5-1-3 Using Estimation and Approximation
Key Concepts
1. **Estimation**: Making a reasonable guess or approximation of a value without performing an exact calculation.
2. **Approximation**: Finding a value that is close enough to the correct answer, often used to simplify calculations.
3. **Rounding**: Adjusting a number to the nearest multiple of 10, 100, or other specified value to simplify calculations.
4. **Significant Figures**: The number of digits considered to be accurate in a measurement or calculation.
Detailed Explanation
Estimation
Estimation involves making a reasonable guess of a value based on available information. It is useful when an exact calculation is not necessary or when the exact value is unknown. For example, estimating the number of people in a crowd by observing the area and density.
Approximation
Approximation is finding a value that is close enough to the correct answer, often used to simplify complex calculations. For example, approximating the value of π (pi) as 3.14 instead of using its exact value of 3.141592653589793.
Rounding
Rounding involves adjusting a number to the nearest multiple of 10, 100, or other specified value to simplify calculations. For example, rounding 47 to the nearest ten gives 50, and rounding 123 to the nearest hundred gives 100.
Significant Figures
Significant figures are the digits in a number that are considered accurate. They include all non-zero digits and any zeros that are between non-zero digits or at the end of the number. For example, the number 123.45 has five significant figures, and 0.00456 has three significant figures.
Examples
Example 1: Estimating the cost of groceries.
Suppose you have items priced at $2.99, $4.49, and $1.79. You can estimate the total cost by rounding each price to the nearest dollar: $3 + $4 + $2 = $9.
Example 2: Approximating the area of a circle.
If the radius of a circle is 7 cm, you can approximate the area using π ≈ 3.14: Area ≈ 3.14 × 7² ≈ 3.14 × 49 ≈ 153.86 cm².
Example 3: Rounding to significant figures.
Round 123456 to three significant figures: The number becomes 123000.
Analogies
Think of estimation as guessing the number of jellybeans in a jar without counting each one. Approximation is like using a map scale to estimate the distance between two cities. Rounding is similar to adjusting the volume on a radio to the nearest whole number. Significant figures are like the important digits in a phone number that you need to remember.
Practical Application
Understanding estimation and approximation is essential in various fields such as science, engineering, and finance. It helps in making quick calculations, simplifying complex problems, and ensuring accuracy within acceptable limits.