Fractions and Rational Numbers – What is the Difference?

Most of us go through years of school math courses and still are confused about some basic things. For example: Why can not you divide by zero? Why is .999 … equal to 1, and not a bit less?

There are loads of these kinds of questions, that would not be a cause of frustration at all, if they were taught reasonably and clearly.

Unfortunately most of these things are supposed to be covered in elementary school, and most elementary school teachers do not have a good understanding of basic math concepts. Instead they are supposedly to teach just a collection of "skills."

One of the simplest concepts that is usually left inadequately explained is the difference between fractions and rational numbers. Let's see if we can clear it up now.

A fraction is a number that expresses part of a whole as a quotient of integers (where the denominator is not zero).

A rational number is a number that can be expressed as a quotient of integers (where the denominator is not zero), or as a repeating or terminating decimal. Every fraction fits the first part of that definition. Therefore, every fraction is a rational number.

But even though every fraction is a rational number, not every rational number is a fraction.

Why? Consider this:

Every integer (all the whole numbers, including zero, and their negatives ….- 3, -2, -1, 0, 1, 2, 3 …) is a rational number , because it can be expressed as a quotient of integers, as in the case of 4 = 8/2 or 1 = 3/3 or -3 = 3 / -1 and so on. So integers such as 4 or 1 can be expressed as the quotient of integers.

But an integer is not a fraction . 4 is an integer, but it is not a fraction. 4 is not expressed as the quotient of integers. The difference here is in the wording.

A fraction is a number that expresses part of a whole. An integer does not express a part. It only expresses a whole number.

A rational number is a number that can be expressed as a quotient of integers, or as part of a whole, but fraction is a number that is (must be) expressed as a quotient of integers, or as part of a whole – there is a difference. The difference is subtle, but it is real.

There are slightly different variations of the definition of a fraction, including, "A fraction is the ratio of two whole numbers, or to put it simply, one whole number divided by another whole number."

That definition also shows that an integer is not a fraction, because an integer is not a ratio. It can be expressed as a ratio, but it is not a ratio in itself; it can be divided by another whole number, but it i s not being divided.

In a nutshell, the fractions are a subset of the rational numbers. The rational numbers contain the integers, and fragments do not.

Professors – Ways to Be Wise When Traveling (Before and During)

Since I am writing this article about how to be wise when you travel (and I am on the road at a professional conference) the ideas are quite fresh in my mind. If you travel as part of your academic responsibility, then you will find these ideas helpful.

Let people know you are on the road so they will not expect you to get back to them as quickly. You can leave this information in a message on your outgoing voice mail and, if you are going to be out for an extended period of time, set up an email responder, as well. (Note: If you are going to be out for one or two days, then my suggestion is NOT to annoy people who email you with a return email that says that you will be out "until tomorrow afternoon" or the like.)

Answer voice mail as you travel so it's not all there when you get back in town. This means you must have a voice mail system that is easily retrievable. Make it a point to return voice mail within 24 hours of when you get the message, if at all possible.

Keep up with your email as you travel. Not doing so is even more onerous than taking the extra steps to be able to access it while you're on the road. There are several possibilities:

  • you can have all your email forwarded to a web-based email system;
  • you can set up your laptop to access all your email (and either leave the original message on the server or not);
  • or you may use what I use, GoToMyPC, which lets me access my home office computer from any other computer. That way, I'm answering my emails just as if I am in front of my main computer. It's convenient because all my deleted, saved, and sent messages are right there where they belong and all attachments I might need to send to someone are easily accessible.

Create a productive environment for yourself when you are traveling. Some ways to do so include:

  • If you are traveling by car, make sure you have all the items that will let you be comfortable and productive while on the road.
  • If you are traveling by plane, then make sure you have noise-cancelling headphones and the kind of work that can easily be retrieved while sitting in an airplane seat.
  • Once you arrive at your hotel, take a few minutes and get it set up. I will often move tables around to create a L-shaped workspace. I have also been known to take 100 watt light bulbs with me when I go to Las Vegas (or stay at a W Hotel) because the lamps are so dim that it's very difficult to see and get reading and work done. Learn what works best for you-and then create that environment as much as you can.

