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

Why, when and where we use numerical approximations.

WHAT IS A ROUGH NUMBER?

An approximate value is an INCORRECT value that is close to the actual value.

HOW CLOSE – HOW MUCH ERROR IS OKAY?

The difference between the actual value and the approximate value is an error.

Although an approximation can often reduce the complexity of a problem, any approximation introduces an error.

We often expect these errors to compensate for each other when adding, subtracting, multiplying, or dividing numbers. However, arithmetic can also add up to small errors. When this happens, many small errors can add up to a giant error.

Nevertheless, NUMERICAL APPROXIMATIONS are used because they SIMPLIFY OUR DAILY LIFE.

We use approximate numbers for countless tasks: to quickly predict travel times, to predict food costs for the week, to predict the height of a neighbor’s tree, to predict next week’s weight in pounds, to grade a test, etc.

Approximate calculations make arithmetic less complex and reduce the time and effort required to process numbers.

Using approximate values will quickly give us a useful answer.

The approximations are practical.

Approximating the number allows us to estimate the course of action immediately without waiting for the exact number.

At the very least, approximations can often show us how to understand and assess the consequences of an important decision without waiting for further research.

But we’ve all experienced how the mistakes of using inaccurate numbers can lead to disaster. For example, using rough estimates can mean you underestimate your costs and run out of money.

WHAT SPECIFIC TYPES OF APPROACHES DO WE USE?

Five uses of approximate methods are discussed below:

1. VALUES…

An approximate value is often given as a range of values.

In every area of life, VALIC values are used that correspond to the exact value.

How much is lunch? Somewhere between $50 and $110 at a high-end restaurant; or $5-$15 at a sandwich shop down the street. How much is your house worth? How much does it cost to repair your car?… 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 calculate the exact value of the square root of 2 because it is an irrational number. 1.4142135623730950488016887242097 is approximate. You have no choice. You must use the square root of 2 approximation.

In addition, most problems do not require 31 decimal places. The square root of 2 is usually rounded to 1.4142. The rounded number is another approximation.

A lot of time in school is spent teaching students how to round up approximate numbers.

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 to calculate this distance.

d = square[h(D+h)]

ï¿½ d = distance to horizon

ï¿½ D = Diameter of Earth

ï¿½ h = height of the observer above sea level

ï¿½ R = Radius of Earth

Using an approximation, this formula can be reduced to:

d = 3.6 * square (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 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?…

For example, consider surveys of likely voters conducted before an election.

Suppose 56.5% of likely voters favor candidate A + or -3%. This is a CUTOUT, which means that the actual number of voters who prefer candidate A is somewhere between 53.5% and 59.5% (AVERAGE 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 high accuracy by trial and error.

The problem is that these approximations often require so many trials that manual calculation is not practical.

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

The trial-and-error approach to solving is important in mathematics, physics, chemistry, electrical engineering, and other fields.

Calculating the square root of five is a simple example of using trial and error.

By trial and error: 1) find the square root of five; 2) multiply your guess times to see how close the results multiplied by five are.

Repeat steps 1) and 2) over and over again until you reach the desired level of accuracy.

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