USES OF ALGEBRA IN OUR DAILY LIFE

Determining probability

How do we determine probability? Let's say there are 12 socks in your dresser drawer. Five are red and 7 are blue. If you were to close your eyes, reach into the drawer, and draw out 1 sock, what is the probability that it would be a red sock? Five of the 12 socks are red, so your chances of picking a red sock are 5 out of 12. You can set this up as a fraction or a percentage that expresses the probability of picking a red sock:

Your chances of picking a red sock are 5 out of 12, or 5 divided by 12, which is about 42%. Not bad, as odds go.

Imagine you're choosing between 2 colleges, 1 in California and 1 in Massachusetts. You decide to flip a coin. Heads, you'll go to California. Tails, you'll go to Massachusetts. When you toss the coin, what is the probability that the head side of the coin will be facing up once the coin hits the floor? There are 2 sides to a coin, and 1 of them is heads, so your odds are 1 out of 2. In other words:

One divided by 2: that's a 50% chance of heads, and therefore also a 50% chance of tails. The odds are equal. You're as likely to go to California as to Massachusetts if you base your decision on a coin toss.

Even if you don't frequent casinos, you probably play the odds all the time. You might invest in the stock market. You might buy auto, health, and life insurance as a hedge against the costs of damage or injury. In many cases in which you are trying to predict the future, you're using the mathematics of probability.

Do you avoid gambling on the stock market or at a casino because you fear heavy financial losses? You may be surprised to hear that you're just as likely to lose money because of your everyday banking decisions. Many people collect only 1 to 3% interest on money in a savings account while simultaneously paying rates as high as 18 to 20% on credit card balances. Over time, this can mean some pretty heavy losses.

With some math smarts and an understanding of simple and compound interest, you can manage the way your money grows (and ideally keep it from shrinking). The principles of simple and compound interest are the same whether you're calculating your earnings from a savings account or the fees you've accumulated on a credit card. Paying a little attention to these principles could mean big payoffs over time.

Understanding the basics

When you put money in a savings account, the bank pays you interest according to what you deposit. In effect, the bank is paying you for the privilege of "borrowing" your money. The same is true for the interest you pay on a loan you take from the bank or the money you "borrow" from a credit card.

Interest is expressed as a rate, such as 3% or 18%. The dollar amount of the interest you earn on a savings account is figured by multiplying the money you deposit (called the principal) by the rate of interest. If you have $100 in an account that pays only 1% interest, you'll only earn $1 in interest. If you shop around for an account that pays 5% interest, you'll earn five times that amount.

In banking, interest is calculated and added at the end of a certain time period. You might have a savings account that offers a 3% interest rate annually. At the end of each year, the bank multiplies the principal (the amount in the account) by the interest rate of 3% to compute what you have earned in interest.

Interest on interest: Compounding

There are two basic kinds of interest: simple and compound. Simple interest is figured once. If you loaned $300 to a friend for one month and charged her 1% interest ($3) at the end of the month, you'd be dealing with simple interest. Compound interest is a little different. With compound interest, the money you earn in interest becomes part of the principal, and also starts to earn interest. If you loaned that same friend $300 for one month but charged her 1% each day until the end of the month, you'd be using compound interest. At the end of the first day, she would owe you $303. At the end of the second day, she would owe you $306.03. At the end of the third day, she would owe you $309.09, and so on.

Compound interest is what makes credit cards and loans so difficult to pay off. The rules of interest are the same ones that increase your savings over time, only with credit and debt, they're in the bank's favor—not in yours. With some rates as high as 21%, collecting interest on credit card loans can be a lucrative business.

Know the Terms: How to Manage Your Credit Cards

APRs. Annual fees. Finance charges. What do all these hidden fees mean, and how can you stay on top of them? Credit cards seem to have a language of their own. Once you understand this lingo, you can learn how to save hundreds or even thousands of dollars in unnecessary fees.

The first credit card concept that you should know about is the minimum payment, or the minimum amount you are required to pay each month. The amount of your minimum payment is usually equal to 2% of your average balance due. So if your average daily balance for a certain month is $500, your minimum payment for that month will be $10. This probably sounds like a good deal until you remember that you're also paying interest on that $500. If you're being charged an interest rate of 18% on the balance of $500, that $10 won't make a dent in paying off your balance. This is how most credit card companies make their money. By requiring that you pay only very small amounts each month, they know that it will take you much longer to pay off your debt, and that you'll end up paying a lot more interest over time.

