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.

Garner Compliments Wearing Rose Gold Jewelry, the Latest Jewelry Trend

Rose gold jewelry, this is appearing since the past 2 to 3 years. This is a blush color metal and has gained prominence during spring and summer. This trend as gold fixtures is bought even as a gift. Today, this precious metal is widely used and women, you must have at least one in your collection.

Adding rose jewelry into your collection means find the pieces of jewelry that look good and go well in combination with your yellow gold and silver jewelry. The gold jewelry looks nice, regardless of the style or metal. Having a personalized rose ring stack is fun. These minimalist competing bands are now available as motifs and in varying thicknesses that it can be a statement ring that is less expensive yet appears dazzling with gemstone studded. The color braided gold band makes it very special.

How it is Made

First, Rose gold appeared in Russia as copper infused metal was incorporated in the jewelry designs by Carl Faberge, the well-known jeweler.

Rose actually is not available directly from the nature. As already known, the pure gold is in same color and the jewelry color changes only owed to the mixing of alloys or other metals. The metal mixing ratios change the gold color.

The metal receives its colors on mixing copper alloys with pure gold and silver is also added in small amount. The mix having more copper means the jewelry shows redder and the one with less copper comes in pink color.

The typical 18k modern-day rose looks featuring gold to 75%, silver 4% and copper to 21%. The fact is that as the content of silver reaches to 5% or closed, you will notice gold appearing more of pink in color.

The beauty of rose jewelry is that it offers a comprehensive look and the jewelry looks simple, yet stylish garnering compliments from your neighbors, love, coworkers, friends and relatives.

Rose Jewelry prominence

Copper is an alloy. Owing to the high copper content only the rose gold jewelry emerges and as the fact is that copper with age gets darker and over time less rosy. The jewelry appears vintage inspired, but certainly is a style statement.

Today, the market has rose jewelry in lots and there are designers trying to bring different styles using this hue of gold. The purple gemstones fit in soft pink makes the rose look stunning and reflect your skin glow.

The necklace pendants in rose gold also look amazing. Rose gold was used in the Victorian era in the pendants and today is a bohemian style. If you are a person loving antique jewelry, turn the gold into some jewelry piece of your taste. Even if a rose jewelry is broken, you can wear it as a brooch or a stickpin, it will shine. The rose gold jewelry is recognized as personalized jewelry offering a feminine touch and charm. Use it for your Zodiac sign and use it as charm pieces in your jewelry, in the necklace or ring, it is sure to look beautiful.

The State Of Paralegal Employment In The United States

In 2004, Paralegal jobs and legal assistant’s jobs amounted to about 224,000 positions in all sectors of the economy of the United States. 84% of these had Paralegal qualifications above both Associate’s and Bachelor’s Degree, with about 10% Paralegals who had higher degrees leaving only about 6% Paralegals who have been trained on the job and working without formal Paralegal education.

As was in the beginning, private law firms controlled the lion share of employing about 70% of all working Paralegals then; today, according to present day employment statistics, this number has fallen as the Paralegal profession continues to expand into deferent fields of human endeavor. Many paralegals and legal assistants these days work for corporate legal departments and various levels of government agencies and None governmental organizations..

The U.S. Department of Justice is the largest employer of Paralegal within the Federal Government, followed by the Social Security Administration and the U.S. Department of the Treasury. State governments and Counties are also increasingly employing Paralegals to help make legal services available to the common man on the street.

More Paralegals are setting up private Paralegal offices in states where the law provides such privileges and work as freelance legal assistants, contracting their services to attorneys or corporate legal departments.

It is hoped that in the nearest future the Paralegal profession would be robust enough to receive country wide approval for Paralegals to practice freely providing legal assistance to all and sundry according to the law. This will be a step in the right direction considering that in the United Kingdom, Solicitors play the same roles as registered legal practitioners.