of b,

where n is a natural number, and a and b are real numbers.

Some examples should hopefully clear up the complexity of what I just wrote.

3 and -3 are square roots of 9, because 3² = 9

4 and -4 are fourth roots of 64, because 4⁴=64

2 and -2 are sixth roots of 64, because 2⁶=64

Further, what these examples should demonstrate to you is that all EVEN roots of postive numbers occur in pairs, with there being one positive and one negative number for each.  Therefore, to distinguish between the two, there has been accepted a common notation:

The positive, or principal, nth root is designated as the root sign with the nth root number over it:   ⁿ√
E.g. ⁴√16 = 2, and NOT -2
To indicate -2, you write the initial expression as -ⁿ√
E.g. -⁴√16 = -2

On the other hand, all ODD roots only occur singly.  Such as 2 is the fifth root of 32, whereas -2 is the fifth root of -32.  (Write it out to see the differences.  -2 x -2 x -2......)

A little bit more nomenclature, just so you always know what is going on now:
What you have always knows as the "square root sign" is technically called the radical sign.
The number underneath the radical sign is called the radicand, and the number n used to indicate the root is called the index.

To summarize all of this, we can revise our original definition to this:

If n is a natural number, and a and b are non-negative real numbers, then

ⁿ√b = a, if and only if b=aⁿ

^{The number a is the prinicpal nth root of b}

Also:

If a and b are negative and n is an odd natural number, then

ⁿ√b = a, if and only if b=aⁿ

^{So there you have it.  You now know what is meant by nth roots.  In my next post, I will go over some of the properties of nth roots that will make your homework questions a lot easier.}

How to Show Inequalities and Absolute Values on a Number Line

This will be a short post, but I wanted to quickly go over something I touched on in my previous post about properties of absolute value, which is how to graphically show absolute value and inequalities on a number line. I mentioned that there are a couple of acceptable ways to do this, so I will try to explain them here.

I want to first go over the concept of inequalities briefly. It is all very intuitive and common sense, but you will see why I want to point this out now before we put inequalities on to a number line.

If I give you the example x < 2, we know exactly what that means. In fact, the words you say when you read that mathematical expression pretty much says it all: x is less than 2. Great. Similarly, the example x < 2 is also as obvious: x is less than or equal to 2. That was the easy part, but the key is to always pay attention to what the sign is saying. Does it say "less than" or "less than or equal to" something. Let me show you why this distinction is important when it comes to graphing inequalities on a number line.

Let's take that first examples again, x < 2. To put this on a number line, there are 3 important things here: the point in question (2), the direction on the number line that makes the expression true (less than), and does the inequality contain the point, or exclude it (less than, or less than or equal to).

If the sign being used is either < or >, without the equals part, you can represent this on a number line as an empty circle at that point. Alternatively, you could show it with a round bracket. This notation implies that you are talking about points that get infinitely close to your number (2, in our example... such as 1.9999999999), but not exactly 2. In addition to that, for all the values that make the expression true you put a line over the number line, or draw a thicker line on top of the number line. If the true values go all the way to infinity, you can put an arrowhead on the line. So for our example of x < 2, we would represent it like this:

or like this:

Now, when we are talking about an expression that has an equals part to it, that is simply represented by a filled circle on the number line, or a square bracket. Note that both kinds of brackets open towards the direction that the expression is true. So, for our second example of x < 2, we could show this on a number line like this:

or like this:

I should probably apologize for the shoddy artistic skills.  ;)

By looking at these number lines, and the ways that we have represented the inequalities, it is very easy to see that we are talking about x < 2 in the first pair, and x < 2 in the second pair.

So, that is most of what I wanted to explain about inequalities and number lines. Now, to apply these ideas to solving an absolute value question, it should be straight forward. And when in doubt, just substitute in numbers for x to see if the inequality is true or false!

