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.

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?

How to Choose Your Motorcycle GPS Systems

Location, location, location are not just words used in real estate. When riding your motorcycle it's a great idea to know where you are and where you are going. Global positioning systems or GPS have come a long way in the last twenty years in helping riders find their way along the highways and byways of traveling. Choosing the right GPS for your motorcycle can depend on what features you desire.

The physical dimensions of the GPS device can be up to 5.7 "x 4.2" giving you a large enough screen to view your roadway, directions, watch a video on screen, or just use the split screen to see your next turn. The buttons for almost anything you need are right there on your large screen. In case you want to find your location at the moment just go to the Menu. From there push the "Where Am I" button on the touch screen and you will be able to read your exact location.

GPS devices are generally pre-loaded with maps of the United States including all cities, towns, highways, and local points of interest. This allows the rider to find any destination, landmark, or address in the continental United States. To help with guidance GPS devices have voice control which allows you to talk to the GPS and ask for directions, turns, or the next filling station on your route. The device also gives you audible turn by turn.
instructions, highways intersections, distance to your turn, and what lane you need to be in to make your next turn.

Wifi capability is available if you need to make reservations at a restaurant, check your stock portfolio, or just look up a local movie theater. The Wifi feature also allows you to get the local weather and traffic information so that you can either get out of the rain if need be or detour around an accident in the road several miles ahead. The colorful touch screen can be a valuable benefit to you when trying to finding your way to local events, route planning, and uploading those points of interest you would like to see along the way.

To give you further enjoyment on your ride the GPS device has Blue tooth Stereo as an added feature. This allows for many hours of listening comfort while you ride. If you enjoy music as most people do you'll want to choose some of your own favorites. An MP3 player gives you the flexibility while you drive. You can easily hook into the side of the GPS with your jack and listen to your music or get directions to your brother's barbecue.

For most of us in today's market we would not want to be without our cell phones. Blue tooth technology has allowed GPS devices to have hands free phone service. Use the touch screen to accept a phone call and talk to the GPS device to find a number and make a call. Hands free phone service is easy and very versatile.

When you've reached your destination GPS device can be removed and used to find your way on that hike up the mountain and back. Your exact location where your bike was left can be found by easily using your touch screen and directions. Global Positioning Systems (GPS) are the best idea for the motorcycle rider.

Can You Really Lose As Much Weight As a ‘Big Loser’ Each Week – And Would You Want To?

With shows like the Biggest Loser getting massive reviews and viewer numbers, clearly its hitting a chord with the general public. The show encourages massive weight loss on a weekly basis. How do we know how much is a reasonable amount of weight to lose? Is it true that you can only lose half a kilo per week. Let's take a look at what is realistic – and achievable – for you …

First off, is it really possible to drop fat as quickly as the contestants on The Biggest Loser? Can the man on the street (or you :) take these kind of losses without a team of trainers, nutritionists and camera men following their every move? Or are the massive and speedy losses on the show nothing more than clever marketing and a not-so-subtublic manipulation of the viewer?

I did a little research to find out just how much weight some of our contestants have stacked up. At the time of my research the show was airing Week 11. Here's a summary of total weight loss for the final four contestants at that time:

Sam: just over 40kg (88 pounds)
Cosi: close to 48kg (105.6 pounds)
Alison: around 36kg (79.2 pounds)
Gary: close to 59kg (129.8 pounds)

The weekly weight loss (divided by 11 just to keep it simple, although obviously they would have lost different amounts each week) is shown here:

Sam: 3.63 kg p / week (7.9 pounds)
Cosi: 4.36 kg p / week (9.6 pounds)
Alison: 3.27 kg p / week (7.2 pounds)
Gary: 5.36 kg p / week (11.8 pounds)

Just in case you're interested, this averages out at a body weight percentage loss of around 2.8 weekly. If you matched this to the 'average' 70kg woman, that would equate to 1.96kg per week, and for a 90kg man it would be around 2.66kg per week.

Does not sound like too much, really, does it? I'm sure many readers out there would be thrilled to lose this amount of weight each week, and not necessarily consider it extreme.

But let's think about that. In my experience most of you have heard over and over that healthy weight loss is around 0.5-1kg per week. In fact, every health expert, resource or study I've mentioned that a 'healthy' (as in mainly fat, and sustainable) amount of weight loss is just that – around half a kilo per week for women, and 1 kilo for men. Extremely overweight people are said to be able to lose 1.5 kg per week.