Finance charges are a murky area, since each credit card company can determine its own method of computing these fees. The finance charge is the interest you pay on the money you owe the credit card company. Many companies compute monthly finance charges by determining what your average daily balance was for the month, then charging interest on that average figure. Other credit cards figure finance charges based on the average daily balance plus any new purchases you make. To avoid paying additional finance charges, find a credit card that figures fees based only on your average daily balance for the current month—not past balances or new purchases. Better yet, pay off your entire balance each month to avoid these fees.

The Annual Percentage Rate (APR) is another important credit card concept. The APR is the finance charge (which can vary month to month) expressed as an annual percentage rate. The APR is normally somewhere between 5% and 21%. One of the things to look for when comparing the APRs offered by two credit card companies is whether the rate is fixed or variable. A variable APR can fluctuate. It might follow the national interest rate, for example. A fixed APR stays the same unless changed in writing by the credit card company. A fixed APR will normally offer the best value over time, especially if you can find a low fixed rate.

You will also need to know what the grace period is for your card. The grace period is the period from when you make a purchase to when the credit card company begins charging you interest for that purchase. Without a grace period, you'll start accumulating interest charges immediately. Your best bet is to find a card with a grace period of 25 days or more and pay off the balance before that period ends.

Getting the Picture: Communicating Data Visually

According to U.S. census estimates, the population of Texas grew from 17,045,000 people in 1990 to 18,378,000 in 1994. The population of Massachusetts grew from 6,018,000 people in 1990 to 6,041,000 people in 1994.

If the population figures above were difficult for you to read and absorb, you're not alone. Reading about data can be awkward. When it's presented like this, it's hard to grasp the essential information and to see the important messages that may be behind the numbers. If this information were presented as a chart or, better yet, as a picture, it would be much easier to understand.

Ratios: Relationships between quantities

That ingredients have relationships to each other in a recipe is an important concept in cooking. It's also an important math concept. In math, this relationship between 2 quantities is called a ratio. If a recipe calls for 1 egg and 2 cups of flour, the relationship of eggs to cups of flour is 1 to 2. In mathematical language, that relationship can be written in two ways:

Both of these express the ratio of eggs to cups of flour: 1 to 2. If you mistakenly alter that ratio, the results may not be edible.

Working with proportion

All recipes are written to serve a certain number of people or yield a certain amount of food. You might come across a cookie recipe that makes 2 dozen cookies, for example. What if you only want 1 dozen cookies? What if you want 4 dozen cookies? Understanding how to increase or decrease the yield without spoiling the ratio of ingredients is a valuable skill for any cook.

Let's say you have a mouth-watering cookie recipe:

This recipe will yield 3 dozen cookies. If you want to make 9 dozen cookies, you'll have to increase the amount of each ingredient listed in the recipe. You'll also need to make sure that the relationship between the ingredients stays the same. To do this, you'll need to understand proportion. A proportion exists when you have 2 equal ratios, such as 2:4 and 4:8. Two unequal ratios, such as 3:16 and 1:3, don't result in a proportion. The ratios must be equal.

Going back to the cookie recipe, how will you calculate how much more of each ingredient you'll need if you want to make 9 dozen cookies instead of 3 dozen? How many cups of flour will you need? How many eggs? You'll need to set up a proportion to make sure you get the ratios right.

Start by figuring out how much flour you will need if you want to make 9 dozen cookies. When you're done, you can calculate the other ingredients. You'll set up the proportion like this:

You would read this proportion as "1 cup of flour is to 3 dozen as X cups of flour is to 9 dozen." To figure out what X is (or how many cups of flour you'll need in the new recipe), you'll multiply the numbers like this:

Now all you have to do is find out the value of X. To do that, divide both sides of the equation by 3. The result is X = 3. To extend the recipe to make 9 dozen cookies, you will need 3 cups of flour. What if you had to make 12 dozen cookies? Four dozen? Seven-and-a-half dozen? You'd set up the proportion just as you did above, regardless of how much you wanted to increase the recipe.

Meters and Liters: Converting to the Metric System of Measurements

Most of the world uses a standard system of measurements called the metric system. This system is based on a unit of measurement called the meter, which gets its name from the Greek wordmetron, "a measure." One meter is equal to 1 ten-millionth of the distance from the equator to the North Pole. It's a standard for measuring length that is derived from the planet we live on.

