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Understanding Approximate Numbers: Why, When, and Where We Use Them
Why, when and where we use numerical approximations.
WHAT IS AN APPROXIMATE NUMBER?
A value that is APPROXIMATE is an INEXACT value that is close to the actual value.
HOW NEAR – HOW IMPORTANT IS A CORRECT ERROR?
The difference between the actual 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 that these errors cancel each other out when numbers are added, subtracted, multiplied, or divided. However, arithmetic can also make small errors worse. When this happens, many small errors can combine to produce one giant error.
That said, NUMERICAL APPROXIMATIONS are used because they SIMPLIFY OUR DAILY LIFE.
We use approximate numbers for a myriad of tasks: to get a quick estimate of commute times, to project our grocery expenses for the week, to guess the height of the neighbor’s tree, to predict how many pounds we’ll weigh by next week, predicting a grade on an essay, etc…
Approximations make arithmetic less complex and reduce the time and effort needed to process numbers.
Using approximations can quickly give us a useful answer.
Approximations are convenient.
Approximation of 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 errors introduced by using inaccurate numbers can lead to disaster. For example, using rough approximations in your calculations may mean that you are underestimating your expenses and running out of money.
WHAT SPECIFIC KINDS OF APPROXIMATIONS DO WE USE?
Five ways to use approximations are described 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 all walks of life.
How much will lunch cost? Somewhere between $50 and $110 at one of the upscale restaurants; or, $5 to $15 at the sandwich shop down the street. How much is your house worth? How much does it cost to repair your car?… 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 digits.
Sometimes rounding a number is strictly practical, like 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 do not have a choice. You must use an approximation for the square root of 2.
Also, it is not necessary to use 31 decimal places for most problems. The square root of 2 is usually rounded to something like 1.4142. The rounded number is another approximation.
A lot of time is spent in school teaching students how to approximate numbers by rounding them off.
3. SIMPLIFY 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 on a clear day? This is called the distance to the horizon.
There is a formula to calculate this distance.
d = square[h(D+h)]
� d = distance to the horizon
� D = diameter of the Earth
� h = height of the observer above sea level
� R = Earth radius
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 bridge of a ship can estimate the distance to the horizon in his head with very little effort.
4. STATISTICS: HOW MUCH IS THE RESPONSE?…
As an example, consider polls of likely voters conducted before an election.
Suppose 56.5% of likely voters favor candidate A + or – 3%. This is an APPROXIMATION, meaning that the actual number of voters in favor of candidate A is between 53.5% and 59.5% (a RANGE OF POSSIBLE VALUES).
5. TRIAL AND ERROR APPROXIMATIONS…
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 the trial and error method.
The problem is that these approximations often require so much trial and error that manual calculation is impractical.
However, using the computer, billions of calculations can be performed in seconds.
Approximation of a solution by trial and error is important in mathematics, physics, chemistry, electrical engineering, and other fields.
Calculating 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 the square root of five; 2) multiply your guess times themselves to see how close the multiplied results are to five.
Repeat steps 1) and 2) over and over again until you achieve the desired degree of precision.
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