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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 APPROXIMATED is an INEXACT value that is 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, every approximation will introduce an error.
We often assume that these errors will compensate for each other when adding, subtracting, multiplying, or dividing numbers. However, arithmetic can also compound small errors. When this happens, many small mistakes can combine to produce one giant mistake.
Having said that, NUMERICAL APPROXIMATIONS are used because they SIMPLIFY OUR DAILY LIFE.
We use rough numbers for a myriad of tasks: to get a quick estimate of travel times, project our grocery spending for the week, guess how tall the neighbor’s tree is, predict how many pounds we’ll weigh next week, 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 give us a useful answer quickly.
The approaches 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 errors introduced by using inaccurate numbers can lead to catastrophe. For example, using rough approximations in your calculations can mean you underestimate your expenses and run out of money.
WHAT ARE SOME SPECIFIC TYPES OF APPROACHES WE USE?
Five ways to use approximations are discussed below:
1. RANGE OF VALUES…
An approximation is often given as a range of values.
A RANGE of values is used that approximates the exact value in each area of life.
How much will lunch cost? Between $50 and $110 at one of the high-end restaurants; or, $5-$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 DON’T HAVE SCHOOL
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 from 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 places for most problems. The square root of 2 is usually rounded to 1.4142. The rounded number is another approximation.
A great deal of time is spent in school training students on rounding approximate numbers.
3. SIMPLIFICATION OF 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 horizon distance.
There is a formula to calculate this distance.
d = square[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 * squares (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 surveys of likely voters conducted before an election.
Assume that 56.5% of likely voters favor candidate A + or -3%. This is an APPROXIMATION, which means that the actual number of voters favoring candidate A is between 53.5% and 59.5% (a POSSIBLE RANGE OF VALUES).
5. APPROACHES USING TRIAL AND ERROR…
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 by trial and error.
The problem is that these approximations often require so much testing that manual calculation is impractical.
However, with 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.
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 by itself to see how close the results multiplied by five are.
Repeat steps 1) and 2) over and over until the desired degree of accuracy is achieved.
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