The metric system has been around for 300 years. France was instrumental in its creation and in 1795 was the first country to adopt it (though in the early 1800s, the emperor Napoleon briefly set it aside in favor of the old system of measurement). The United States remains one of the few countries that has not yet adopted the metric system as the standard for measurement.

Converting to metric values

Remember that mouth-watering chocolate chip cookie recipe? What if you wanted to send it to a friend in Portugal? You could send him the recipe with the measurements given in cups and teaspoons and hope it worked out for the best. Or you could convert the recipe to metric values, guaranteeing that the cookies would taste as delicious in Portugal as in the U.S.

To convert from cups to the appropriate metric measurement, liters, you need to know how many cups are in a liter. The table on this page shows some common conversion values. You can see that 1 cup is equal to 0.24 liters. To convert your 1 cup of flour to liters, you'd multiply 1 by 0.24. The chocolate chip cookie recipe calls for 0.24 liters of flour.

What about sugar? In the original recipe, you need 1/3 of a cup of sugar. To make the conversion easier, convert the fraction 1/3 to a decimal: 0.33. Now multiply 0.33 by 0.24. Your friend will need 0.08 liters of sugar to make your cookie recipe.

7 WAYS MATHS CAN SAVE THE WORLD

Can numbers, algebra and trigonometry save the planet? This was the question put to experts during a panel discussion at Imperial hosted by the Grantham Institute and the Mathematics of Planet Earth CDT. CDT students Paula Rowińska and Tom Bendall report back on seven ways that mathematicians are already working towards securing our planet’s future.

From meteorology to economics, a wealth of scientific research will be necessary to improve our understanding of climate change, its impacts and what we can do to prepare for them. Scratch beneath the surface and you’ll find mathematicians doing their bit to save the planet in a multitude of ways:

1. Designing better weather forecasts and climate models

Accurate weather forecasts predict when and where extreme weather may strike, whilst climate projections are key to identifying weather patterns changing on a longer time scale. Our ability to predict weather and climate has advanced in leaps and bounds in the last few decades, thanks to maths. Modern weather forecasts rely on computers to solve the complex equations that simulate the atmosphere’s behaviour – from global processes that influence the flow of the jet stream down to local rain clouds.

Mathematicians play an important role in this process, working with a set of equations that describe the atmosphere, taking into account temperature, pressure and humidity. Global Circulation Models (GCMs) describe the interactions between oceans and atmosphere to look at what the average conditions could be in decades to come.

2. Getting ‘bang for buck’ out of supercomputers

The computers used to model weather and climate get more powerful every year – but sheer processing power isn’t everything. Maths makes these computers far more effective both through contributing to technological improvements in areas like quantum computing, and by rethinking the algorithms used in computer programs. For instance, new research allows the computer to automatically zoom its attention in on areas where the weather is particularly interesting, such as around storms.

Optimising computers’ performance can also reduce their energy demand. For example, the Met Office’s Cray supercomputer runs on 2.7 MW of electricity, so even modest efficiency gains could have a massive impact on its overall energy consumption.

3. Making the most of renewable energy sources

Optimising the layout of wind turbines enables them to harvest more energy

Renewable energy sources lie at the heart of a low-carbon world. By choosing optimal locations for wind or solar farms and designing the most effective layouts for tidal and wind turbine arrays, mathematicians ensure that these technologies harvest the maximum energy as efficiently as possible.

Mathematicians contribute to research into energy supply and demand that ensures networks incorporate higher proportions of weather-dependent energy sources such as wind or solar power, making sure that the lights stay on in years to come.

4. Preparing for change

The effects of climate change will be felt on many levels, and knowledge is key to safeguarding human health and livelihoods as we adapt to changing circumstances. Mathematicians use their understanding of probability and uncertainty to advise policymakers on the likelihood of heatwaves, floods or other changes in weather patterns, and help them to plan accordingly.

Businesses also need detailed information on how climate change might affect them. The food industry for example is highly dependent on agriculture, and could use advance warning of an upcoming drought for instance to prepare themselves for smaller yields. Mathematicians try to predict who might be at risk so they can prepare for the future.

Moreover, mathematical simulations are a valuable tool for estimating the possible consequences of specific actions, by playing out different scenarios. This too can help policymakers choose one course of action over others. By presenting the hard numbers, mathematicians with an environmental conscience can seek to influence the ways businesses operate.

