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.

Instant High Paying Jobs For Teens

So you have a teen on the hunt for a half decent job but what are their options these days. Everyone is telling them to baby sit or get a paper delivery job but both are time consuming and pay third world country rates when it comes to money in their pocket. It is even harder trying to land a job position that has been advertised somewhere as employers receive hundreds and hundreds of applications when advertising in the current job market.

A recent study shown that it is now taking the average person 4 – 7 months to find new employment and they expect that period of time to lengthen substantively over the next few years.

So what can I do you say, well thanks to modern technology all this is about to change. The days of searching endlessly for employment are fast coming to an end as the internet slowly takes over and starts filling this increasing need in the teen job market. When it comes to jobs for teens it is now possible for them to land their dream job in under ten minutes no matter who they are. When they take a job online there are no lengthy, intimidating interviews to go through and their application is always accepted no matter where you live or what qualifications they may or may not have. When you compare this to the difficulty of applying for and getting a normal advertised job it is no wonder that thousands of teens are switching to paid online jobs.

Paid online work beats traditional jobs hands down every time. Online jobs these days mostly consist of completing online survey's as these are generally higher paying when it comes to the types of work you can do online and they make fantastic jobs for teens. Most of these places are free to join and once they have signed up they can then start taking surveys and get very well paid for it. Most teens that are doing these surveys are at least $ 10 for a twelve minute survey with the money getting transferred to their account on completion.

The surveys themselves are extremely easy to complete and there is an unlimited number of different types of surveys that can be taken. There are no set times that they have to work, teens can quickly and easily work at their own pace as often or as little as they like each week, it's up to them. By just taking 3 surveys per day, 5 days per week, a teen is capable of earning $ 150.00 per week. Just try finding a job anywhere else that pay's as well as this that will hire them instantly.

Which survey companies they end up joining makes a huge difference to how much money they make. Pick the wrong ones and they could end up working for $ 2 – $ 5 per hour and getting very frustrated. Out of all the "Free To Join" paid survey sites online today there are only a small list of survey companies that have the genuine higher paying surveys available on a daily basis. Joining all of these top 7 companies ensures that they have and endless supply of high paying surveys to take every day when they log in.

So the choice is up to them, with the state of the current job market are they going to keep blogging.

Cheap Holiday Deals to Stalis Provide Perfect Holidays For All

Family holidays to Stalis are about as popular as couples holidays, beach holidays, 18 – 30′s holidays, and mature holidays to Stalis. In fact, it is difficult to find a group that can’t appreciate holidays spent in Stalis. Greek island holidays seldom have as much to offer visitors as you will find in the resort of Stalis-a resort that offers a little bit of everything you could possibly want from a beach holiday and a little bit more.

First of all, you should check out various package holidays to Stalis. You might be able to learn a little more about the possibilities that holidays to Stalis present for the entire family simply by taking in the long list of package options that are available to you. More importantly, in these economic times, packages are a great way to make cheap holiday deals to Stalis and other locations on Crete.

The nightlife in Stalis is rather lively though not nearly as dominating as the nightlife and club activities are in Malia. Stalis is a more well-rounded option for holiday fun than those resorts that are designed to appeal to party crowds or even those that attract more mature holiday crowds. What you have in Stalis is a healthy mix that makes it a worthy choice for family holidays.

The beaches of Stalis provide excellent opportunities for families that enjoy beach holidays and are content to spend the entire time on the beach. There is much more to this charming resort than a few beaches though so you don’t risk running out of things to do. Popular excursions to enjoy from Stalis include mini trains to Malia, boat trips to Santorini, and the traditional villages that can be found further inland. Greek island holidays are about experiencing the best Greece has to offer. Don’t miss out on the cultural treasure chest waiting for you away from the primary resorts.

Cheap holiday deals to Stalis are a great way to enjoy the very best of Crete. Don’t miss out on package holidays and prices that are lower in price now than they have been in years. Family holidays to Stalis may be the perfect way for you and your family to enjoy Greek island holidays. Make your plans today and you could be enjoying the beautiful views and gorgeous waters of Crete before you know it.

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!