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## Understanding Approximate Numbers: Why, When, and Where We Use Them

Why, when, and where do we use numerical estimation?

What is the estimated number?

A value that is APPROXIMATE is an INEXACT value that is close to the actual value.

How close – how big is the error?

The difference between the actual value and the predicted value is the error.

Although an approximation can often reduce the complexity of a problem, every approximation will introduce error.

We often assume that these errors will offset each other when adding, subtracting, multiplying, or dividing numbers. However, arithmetic can accommodate even small errors. When this happens, many small errors can combine to produce a huge error.

That said, numerical approximations are used because they simplify our daily lives.

We use estimated numbers for a myriad of tasks: to get a quick estimate of travel time, project our grocery expenses for the week, estimate how tall the neighbor’s tree is, predict how many pounds we’ll weigh in the next week, predict a grade. A test, etc.

Approximation makes arithmetic less complicated, and reduces the time and effort required to process the numbers.

Using assumptions can quickly give us useful answers.

The assumptions are reasonable.

Estimating a number allows us to immediately evaluate a course of action without waiting for the exact number.

At the very least, inference can show us how to perceive and understand the impact of an important decision without waiting for further study.

However, we have all experienced how the errors introduced by using unqualified numbers can lead to disaster. For example, using rough estimates in your calculations means that you underestimated your expenses and ran out of money.

What are some special types of assumptions we use?

The five methods used in estimation are discussed below:

1. A range of prices…

An estimate is often given as a range of values.

A range of approximate values used in every field of life.

How much will lunch cost? somewhere between $50 and $110 at one of the higher end restaurants; Or, $5 to $15 at the sandwich shop down the street. How much is your house worth? How much is it to repair your car?… and so on.

2. Round prices… Sometimes you have no choice

A number is often approximated by rounding 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, one cannot calculate the exact value for the square root of 2 because it is an irrational number. 1.4142135623730950488016887242097 is an estimate. You have no choice. You should use the approximation for the square root of 2.

Additionally, for many problems there is no need to use 31 decimal places. The square root of 2 is usually rounded like 1.4142. Rounding numbers is another approximation.

Much time in school is spent training students how to approximate numbers by rounding them.

3. Simplifying Formulas…

Assumptions are used to make formulas more useful.

For example, if you are on the deck of a ship, how far can you see in clear weather? This is called horizon distance.

This is a formula to calculate the distance.

d = sqrt[h(D+h)]

ï¿½ d = distance to horizon

ï¿½ D = Diameter of Earth

ï¿½ h = height of observer above sea level

ï¿½ R = radius of the earth

Using approximation, this formula can be reduced to:

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 an approximation formula, an officer standing on the deck of a ship can estimate the distance to the horizon in his head with little effort.

4. Statistics: How close is the answer?…

As an example, consider a survey of likely voters taken before an election.

Suppose that 56.5% of the likely voters support candidate A + or – 3%. This is an APPROXIMATION which means that the actual number of voters who support candidate A is between 53.5% and 59.5% (range of possible values).

5. Estimates 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 non-linear equations can be approximated with a high degree of accuracy using the trial and error method.

The problem is, these estimates often require so many tests that hand calculations are not practical.

However, billions of calculations can be completed in a few seconds using a computer.

Approximate solutions by trial and error are 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 at the square root of five; 2) Multiply your guess times to see how close the multiplication results are to five.

Repeat steps 1) and 2) and repeat until you achieve the desired degree of accuracy.

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