If we are talking about |x| < 2, we saw in my original post on absolute values that this simply means that x is 2 units away from zero. It doesn't specify only positive numbers, or negative numbers, or any type of restrictions on the question. Therefore, 2 units away from zero in both directions encompasses the interval of -2 to 2. So, to show this on a number line is simply two open circles (on -2 and 2), and a thick line connection to two points to represent that all number within that interval make that inequality true. Try it and see for yourself! 1.999934958... true. -1.21111... true. 0.00000000003... true. The absolute value of these values for x are all less than 2.

I don't think I need to give anymore examples to explain this any further. It's pretty simple stuff, right? If you do need a bit more explanation or you would like additional examples, don't hesitate to drop me a line and I'll see what I can do!

Properties of Absolute Value

In my previous post, I went over the definition of absolute value and gave you a bunch of examples to show you how to work with absolute values in your math homework. In this post, I'm going to add to that by describing some more properties and rules of absolute values. Then I'm going to give you a couple of practice questions like those you would likely find in your math textbook or on a math test. By the end of it, you will be a pro at working with absolute values!

These properties are all extensions of what I taught in my last post, so if you need a refresher, head back to that one and give it a review. There's not many, so it shouldn't be too difficult for you to think about each point for a minute and see how it relates to what you already know.  Here we go:

1)  |x| > 0, where x is a real number.
This one should be straight-forward.  We talked about how absolute value is always positive, as it represents a distance on the number line between a number and zero. Therefore, we can write is this way. The absolute value of x is always greater than, OR EQUAL, to zero. (Yes, you can technically say that the absolute value of zero is zero. The distance on the number line from zero to zero is zero units. Weird, I know...)

2)  x < |x| and -x < |x|, where x is a real number.
This is, again, another way of showing that the absolute value of a number (positive or negative) will always be positive. If x is negative, the absolute value is positive, which is greater. If x is positive, the absolute value is positive, and so they're equal. Substitute in some numbers for x and you can prove for yourself that those expressions are always true.

3)  |x|^2 = x^2, where x is a real number.
The absolute value of x squared is equal to x squared. This is because whenever you square a real number, your result is positive. 2 times 2 is positive 4, but so is -2 times -2... positive 4! Again, plug in some numbers to prove it to yourself.

There are also a couple of arithmetic rules for when using absolute values. These are important to learn, and important to not confuse or get wrong! I'm not going to go in depth to explain these here, but you will learn through examples how and where to use them.

4)  |ab| = |a| |b|, and |a/b| = |a| / |b| (where b is not zero) for all real numbers.
5)  |a+b| < |a|+|b|

For this last one, refer to a number line:

6)  For two points on a number line, a and b, the distance between them is |a-b| = |b-a|.
This one is intuitively easy to understand.  Basically, it says the distance between a and b is the same as the distance between b and a. It almost goes without saying!

Also, I should mention that I will put up a post soon about representing absolute values and inequalities graphically on a number line. There are a few different ways to actually show these visually.

OK, I said I would give some practice questions, so here I will put a few different kinds of questions that all deal with absolute values.

1. Evaluate the expression: |-5-4| + |-3|.
This is |-9| + |-3|, which is 9 + 3, which is 12.

2. Rewrite the statement using absolute value notation: The distance between x and 2 is 10.
When you're talking about distance, you're talking about the difference of two points. Therefore, we can write this as |x-2| = 10.

3. Rewrite the statement using absolute value notation: The distance between x and 7 is less than 20.
Same thing here, only using a less than sign. |x-7| < 20.

Those are a couple of typical examples. Let me know if you have any questions in particular, and I'll try to address them as soon as possible!

Also, please remember to hit the Google +1 button below if you liked my post! It will help others to find my site easier! Thanks!

Absolute Value

"Absolute value" sounds like a complicated mathematical concept, but really, it is quite simple. The concept is straight-forward, and most arithmetic that uses is can be quite easy as well. It can be thought of as a tool for measuring distances between two number. "Absolute value" simply refers to how far away a value is from zero. It is easiest to understand this concept by using a number line:

If we consider the value of 3, we can easily see that it is 3 units away from zero. Simple, right?

Similarly, if we consider the value of -4, we can also easily see that it is 4 units away from zero.  It is not 'negative 4' units away. It is just 4 units away.