If we take this as a gospel (for now), then clearly the BL contestants are losing weight much faster than they should be in order to keep it off. According to Anne Collins, fat loss expert extraordinaire,

'When we lose weight we do not just lose fat. We lose a combination of body fat, and muscle tissue. Studies show that when we diet, the weight we lose is on average 75 percent fat and 25 percent muscle. (Also) remember water accounts for about 70 percent of the total body weight of an average person, with muscle tissue containing roughly 75 percent water (plus 20 percent protein and 5 percent minerals), and body fat containing roughly 50 percent water. '

How is this relevant?

When you lose weight quickly, your body will shed muscle and water before it lets go of fat stores. This is because your body views rapid weight loss as a threat to your survival – your physiology assumes that you must be low on resources (food), or on the run. Why else would you starve or over-exercise yourself?

In order to preserve life (yours!) Your very intelligent body gets rid of the less important stuff – like muscle. Yep, that's the metabolism boosting stuff in case you forgot. After all, fat stores will keep you alive for longer.

So the long and short of it is that losing weight too quickly will not only mean you lose muscle and water over fat, but you will also (at least semi) permanently lower your resting metabolic rate, meaning that the same amount of food you used to eat will cause you to gain more weight.

This is where the concept of the 'yo-yo effect' comes in to play – the idea (reality) that most diets result in a bounce-back effect that leaves the sorry soul in question more overweight than when they started.

NOT good.

But that's not really the full story. Here's where it gets even more confusing (or interesting, for the thinkers among you).

o If your body is under a lot of stress, you can not safely lose weight. Basically, your body will not allow you to. This is because stress is a threat to your survival – and your body can not distinguish between our modern day stress of a demanding job or unsettled relationship, and the stress of a threat on your life. Therefore, your body views all stress as a threat to your survival – and attempts to give you a fighting chance by keeping a protective layer of fat that will not go away regardless of what you eat or how you exercise.

o According to William W. Wolcott, author of The Metabolic Typing Diet, 'It is not uncommon for people to gain weight in response to stress. Weight gain literally represents a protective layer, an insulator, against the pain and suffering of the stress. In such cases, diet, nutrition, and exercise are powerless to resolve the problem '. Solution: put steps into place to detoxify stress. This includes eating correctly, but also taking time out.

o Ironically, cutting back calories to an extreme or doing high amounts of cardio exercise can also prevent you losing weight.

o Even if you are losing weight at the 'safe' rate of, you will still find your lean muscle mass decreases to some extent. In the long run this means you will lower your metabolism and possibly gain weight back. The only way to counteract this is to perform resistance exercise. I'd recommend at least 1-2 half hour sessions per week, for maintenance, or more if you're looking to gain a significant amount of muscle.

o A final spark of hope – Over the years I have known clients to 'lean up' in an incredibly fast manner. Without losing a significant amount of muscle – and I'm not just talking abut extremely overweight people. I'm going to finish this article with my hot tips for maximizing weight loss from fat.

MAXIMISE YOUR FAT LOSS

1. Eat correctly. Trust me on this – pill popping is NOT how the human body achieves true health and weight loss. Choose natural, minimally processed sources of carbohydrates, fat and protein.

2. Sleep correctly. Numerous studies have shown that building up a sleep debt will result in rampant stress hormones that cause your body to store fat. Regardless of how you eat or exercise. I've written many articles on the topic myself.

3. Think correctly. I do not care how 'airy fairy' this sounds – your thoughts do have power. If you believe and tell yourself that you can not lose weight, you WILL NOT lose weight. Positive thinking on its own may not cut it, but it sure is not going to hurt. Build confidence and promote positive action by telling yourself that you can and will improve your health and weight.

4. Reduce stress. We've spoken about how stress hormones cause you to store fat. If you really can not change your lifestyle, put steps into place to relax and recuperate. Using your one free hour pounding the treadmill is not always the best option. But ask yourself this. Can you REALLY not change your lifestyle? If your life depended on it I bet you would walk out of the job, relationship, situation or whatever it might be. Guess what? Your life does depend on it.

I guess when all is said and done, the rapid weight loss of Biggest Loser contestants may not be possible or ideal for most people. This next I believe the show does a brilliant job of inspiring people to reach, to fight, to do whatever it takes to achieve their dreams. I think if you asked them, the BL contestants would tell you that they had definitely been living life.

How 'bout you?

Life is Now. Press Play.