Why, when and where we use numerical approximations.
WHAT IS AN APPROXIMATE NUMBER?
An APPROXIMATE value is an INEXACT value close to the actual value.
HOW CLOSE – HOW BIG IS A MISTAKE?
The difference between the actual value and the approximate value is the error.
Although an approximation can often reduce the complexity of a problem, each approximation will introduce an error.
We often assume that these errors will offset each other 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 one giant error.
That said, NUMERICAL APPROXIMATIONS are used because they SIMPLIFY OUR DAILY LIVES.
We use ballpark numbers for a myriad of tasks: to get a quick estimate of travel times, to project our grocery expenses for the week, to guess how tall the neighbor’s tree is, to predict how many pounds we’ll weigh next week, to predict a grade in a test, etc.
Approximations make arithmetic less complex and reduce the time and effort required to process numbers.
Using approximations can quickly give us a useful answer.
Approximations are practical.
Approximating a number can allow us to evaluate a course of action immediately, without waiting for an exact number.
At a minimum, 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 errors introduced by using inaccurate numbers can lead to catastrophe. For example, using rough approximations in your calculations may mean you underestimate your expenses and run out of money.
WHAT ARE SOME OF THE SPECIFIC TYPES OF APPROACHES WE USE?
Five ways in which approximations are used are discussed below:
1. RANGE OF VALUES…
An approximation is often given as a range of values.
A RANGE of values that 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 does your car repair cost?… and so on.
2. ROUND 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 calculate 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.
Also, it is not necessary to use 31 decimal points for most problems. The square root of 2 usually rounds to something like 1.4142. The rounded number is another approximation.
A great deal of time in school is spent teaching students how to approximate numbers by rounding them.
3. SIMPLIFYING FORMULAS…
Approximations are used to simplify formulas and make them more useful.
For example, if you are on the deck of a ship, how far can you see in good weather? This is called the distance to the horizon.
There is a formula to calculate this distance.
d=sqrt[h(D+h)]
� d = distance to the horizon
� D = diameter of the Earth
� h = height of the observer above sea level
� R = radius of the Earth
Using an approximation, this formula can be reduced to the following:
d = 3.6 * sqrt(h)
�d is the distance to the horizon (in kilometers)
� h is the height above sea level (in meters)
Using the approximate formula, an officer standing on the deck of a ship can mentally estimate the distance to the horizon with very little effort.
4. STATISTICS: HOW CLOSE IS THE ANSWER?…
As an example, consider surveys of likely voters conducted before an election.
Assumes 56.5% of likely voters favor candidate A+ or -3%. This is an APPROXIMATION, meaning that the actual number of voters favoring Candidate A is between 53.5% and 59.5% (a RANGE of POSSIBLE VALUES).
5. TRIAL AND ERROR APPROACHES…
Some calculations are so complex that they cannot be solved analytically.
But that doesn’t mean they can’t be solved.
The solution of many nonlinear equations can be approximated with a high degree of accuracy using trial and error.
The problem is that these approximations often require so many trials that manual calculation is impractical.
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.
Finding the square root of five is a simple example of how trial and error can be used.
To use trial and error: 1) guess the square root of five; 2) Multiply your guess by itself to see how close the multiplied results are to five.
Repeat steps 1) and 2) over and over until the desired degree of accuracy is achieved.
