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.

How to Talk to Your Spouse Who Says You Work Too Much

Strong marriages and relationships require attention from both partners. This balance of attention to spouse versus earning a living or furthering your career can be difficult to find. And, once you have found a balance that works in your relationship, things change and you have to find the balance again.

Why do some people overwork? The Overworking Spouse may be under considerable stress on the job, or may have a boss that demands long hours. Or maybe there are layoffs coming up and the less productive workers are the first to go. Maybe one partner has to work long hours just to support the family. On the other hand, the overworking spouse may be strongly identifying with career advancement to the exclusion of the marriage. Of course, we should be involved in work that is worthwhile, fulfilling and financially rewarding but over focus can put the marriage at risk.

What can happen when you put your work/career first? Your spouse may be feeling emotionally disconnected from you and lonely. There may be a buildup of resentment which can lead to anger and finally bitterness-towards you. This situation often leads to unhappiness and discord for the entire family.

As a psychologist who has worked for 20 years counseling couples in my therapy practice in North St. Paul, MN, I met with Amy and Josh with just this complaint. Amy was complaining that she was overburdened because Josh worked too much and left her with the responsibilities of home, children, and her own 40-hour job. He knew she was right but he felt defensive when she tried to tell him how to manage his work schedule. He’d started complaining that when wasn’t very affectionate lately and always blamed her disinterest in their physical intimacy on fatigue. He asked to meet with me individually after Amy had had a session to talk about her “side”of the problem. He was appreciative that she was not as naggy as she had been in the past, but he still felt blamed and criticized.

When Josh and I met, we talked of the reasons that he was spending so much time at work. The usual reasons people overwork is that they are feeling under considerable stress on the job, or may have a boss that demands long hours. Or maybe there are layoffs coming up and the less productive works are the first to go. Maybe one partner has to work long hours just to support the family. On the hand, the overworking spouse may be strongly identifying with career advancement to the exclusion of the marriage.

Drs. John and Julie Gottman, in their book 10 Lessons to Transform Your Marriage, suggest the following questions:

1.What does your work mean to you?

2.What pleasure or satisfaction does work bring to you?

3.What need does working fulfill in your life?

4.Does your work related to some personal legacy you would like to contribute to the world?

In a discussion with Josh, he pointed out that he was trying to make headway in his career now, while he was still young and had the energy. He knew his wife was doing a good job parenting their two small children (3 years and 9 months) so he wanted to focus on earning good money for his family. At first, he couldn’t understand why Amy was complaining so much because he was doing all this work for her and their children. He felt it was his responsibility. He wanted to provide his children with a fund for their education.

I suggested he consider the questions listed above. He said that his work is very satisfying in that he felt very confident and capable at work. He also wanted to meet his father’s expectations who had had a successful career. His older two brothers had moved from job to job and he knew his father was disappointed that they had difficulty supporting their families.

It was important for Josh to understand the underlying reasons for his excessive hours at work. He did not have a demanding boss and he was in no threat of a layoff. He just wanted to do the right thing. He had not thought of himself as having value over and above that of a worker and wage earner. He resigned himself to have a discussion with Amy. He said he would try to listen carefully about what Amy was longing for when she complained to him. I suggested he try to imagine the experiences she wanted to have with him and not to focus on her criticisms.

When Amy and Josh came for the next couple’s session, they wanted to further discuss the overworking situation. (They no longer saw the problem as residing inside one of them but as a problem the two of them need to figure out.) Any was able to explain to him what she missed when he work so much. He came to get a sense that his contribution to their family was not solely a financial responsibility. He heard her tell him he was also loved, appreciated, and needed as a friend, confidant and co-parent. She especially enjoyed his easy humor with their children and felt he was the only one who would be able to provide that.

This was a difficult issue for this couple and it was not resolved quickly. They were able to discuss their needs and wants in a different way when they addressed the questions about the underlying reasons for their positions.

Getting Low Auto Insurance Rates – Cheap Car Insurance Policies

First of all, avoid being in an accident. Having a clean accident-free driving record is a big plus for negotiating lower rates from your auto insurance provider. That you are a good driver makes the risk level for the insurance agency to be relatively low thereby giving you more power to make certain demands especially on lower rate considerations.

Also having a clean driving record will make you top choice and a better client to the insurance companies around.

You should endeavor to understand some of the value added services that auto insurance companies provide. Most times, these services help you save both time and money. A good insurance company will certainly love to have smiling customers and one of the greatest means for achieving this is to increase service value while maintaining price points or even lowering the overall price.

You can save more if you have multiple policies running on the same account. For example you can have a life insurance policy, home renters insurance and your auto insurance policy all being active in the same account. This will typically prompt for a lower rate negotiation.

Insurance companies will cross check on some of your personal behaviors and habits. If you have a record of getting drunk on alcohol or excessive speeding with friends and over loading vehicles, then your record is that of a high risk for the insurance companies. Having a non-destructive record will certainly give you more points for lower rates. The first step for you to get your lower rates is to get free auto insurance quotes from trusted agents online.

Where To Start?

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.