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.

What Are the Top 10 Tips for Small Business Branding?

Managing a business is not an easy job but promoting your business may be as hard as anyone else could ever imagine. And some small business owners thought that they have only limited things in order to manage and promote their business then they are definitely wrong. And some think that branding is for big businesses only then I will prove you wrong. And despite the business size, branding can be as effective as it can be. Here are the top 10 tips for small business branding.

The first one is to focus on the thing that you would like to establish, people won’t be attracted if you promote everything and anything under the sun, being an expert on one thing is what matters the most for the clients, making clients confused can never be a good idea, make them think on one important thing that stands your business.

The second one, is consistency on your performance. No one would love to patronize a company that doesn’t have consistency on its performance or on its quality. If you commit something to some clients, make sure you’ll do it, first impressions really lasts so better make sure to impress them positively and maintain it in order for you to have their loyalty.

The third one is to make an overview of your business at least on a one page document, letting them know how your business is running and what are your goals, mission and vision are very important tools in achieving for their approval, just be sure to make it brief and concise, no one will last if you write on plenty of flowering words, better to put on true informations.

The fourth one is, good treatment between you and your employees and between clients and employees as well, nothing can beat that.

The fifth one is, to remember that branding needs straight direction and longer maintenance and that there should be straight connection between the actual and the desired image.

The sixth one, join locally organized events and be one of the committees, it’s a good way to establish your image and business.

The seventh one is to blog, blogging is one of the great tools in establishing your business, making other people have an idea about it.

The eight one, you don’t have to depend everything to your logo, logo doesn’t make up the whole part of the business, so never spend too much time and effort on it, its just a part of the business.

And having your own company website is on the ninth place because obviously almost all people uses web, and the last one is not to depend too much on blogs, you must explore more over.

That’s the top 10 tips for small business branding Just don’t be afraid to give a try and to plan big for your business. And searching for branding tips can be easily done when you are eager to do it.

The iPhone 4 – New Features, New Design, Brand New All in One Gadget

The new iPhone is finally here. More technology packed then ever before, it sets new benchmarks for other mobile phones manufacturers that have been overwhelmed by the success of Apple's iPhone.

But I guess you're more interested in the features of then new phone rather than the impact it has on the business world so let me share with you the novelties it brings along.

Design:

The new iPhone is thinner, sleeker and better looking then its predecessors. The stainless steel looks good and refined giving it a more futuristic look. The metal buttons add to the overall feel of sturdiness without dramatically increasing the weight. It is heavier then the iPhone 3GS by only 0.1 ounces and it feels great in the hand.

Technology:

But the most obvious improvement is by far the new screen. The "Retina" is a 3.5 inch display which looks absolutely superb, mainly because its pixels are only 78 micrometers wide. It has four times more pixels then the previous version and it is by far the best display I have ever seen on a handheld device. The zoom is flawless and not a single bit of detail is lost in the process.

The "camera system" (as Steve Jobs called it) is comprised of a 5MP capturing device on the back and a VGA one on the front. The image can be shifted between cameras at any time during a video call. Apple calls this "Face Time" and although it has been praised by company officials it is pretty hard to believe it will revolutionize mobile communications or set a trend for other manufacturers to follow. It is undetected a clever feature but it is overshadowed by other technological marvels such as the flawless screen.

iMovie:

iMovie is an app that allows complex video editing like nothing I have seen before on a mobile phone. You can trim out a section of a movie, add a musical soundtrack to any clip, insert pictures in movies and many more. It costs $ 4.99 in the Apple applications store but it is really worth the money.

Conclusion:

The iPhone 4 is a great device with major improvements from the previous versions. Technology packed but also a looker it will unduly become a favorite among geeks and fashion savvy teens alike. A true all in one that certainly describes your attention.

Container Gardening Basics For Success

Container Gardening is becoming more and more popular as the population of cities and suburbs increases. Your "garden" is movable and so it is easier to manage pests and garden environments. It also brings the time that you need to tend to your plants because you can place them where they are easier for you to reach. This is especially good for people who are handicapped but still want a garden.

There are some things that are necessary in order to have a successful container garden.

First, you must have a container. You can find them in almost every size, shape and they can be made of many different materials. And, they can be very inexpensive, especially when you create them from "found" objects. You must make sure that the container that you choose has adequate drainage.

Make sure that your container is appropriate for your plant's full-grown size. You do not want to have to keep changing pots as your garden grows.

Soil is the next thing you have to have for your garden. I'm sorry, but you can not go outside and dig up some dirt. It will be much too heavy and will probably contain lots of bugs and pests that you do not need. The soil must be of good quality to keep your plants healthy and growing well. Buy potting mix that drains well but still is able to retain adequate moisture. The mix should not be so light that it will not hold the plant and root system in the container without propping the plant up.

You can mix your own potting soil by using one part compost; one part perlite and one part garden loam. Be sure that you inspect the loam for pests before using it.

Choose plants that do not have very large root systems. Those plants will soon get too large for the pot that you have planted them in, even to the point of breaking it. As I mentioned before, not sizing the plant to the pot (or the pot to the plant), can cause plants to be spindly and root-bound.

Tomatoes are a good choice for a new gardener because they are easy to grow and have a strong, but small, root system. Other vegetables that are appropriate are peppers, lettuce, spinach, radishes and eggplant.

Herbs are also an excellent choice for container gardens. Herb gardens do not require much space and they are extremely easy to grow, even for the novice gardener. Like most plants, they do require adequate drain to grow them successfully.

Some people use container gardens as decorating accents as well as growing vegetables for the dinner table. Choosing containers that fit with the style of the area you want them in terms of its ambiance. You might even want to use them in a specific area inside your home. Sunrooms come to mind as a great place to grow plants. Do make sure there is adequate sun and shade for your plants.