The Triangle Inequality Theorem

This is a post based on a question.

I am looking to write a proof (without breaking it into cases) for the following:

Prove that |x+y| ≤ |x| + |y| using the following assumptions:

-|x| ≤ x ≤ |x|

if c ≤ r, then c ≥ -r

if r ≥ -c, and r ≤ c, then -c ≤ r ≤ c

Before I go on, I have to apologize. I was unable to come up with a proof of my own (I kept getting stuck), so I searched the internet (this property is famously known as the "Triangle Inequality", and has applications in number theory, calculus, physics, and linear algebra) and found two different proofs that appeared side-by-side on numerous sites. I have copied and pasted these proofs into this post BUT I must warn you that I myself do not believe either of them. Though these proofs seem unanimously accepted by the people of the internet, they do not seem correct to me (I will explain why later). I am doing my best to answer your question, but just know that.

The red question marks indicate areas where the proof fails for me. If you can find an explanation for these areas, let me know.

Technical Difficulties

Blogger is not allowing comments on Pages at the moment (there are about a thousand people in the forums complaining about the glitch), which means that no one can leave a comment/question in my "Click Here to Post a Question" area. For now, ask questions as a comment to this post or any other posts. Hopefully this issue will be resolved soon.

Find a (Math) Tutor Near You

When I first created this blog, I wanted to keep it simple because I personally don't like it when I'm browsing other blogs and they take forever to load.

However, I've added another useful tool to the sidebar (in addition to the widget I added yesterday) which I hope you will use. I just discovered a revolutionary website called WyzAnt. This site gives tutors and students a place to find each other without going through the hassle and creepiness-factor of websites like cragslist. All of the tutors must complete an extensive application process and YOU, the student, can have THEM, the tutor(s), background-checked. How neat is that?

WyzAnt not only for math, either. There are tutors there that specialize in everything from SAT-prep to violin lessons to Mandarin Chinese.

If you think that you could benefit from meeting regularly face-to-face with a tutor in your area, type your zip code (or a nearby zip code, if you're internet-scared like I am) into the widget I added to the sidebar. Otherwise, just browse on over to the site anyway and check it out.

Sorry, international friends, WyzAnt is restricted to only the U.S. at the moment (I think, though I didn't see anything that explicitly said that).

Basic Integration By Parts

This is a post based on a question.

If it isn't a problem, can I ask you how to integrate by parts?

The properties of integrals let us split up addition, but not multiplication.

So, when we have two functions MULTIPLIED together, we must use the technique of Integration By Parts.

(Note: In this post I will show the formula and explain how to use it, but I will not show why it works. If anyone would like to see where the formula comes from, please let me know in a comment.)

Notice that there is an integral in the answer! This means that to evaluate the problem completely you must also do that integral at the end. This also means that there is potential for the process repeating over and over (you get the answer, but then you have to use Integration By Parts again), but I titled this post Basic Integration By Parts for a reason; repetition would be Advanced Integration By Parts.

Anyway, this formula is critical. After extensive practice, you will know it by heart just because you will use it so much. Before I was entirely familiar with it, however, I had a chant that helped me remember it. All I said was "u v minus v du," but it was enough for me remember it forever.

Now, let's do an example.

Mathway Widget

Scroll all the way to the bottom of the page. See that mathy-looking box? That's the Mathway Widget. I'm not entirely sure how it works yet, but I'll update this post with some examples of it once I find time to play around. For now, it's there for you to play around with. Enjoy!

Weekend Pay Problem

This is a post based on a question.

javis works in a garage for 8 hours. if he works on saturday he is paid time and a quater, if he works on sunday he is paid time and a half. last week-end javis for four hours on saturday and 4 hours on sunday. how much is he paid last week-end

NOTE- time and a half means for every hour you work you get paid for one and a half hours.

Let's look at a chart that will help with this problem:

We can also do the problem using a more algebraic approach:

 



Variable vs. Constant: Using Letters to Represent Numbers

If you are far along enough in your math education that you have encountered letters, then this post is for you.

There are cases in math where we want to work with a more abstract form than a number. See, to us, a number represents only one single quantity. We don't look at the number 4 and say "well that's the number of toes I have." (Assuming that we are human beings that have all of our appendages in the proper places…otherwise, we might say that.) A number to us is a concrete, undeniable amount.

But what if we want to talk about many different numbers? There are times in math when we want to think about a single number without knowing exactly what that number is. Or maybe we need to be able to think abstractly about a whole bunch of numbers at once. That's where letters come in.

We don't have to use letters. We could draw little pictures of elephants if we wanted to, and have the pink ones represent even numbers while the green ones represent odd numbers. The reason we use letters is simply convenience: they are right there, quickly accessible in our minds and we already know how to draw them.

There are two different uses for these letters, which I mentioned above. Now I will go into detail.

Geometry of a Line

This is a post based on a question.

In the diagram, points, V, W, X, Y, and Z are collinear, VZ= 52, XZ= 20, and WX= XY = YZ. Find the indicated length: WX

Negative Numbers

I see issues with negative numbers all the time, even from calculus students.
I think I can pinpoint three key things that really helped me understand negative numbers. Hopefully they will help you, too.