5. Making sense of ‘big data’

Collecting billions of pieces of data in environments, from ice sheets to cities, can deliver precious insights into our planet’s physical processes, human behaviour and everything in between. Climate scientists rebuild the history of our planet’s atmospheric composition by analysing the tiny bubbles trapped in ice records, in order to anticipate the scope of future changes. But without the statistical methods that mathematicians bring to analyse this data and assess its reliability, the information has less value.

6. Developing new technologies

Mathematical modelling is key to the development of new technologies such as CCS

New technologies are key to a low carbon future. carbon capture and storage (CCS), for instance, could safely lock away greenhouse gases emitted by fossil fuel-fired power stations, and is likely to play a key role in averting dangerous levels of global warming. Detailed mathematical models make this research possible by using sophisticated logistics methods, network analysis, statistical modelling and many other mathematical tools.

7. Making maths accessible to everyone

Crucially, maths can’t save the planet on its own. Many of the global challenges we face are multi-disciplinary: overcoming them requires mathematicians to collaborate with scientists and engineers in different fields. And although the basic science behind climate change is well understood, convincing the general public and decision makers to take action to reduce carbon emissions is very much a work in progress. With their firm grasp of concepts such uncertainty and probability, Mathematicians are uniquely placed to communicate the science, data and forecasts, and ensure that this information is meaningful to the people who need it.

For maths to have a real impact on our planet’s fate, mathematicians therefore need to communicate the importance of their work clearly and effectively, knowing when to swap complicated equations for persuasive story-telling, pictures, games or genuine interaction. Opening up maths up for the world to understand might just be the best way that we can come to our planet’s rescue.

10 EVERYDAY REASONS WHY STATISTICS ARE IMPORTANT

Statistics are sets of mathematical equations that are used to analyze what is happening in the world around us. You've heard that today we live in the Information Age where we understand a great deal about the world around us. Much of this information was determined mathematically by using statistics. When used correctly, statistics tell us any trends in what happened in the past and can be useful in predicting what may happen in the future.

Let's look at some examples of how statistics shape your life when you don't even know it.

1.Weather Forecasts

Do you watch the weather forecast sometime during the day? How do you use that information? Have you ever heard the forecaster talk about weather models? These computer models are built using statistics that compare prior weather conditions with current weather to predict future weather.

2. Emergency Preparedness

What happens if the forecast indicates that a hurricane is imminent or that tornadoes are likely to occur? Emergency management agencies move into high gear to be ready to rescue people. Emergency teams rely on statistics to tell them when danger may occur.

3. Predicting Disease

Lots of times on the news reports, statistics about a disease are reported. If the reporter simply reports the number of people who either have the disease or who have died from it, it's an interesting fact but it might not mean much to your life. But when statistics become involved, you have a better idea of how that disease may affect you.

For example, studies have shown that 85 to 95 percent of lung cancers are smoking related. The statistic should tell you that almost all lung cancers are related to smoking and that if you want to have a good chance of avoiding lung cancer, you shouldn't smoke.

4. Medical Studies

Scientists must show a statistically valid rate of effectiveness before any drug can be prescribed. Statistics are behind every medical study you hear about.

Genetics

Many people are afflicted with diseases that come from their genetic make-up and these diseases can potentially be passed on to their children. Statistics are critical in determining the chances of a new baby being affected by the disease.

6. Political Campaigns

Whenever there's an election, the news organizations consult their models when they try to predict who the winner is. Candidates consult voter polls to determine where and how they campaign. Statistics play a part in who your elected government officials will be

7. Insurance

You know that in order to drive your car you are required by law to have car insurance. If you have a mortgage on your house, you must have it insured as well. The rate that an insurance company charges you is based upon statistics from all drivers or homeowners in your area.

8. Consumer Goods

Wal-Mart, a worldwide leading retailer, keeps track of everything they sell and use statistics to calculate what to ship to each store and when. From analyzing their vast store of information, for example, Wal-Mart decided that people buy strawberry Pop Tarts when a hurricane is predicted in Florida! So they ship this product to Florida stores based upon the weather forecast.

9. Quality Testing

Companies make thousands of products every day and each company must make sure that a good quality item is sold. But a company can't test each and every item that they ship to you, the consumer. So the company uses statistics to test just a few, called a sample, of what they make. If the sample passes quality tests, then the company assumes that all the items made in the group, called a batch, are good.

10. Stock Market

Another topic that you hear a lot about in the news is the stock market. Stock analysts also use statistical computer models to forecast what is happening in the economy.