Join loyalty programs whenever possible and then maximize your use of those particular airlines and hotels. There are small (and sometimes large) amenities that go along with being a member of a program. If at all possible, aim for the elite status which really gives you worth perquisites (including leverage) when traveling.

You're a professor. You're smart. You need to travel to conferences, to conduct your research, to learn, to share your knowledge, and to expand your horizons. So now, be wise about traveling by putting one or more of the ideas in this article into practice and move closer to peaceful productivity as a way of life.

Since I am writing this article about how to be wise when you travel (and I am on the road at a professional conference) the ideas are quite fresh in my mind. If you travel as part of your academic responsibility, then you will find these ideas helpful.

Let people know you are on the road so they will not expect you to get back to them as quickly. You can leave this information in a message on your outgoing voice mail and, if you are going to be out for an extended period of time, set up an email responder, as well. (Note: If you are going to be out for one or two days, then my suggestion is NOT to annoy people who email you with a return email that says that you will be out "until tomorrow afternoon" or the like.)

Answer voice mail as you travel so it's not all there when you get back in town. This means you must have a voice mail system that is easily retrievable. Make it a point to return voice mail within 24 hours of when you get the message, if at all possible.

Keep up with your email as you travel. Not doing so is even more onerous than taking the extra steps to be able to access it while you're on the road. There are several possibilities:

  • you can have all your email forwarded to a web-based email system;
  • you can set up your laptop to access all your email (and either leave the original message on the server or not);
  • or you may use what I use, GoToMyPC, which lets me access my home office computer from any other computer. That way, I'm answering my emails just as if I am in front of my main computer. It's convenient because all my deleted, saved, and sent messages are right there where they belong and all attachments I might need to send to someone are easily accessible.

Create a productive environment for yourself when you are traveling. Some ways to do so include:

  • If you are traveling by car, make sure you have all the items that will let you be comfortable and productive while on the road.
  • If you are traveling by plane, then make sure you have noise-cancelling headphones and the kind of work that can easily be retrieved while sitting in an airplane seat.
  • Once you arrive at your hotel, take a few minutes and get it set up. I will often move tables around to create a L-shaped workspace. I have also been known to take 100 watt light bulbs with me when I go to Las Vegas (or stay at a W Hotel) because the lamps are so dim that it's very difficult to see and get reading and work done. Learn what works best for you-and then create that environment as much as you can.

Join loyalty programs whenever possible and then maximize your use of those particular airlines and hotels. There are small (and sometimes large) amenities that go along with being a member of a program. If at all possible, aim for the elite status which really gives you worth perquisites (including leverage) when traveling.

You're a professor. You're smart. You need to travel to conferences, to conduct your research, to learn, to share your knowledge, and to expand your horizons. So now, be wise about traveling by putting one or more of the ideas in this article into practice and move closer to peaceful productivity as a way of life.

Discover Exclusive Multi Level Marketing Secrets

You will have heard of Tony Robbins – back in the 80s he popularized the concept of “modeling” as the best way to excel in all areas of life.

The general idea is that if you find any person having the same result you desire, then all you need to do is find out exactly what that person is doing to get that result, copy them, and you will have an excellent chance of producing the same result yourself – if you put in the time and effort necessary to make it happen.

When you find your way to success, your downline will be well placed to copy that success, ultimately getting wealthy themselves and contributing to your wealth.You would like people to replicate you and your success so you will enjoy even greater success independent of your time and effort.

To explain, you truly can find the “multi level marketing secrets” you are searching for by searching out what secrets top producers are using, copy them, and you can quite easily predict the same results for yourself.

Multi Level Marketing Secrets Revealed

The web makes is very easy to find out exactly what the top MLM producers are doing to get their results.By employing a tool called the Way Back Machine, you’ll be able to take a look at their websites, and see how they have developed and progressed over a period of time.

You can follow any of these top producers on their blogs and by watching their YouTube videos so that you can get an idea of how they recommend their products and services.Subscribe to their newsletters and you may also discover a few of their multilevel marketing secrets.

The trick is to look below the surface and see how they are doing their sales and marketing promotions.

How do these top guys promote their own sites, the videos they produce, and their coaching materials? What strategies are they using, are they creating articles, making webinars, and sending out videos in their e-mails?