 

Here is a free online game to practice negative numbers:
http://www.mathsisfun.com/numbers/casey-runner.html

My First Video

This is a video that expands the lessons Factoring and FOIL. I work through an example, then check my work.

I would love some feedback. Is this a good way to teach lessons? Does it help you understand any better? Does it confuse you? Is it an awful video in general?

Factoring

Factoring is the exact opposite of FOIL-ing (see last post). Instead of multiplying out two binomials to create a trinomial, we will be breaking a trinomial into two binomials.

This is probably the most important skill you will learn in algebra (not counting all of the skills that are used for this one). So when your teacher says it's time to begin the chapter on factoring, sit up straight in your seat and pay attention.
While there are something like 8 different factoring methods, I am only going to present one. This is for a reason: this method works for every single factorable trinomial (yes, there are many methods that only work in specifc cases but I will not be showing those to you unless you ask nicely).

Here is another example, which introduces the case of a trinomial with negatives:

FOIL

This is an algebra lesson.

To multiply two binomials (polynomials with two terms), you must use a distribution method called FOIL, which stands for First Outer Inner Last. Multiply the two first terms, then add the product of the two outer terms, then add the product of the two inner terms, then add the product of the two last terms.

To give this a more visual approach, here is what it looks like using letters instead of numbers:

Memorization

This is a post based on a question.

I'm going to sixth grade and I still haven't memorized my multiplication tables. Help me?

This is more of a series of helpful hints than an actual lesson.
While the vast majority (and most exciting parts) of math involves critical thinking, problem solving, and multi-step processes, there are certain parts that just require some simple memorization. For example, once you have the concept of what multiplication actually is, it is immensely helpful (perhaps even necessary) to simply memorize multiplication tables for all combinations of numbers 0-12. Likewise, in Calculus it can greatly improve your test-taking speed by just memorizing certain basic derivatives and antiderivatives.

Before I go any further, I must make an extremely important point: the goal of memorization is not to substitute for understanding, but to increase mental speed. That is, you should know what I mean when I ask what 4x6 is, and not just know the answer. You must first deeply understand the concepts before memorizing the numbers.

I have a few tried-and-true methods for memorizing simple facts, such as times tables and derivatives. It is best to try all of them and see which works best for you. Everyone's mind works differently and there are numerous learning styles to take into account when selecting a method.

Please Respond to My Poll

I have added a poll to the sidebar; I am very interested in what my viewers think.

I am considering opening a YouTube account and making lessons as videos instead of typing them. It would be faster, and I have a nice new HD camera and some basic editing software to put together some nice videos. I would write on notebook paper or a whiteboard and narrate as I teach, similar to many of the math lessons already on YouTube (the difference is that I would be responding directly to YOUR questions).

If you are interested in seeing some of these videos, opposed to the idea, or even neutral, please let me know by responding to my poll. You may answer more extensively in the comments section below this post.

The Math Geek Is Back

I am back from vacation and ready for questions. So put me to work. What are you having trouble with?

Vacation

The Math Geek will be on vacation until next Saturday, 8/6/11. Anyone may ask questions during this time, but they will not be answered until my return. I apologize for any inconvenience.

Wolfram Alpha

This website is an invaluable resource for calculus (or any math or any other class, really) students: http://wolframalpha.com/

Find answers to questions in an immense variety of areas: Mathematics, Statistics, Physics, Chemistry, Engineering, Finance, Weather, Music, Nutrition; it can do almost anything!
Take a look at examples of many things it can do: http://www.wolframalpha.com/examples/

The Power Rule

This is a calculus lesson based on a question:

 Would you be able to explain how to derive a square root? 

The power rule is used to find the derivative of (or "differentiate" or "find the instantaneous rate of change of") an algebraic expression of the form x to a numerical power, or xⁿ.

The derivative of a function ƒ(x) = xⁿ is ƒ'(x) = n * x^n-1. If it is easier to remember think of it as dragging the exponent down (to multiply the x by) and then subtracting one in the exponent.

Here are some examples to clarify:

Fractions

In this lesson I will show you how to add, subtract, multiply, and divide fractions. Then I will show you how to simplify your answers.

Addition/Subtraction

These are done almost the same way. First, you find a common denominator (you need the same number on the bottom of both fractions). Then, you add or subtract across the top of the fractions and keep the bottom the same.

The hardest part of this is the common denominator. If you are good with numbers, you are trying to find the smallest number that both denominators can multiply to. If you are not good with numbers (don't know the times tables or need to use a calculator often), simply multiply the two bottom numbers. REMEMBER: whatever you do to the bottom you must also do to the top, and whatever you do to the top you must also do to the bottom.

I'll provide a few examples.

Order of Operations

I'm starting off with the order of operations because I feel that this is the beginning of math. Yes, there are the actual operations of addition, subtraction, multiplication, division, and exponents. Those are what I consider pre-math.

The purpose of the order of operations is to simplify an expression (hint, if you ever come across a question on a homework or test that says "simplify" or "simplify the expression," this is what it is talking about). In other words, we want to make something ugly into something pretty.

In order to do that, we use the order of operations, which is:

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction