Why, When, and Where we use numerical approximations.
WHAT IS AN APPROXIMATE NUMBER?
A value which is APPROXIMATE is an INEXACT value which is close to the real value.
HOW CLOSE – HOW BIG AN ERROR IS OK?
The difference between the real value and the approximate value is the error.
Although an approximation can often reduce the complexity of a problem, every approximation will introduce an error.
We often assume these errors will offset one another when numbers are added, subtracted, multiplied, or divided. However, arithmetic can also compound small errors. When this happens, many small errors can combine to produce a giant error.
That said, NUMERICAL APPROXIMATIONS are used because they do SIMPLIFY OUR DAILY LIFE.
We use approximate numbers for a myriad of tasks: to get a quick estimate of travel times, project our grocery expense for the week, guess how tall the neighbor’s tree is, predict how many pounds we will weigh by next week, predict a grade on a test, etc.
Approximations make arithmetic less complex, and reduce the time and effort needed to process numbers.
Using approximations can give us a useful answer quickly.
Approximations are practical.
Approximating a number can allow us to evaluate a course of action immediately, without waiting for an exact number.
At the very least, approximations can often show us how to understand and appreciate the implications of an important decision without waiting for further study.
However, we have all experienced how the errors introduced by using inexact numbers can lead to a catastrophe. For example, using rough approximations in your calculations can mean you underestimate your expenses and run out of money.
WHAT are some of the SPECIFIC TYPES OF APPROXIMATIONS WE DO USE?
Five ways approximations are used are discussed below:
1. RANGE OF VALUES…
An approximation is often given as a range of values.
A RANGE of values which approximates the exact value is used in every area of life.
How much will lunch cost? Somewhere between $50 and $110 at one of the high end restaurants; or, $5 to $15 at the sandwich shop down the street. How much is your house worth? How much is your car repair?… and so on.
2. ROUNDING VALUES… SOMETIMES YOU HAVE NO CHOICE
A number is often approximated by rounding it to a certain number of significant figures.
Sometimes rounding a number is strictly for convenience, such as rounding 999 to 1000.
Sometimes, there is no choice.
For example, the square root of 2 = 1.4142135623730950488016887242097 (and so on). However, no one can compute an exact value for the square root of 2 because it is an irrational number. 1.4142135623730950488016887242097 is an approximation. You have no choice. You must use an approximation for the square root of 2.
In addition, it is not necessary to use 31 decimal points for most problems. The square root of 2 is usually rounded to something like 1.4142. The rounded number is another approximation.
A great deal of time in school is spent training students how approximate numbers by rounding them.
3. SIMPLIFYING FORMULAS…
Approximations are used to simplify formulas to make them more useful.
For example, if you are on the deck of a ship, how far can you see in clear weather? This is called the distance to the horizon.
There is a formula for calculating this distance.
d = sqrt[h(D+h)]
ï¿½ d = distance to the horizon
ï¿½ D = diameter of the Earth
ï¿½ h = height of the observer above sea level
ï¿½ R = radius of the Earth
Using an approximation, this formula can be reduced to the following:
d = 3.6 * sqrt(h)
ï¿½ d is the distance to the horizon (in kilometers)
ï¿½ h is the height above sea level (in meters)
Using the approximate formula, an officer standing on the deck of a ship can estimate the distance to the horizon in his head with very little effort.
4. STATISTICS: HOW CLOSE IS THE ANSWER?…
As an example, consider polls of likely voters taken prior to an election.
Suppose 56.5 % of likely voters favor candidate A + or – 3%. This is an APPROXIMATION which means that the real number of voters which favor candidate A is somewhere between 53.5% and 59.5% (a RANGE of POSSIBLE VALUES).
5. APPROXIMATIONS USING TRIAL AND ERROR…
Some calculations are so complex they cannot be solved analytically.
But that does not mean they cannot be solved.
The solution to many non-linear equations can be approximated with a high degree of accuracy using the trial and error method.
The problem is, these approximations frequently require so many trials that hand computation in not practical.
However, using the computer, billions of calculations can be completed in a few seconds.
Approximating a solution by trial and error is important in mathematics, physics, chemistry, electrical engineering, and other fields.
Computing the square root of five is a simple example of how trial and error can be used.
To use the trial and error method: 1) guess at the square root of five; 2) multiply your guess times itself to see how close the multiplied results are to five.
Repeat steps 1) and 2) over and over until you achieve the desired degree of accuracy.