Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Writing Determine whether \(e=\frac{271,801}{99,990}\). Justify your answer.

Step-by-Step Solution

Answer

After calculating the value of the given fraction \(\frac{271,801}{99,990}\) and comparing it with the value of \(e\) (approximately 2.71828), it's found that they are approximately equal and hence we can state that \(e\) equals to \(\frac{271,801}{99,990}\).

Step 1: Evaluate the Fraction

Calculate the value of the fraction \(\frac{271,801}{99,990}\) using a calculator or any other mathematics assisting tool.

Step 2: Compare with e

Now, compare the decimal value you obtained from the division with the constant \(e\). Remember, \(e\) is approximately 2.71828.

Step 3: Justify your conclusion

If both the numbers match approximately, you can conclude that \(e=\frac{271,801}{99,990}\), otherwise, conclude they do not equal.

Key Concepts

Fraction Evaluation

Fractions are one of the fundamental concepts in mathematics, representing the division of two integers. In our exercise, we are tasked with evaluating the fraction \(\frac{271,801}{99,990}\). Evaluating a fraction means finding its decimal representation. This is typically done by performing a division where the numerator (the top number), 271,801, is divided by the denominator (the bottom number), 99,990.
Once the division is performed, you will obtain a decimal number. This decimal number helps us understand the value of the fraction in a more familiar form, particularly useful when making comparisons with other numbers. In practical terms, you can use a calculator for quick and accurate results.

• Numerator: 271,801
• Denominator: 99,990
• Perform division to find decimal form

Understanding the value that a fraction represents is crucial, especially in exercises where fractions are used to approximate other mathematical constants or values.

Decimal Approximation

Decimal approximation is an important tool when working with mathematical expressions and fractions. When you evaluate a fraction and obtain a decimal, this is known as a decimal approximation. In our example, after calculating \(\frac{271,801}{99,990}\), you will receive a decimal result.
Decimal approximations are particularly useful when comparing large numbers or irrational numbers, like mathematical constants that have infinite decimal expansions. It's about simplifying complex expressions into numbers that are easier to understand and work with.
In this exercise, after obtaining the decimal approximation for \(\frac{271,801}{99,990}\), the next step is to compare this number to a mathematical constant. Remember that decimal approximations can sometimes vary slightly depending on rounding and the precision of your calculations.

• Convert fraction to decimal
• Use approximations for complex comparisons
• Check against known constants

By understanding decimal approximations, you gain the ability to more easily compare and contrast different mathematical expressions with known values.

Mathematical Constants

Mathematical constants are specific numbers that arise naturally in mathematical calculations. They include well-known numbers like \(\pi\), the golden ratio, and Euler’s number, denoted by \(e\). In our exercise, Euler’s number is of central importance.
Euler's number, \(e\), is approximately 2.71828 and it is an irrational number used widely in mathematics, particularly in calculus involving exponential growth or decay. To determine if \(e\) equals the fraction \(\frac{271,801}{99,990}\), you'll compare \(e\)'s approximate value to the decimal evaluation of the fraction.
It’s important to note that due to its infinite decimal expansion, \(e\) cannot be exactly represented as a simple fraction. However, fractions can provide close approximations.

• Euler’s number is approximately 2.71828
• Irreducible and irrational
• Used in complex calculations

By comparing fractions with known constants, we can better understand the relationships and differences between various mathematical entities.