It doesn't matter if you're on the positive or negative side of zero, whatever value that you are considering is a positive number of units away from zero.

In mathematical notation, absolute value is denoted as two straight lines that surround the value, like this: |x|. So, for our examples above, we can state mathematically that:

|3| = 3

|-5| = 5

These expressions say the same thing as our words did above.  3 is 3 units away from zero, and -5 is 5 units away from zero.

Now, if we want to go a step further and actually perform some arithmetic with the concept of absolute value, this is quite simple to do as well. Just as you have likely learned in your Order of Operations lessons (solve expressions in this order: brackets, exponents, division/multiplication, addition/subtraction), when working with absolute value expression, you solve for the expression within the absolute value sign (similarly to solving inside the brackets first).

So, if we have the expression |2+5|, we solve inside first to get |7|, and then we just learned that |7|=7.

OK, let's try one a bit tougher. |4-9|. Solving inside first gives |-5|, for which we already saw |-5|=5.

What about |-8-12|? Solve inside to get |-20|, and then |-20|=20.

Now, let's throw in a twist.  Just like when working with brackets, where you can have brackets inside brackets and you solve them from the inside out, you can have the same thing here.

Try ||-3|+|7||.

Solving the inner values first gives: |3+7|, which gives |10|=10.

Similarly, try ||-5|-||2-4|-8||. (This is tough to type on a blog, but you would likely find in your math book that the matched sets of bars to be the same size, and inner sets would get smaller.)
||-5|-||2-4|-8||
=||-5|-||-2|-8||
= ||-5|-|2-8||
= |5-|-6||
= |5-6|
= |-1|
= 1

That's a tougher one, so if you can follow through that, you are well on your way.

Now that you understand the concept of absolute value, I will put the algebraic definition of it here:

|x| = {  x   when x > 0

          -x   when x < 0

Think about that definition, and you will see that it says exactly the things that we've been practicing. It just looks a lot fancier and more complicated. But, if you followed through all those examples and they made sense to you, then you are now and expert at working with absolute value!

In my next post, I will teach you a few more properties about absolute value, and give you more examples to think about.

The Batman Equation

I can't verify that this is completely legit, but it certainly is very cool!  HardOCP has posted on their website a screen shot of an extremely elaborate mathematical formula, and its corresponding plot on a graph.  And, lo and behold, it's a Batman graph!

I wonder how long it will be before we have the Superman shield plotted out as well!  Though, I would imagine that as complicated as this Batman equation appears, it is probably just a piecemeal equation composed of lines and curves over specific domains.  The Superman equation would likely be far more complicated, as its shape is far more intricate.  If anyone has any free time and is able to put one together, send it my way and I'll be the first to post it for you!  :)

If ever there was a reason to learn how to make charts, this Batman graph is a pretty good one!  Though, judging by the complexity of the Batman equation (in the top part of the pic below), you would likely want to have a good graphing solver program.  Some of the most popular graphing calculators are made by Texas Instruments.  These TI calculators, such as TI 83, allow you to plug in your equations and have a big screen that will then display what your curve looks like.  These kinds of math graphing calculators would be invaluable if you need to work with complex mathematical formulas and their graphs.

*Edit: I'm surprised at the amount of traffic this blog posting generates!  When I first posted, I had no idea that so many people would end up searching for a Batman equation, and my site would rank so highly for it!  Check out this website as well for really cool visuals created by mathematical equations.  (Batman makes an appearance there as well.)  If nothing else, I hope that this demonstrates how useful math can be to do really cool things!  As I mentioned above, the equation shown above the Batman graph seems to be a piecemeal function, which utilizes a different equation over different domains of the function.  As you can see, piecemeal functions can produce some very interesting designs when graphed out.  Similarly, polar equations can produce very elegant designs as well!  A polar graph is based on an entirely different coordinate system, though it's just another example of how what seems to be just complicated strings of numbers and operations can come together to make something really interesting!  This general math concept should be taught to students early on, to hopefully catch their interest in pursuing further studies in mathematics!  After all, someone needs to develop the Superman equation!