What methods are they using to get highly targeted traffic to their site? You can see how many back links they’re getting and also check to confirm how they’re ranking in all major search engines. Is lots of the traffic coming from one precise area like YouTube? Are they using paid advertising through Google AdWords or via Facebook marketing?

What methods are they using to market the content of their internet sites and how are they deciding what kind of content to put on them?

Hidden Multi Level Marketing Secrets

Keep in mind that you want to uncover the sales and marketing processes of these top producers. What are they doing to get the results you are looking to get yourself?

So, the general concept is to actually dig deep into, not only WHAT they are saying, but HOW they are saying. Find out if they’re using a blog and, if so, how often are they posting to it.

What do they do after making a post? Do they e-mail their customers or syndicate the new content by driving back links to it? Why do they produce 2 minute videos vs. ten minute videos? Do they outsource most of their work or not?

As soon as you find out their “secrets to success” all you have to do then is take similar action.

Just as an example, during the last ninety days we used a collection of straightforward articles to push our networking opportunity. How were the results? Well, we easily generated over eight hundred qualified leads, pocketed a few thousand dollars in varied commissions, and we joined up twelve people into our primary MLM business.

How did we do it?

We follow a proven” funded proposal” multi level marketing secret we discovered by spying on top producers. Now it’s your turn.

Click below now to find out about OUR multi level marketing secrets.

3 Quick and Easy Ways to Make $1000 a Week Online (Without Spending a Dime on Traffic)

Who else wants to learn how to make real money online? Are you sick and tired of being lied to? Of being deceived, duped and disappointed? If you are like most of the people who enjoy our articles about EASY online wealth creation, the simple truth is that you are just FED UP.

But…I want you to do me a favor.

  1. Imagine being able to generate a veritable AVALANCHE of traffic to any site, service or offer…and NOT having to pay a dime to do it.
  2. Imagine having your PICK of highly qualified, fresh leads that are already interested in your offer…BEFORE they arrive at your site.
  3. Imagine being able to build a BOOMING business on the broad back of FREE traffic……without ever relying on shady tricks or techniques that will NEVER work (nor make you feel good inside even if they did)

The truth is, I’ve mastered a very specific traffic generation strategy that I call content marketing…and the very best part is just about ANYONE reading this right now can do what I’m about to lay out for you.

Why should you believe me? I’ve written over 5,000 (yes thousand..:-) pieces of unique content that have generated MILLIONS of readers, tens of thousands of subscribers in many different niches and an empire of blogs, web sites and affiliate product promotions that you can COPY…simply using the 3 simple techniques below.

Ready? Here are my top 3 ways to generate a 4 figure WEEKLY income using nothing but free traffic derived ENTIRELY from content marketing methods. And remember, you ONLY need to make about $150 a day to get to that number, which may FEEL like a lot right now, but it’s not…and you CAN do it if you follow through.

1 – Root Level Redirection: Pick ONE high EPC product (I like an EPC of 80 or above if you are using one of the main affiliate networks like CJ) and simply install a root level re-direction that forwards the visitor from YOUR offsite content directly to the vendors landing page, with your cookie.

This works very, very well for a whole hodge podge of different products…and “free trial” offers can often convert at 3 or 4% using this approach. (and usually pay out 35 dollars PLUS per sign up…which adds up hyper fast)

2 – One page review sites: I love this approach, and this WILL convert better than the root level re-direction above…PLUS you get the added benefit of search engine optimization and natural traffic from natural search as well. (rather than just from the content directories themselves…which is pretty much where you’re going to be limited in our first example)

The key? Simply expand your product selection to 3 products, rather than one! Very easy…and while the actual click through rates are going to be less….you will probably make MORE money using this approach for sure.

3 – Relationship Marketing: Build a list. Build rapport…and relationships. And then sell your OWN suite of services. Become a coach. Or consultant. Personal services is the absolute BEST way to make a fortune online that endures in value…and won’t disappear if the products you’re promoting come down. (as CAN happen in our first two examples, and can also be a pain in the rear end AND your bank account when it does)

Honestly?

You can make 3, 4 or even 500 dollars a day as a coach, often taking whatever you are MOST personally passionate about into perpetual piles of profit…AND making it happen in a hurry!