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.

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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.

Why Healthy Food For Diet?

People already know that they need healthy food for diet. The problem is, not many of them really understand the meaning of it. Most of them are more interested in the diet itself than more the healthy food. Well, we need to get it right because there will be no result for our diet without we eat the right kind of healthy food.

Before we talk about the kind of food, we need to understand the meaning of diet. Diet is a way of controlling the amount of food and the kind of food entering our body in order to get a healthy body. So there are two things that we need to keep in mind, the amount and the kind of food. Actually by eating the right kind of foods, we do not need to worry to much about the amount.

Why? Because by choosing the right kind of food that can actually burn fat and help us gain muscle, and avoiding the kind of food that only encourage our body to store fat, we do not need to worry anymore either we should eat more or less. The more we eat, the more we burn fat and gain muscle.

For an optimum health, we also need foods that contain carbohydrates, protein, vitamins, minerals, and fat. We need all those nutrients so that our body can perform an optimum metabolism. So if you are following a diet or weight loss plan that recommend you not to eat one of them, you better STOP NOW.

We can actually find these foods anywhere. Lean meats and steaks, raw fruit, raw nuts, vegetables, olive oil are sources of healthy food. The key is choose raw or less processed foods.

Top 11 Reasons to Use Social Media in Business

Over the past few years, to say social media has exploded would be an understatement. Members of social media sites have discovered it's the easiest way to find, learn about, and share information that directly influences their purchasing decisions. It is this reason alone that every business should be on multiple social media sites in an effort to reach as many people as possible.

Below are the top 11 reasons to use social media in business:

  1. Build relationships – personal and professional; connect with all types of people – past and present coworkers, current clients, prospects, vendors, competitors, and marketing professionals to build your network
  2. Share your blog posts; share interesting and useful articles, other blogs, webinars, podcasts, presentations, and poll / survey results with your fan base
  3. Gather valuable information – research similar products / services, competitors, prospects, marketing tips, blog ideas, create your own poll / survey – the amount of information available is endless
  4. Join a social network of like-minded individuals or start a new interest group where you can further brand your product / service, build relationships, and share ideas and information
  5. Stay current with the latest trends – write a blog or join a discussion on a trending topic; become aware of new social media sites that might fit the needs of your business
  6. Land a great new job – using your new connections on various social media sites, you can get the inside scoop on companies that are hiring, as well as resume and interview tips
  7. Discover a new passion in life, become a volunteer, or start a fundraising event
  8. Get found by prospects by using inbound marketing tactics – spread the word about your product / service through social media sites; to form a complete marketing plan, combine this with outbound techniques (interruptive marketing)
  9. Build business reputation and brand – constantly add value to all your posts and become a trustworthy leader in your industry; to quote Bob Burg, "All things being equal, people will do business with and refer business to those people they know, like, and trust."
  10. Build a marketing department inexpensively – for start-ups, employing inbound marketing tactics is a cheap way to start marketing your product / service; most social media sites are free to join, while others have a minimal monthly or annual fee
  11. Help increase conversion rates – use social media and inbound marketing to steer more organic traffic to your website and convert more leads into sales; the average conversion rate is 2.3%, but varies by industries according to ClickZ, Marketing News & Expert Advice, http://www.clickz.com/clickz/column/1691779/benchmarking-average-conversion-rate .

I recommend choosing a useful of sites to manage and start getting your feet wet. Not every social media site available will be right for you or your business and it would overwhelming to try them all. For example, consistently maintaining a blog, Facebook, LinkedIn, and Google+ pages, a YouTube channel, as well as a Twitter account might be a great place to start. Depending on the type of business, a Pinterest, Tumblr, or Flickr account might or may not be worthwhile, but obviously they are great certain brands.

If you need help getting started, there are countless different types of resources. I have used the An Hour a Day book series on Facebook Marketing, Twitter Marketing, and Social Media Marketing. They address each entity in a step-by-step fashion that is very user-friendly. I also find Hubspot to be extremely helpful. They are continuously publishing user guides that contain valuable information, tips, and tricks. Lastly, when creating your profile on the various social media sites that best fit your marketing plan, it's a good idea to connect with other marketers to check out what they are doing to boost business.

Do you have a tip or trick to share with newcomers to social media? Share it with us; you are more than welcome to post questions and / or comments here!