Problem 8 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To two decimal places, the value of \(e\) is ___.

Step-by-Step Solution

Answer

The value of \( e \) to two decimal places is 2.72.

Step 1: Identifying the Value of e

The value of the mathematical constant \( e \) is approximately known and is similar to the constant \( \pi \). It has a non-repeating, non-terminating decimal expansion.

Step 2: Approximating to Two Decimal Places

The approximate value of \( e \) is 2.71828. To find \( e \) expressed to two decimal places, we must round this value.

Step 3: Rounding to Two Decimal Places

To round 2.71828 to two decimal places, look at the third decimal place. Since it is 8, we round the second decimal place up from 1 to 2.

Key Concepts

Understanding the Value of e

The mathematical constant known as "e" is a fundamental number in mathematics, much like the well-known constant \( \pi \). The value of \( e \) plays a crucial role in calculus and complex analysis. It is widely used in exponential growth and decay models, as well as in many formulas involving logarithms.

What sets \( e \) apart from other numbers, however, is its unique properties in terms of growth rates. It serves as the base for the natural logarithm, which is defined as the logarithm to the base \( e \).

In simpler terms, \( e \) is the number whose natural logarithm equals 1. This property of \( e \) is what makes it special in mathematics. The continued fraction representation of \( e \) is also simple and unique, contributing to its significance. While it's approximately 2.71828, this decimal goes on indefinitely without a repeating pattern.

Decimal Expansion of e

The term "decimal expansion" refers to the representation of a number in a decimal form, where the sequence of digits continues after the decimal point. The value of \( e \) has a non-repeating and non-terminating decimal expansion. This means that when \( e \) is expressed as a decimal, it goes on forever without repeating any sequence of numbers.

Because of this property, \( e \) is classified as an irrational number. Irrational numbers cannot be exactly represented as fractions, and their decimal expansions are infinite and non-repetitive.

This non-repetitive nature of their decimal expansion makes every digit significant. For practical purposes, the value of \( e \) is usually approximated to a certain number of decimal places depending on the precision required for the calculation.

• The first few decimal places of \( e \) are 2.71828.
• However, there are many ways to compute \( e \), including series expansions and continued fractions.

Approach to Rounding Numbers

Rounding numbers is a common mathematical technique used to simplify numbers while still keeping them close to their original value. This involves reducing the number of digits right of the decimal when absolute precision isn't necessary.

To round a number, focus on the digit to the right of the desired place. For this exercise, we rounded the value of \( e \) to two decimal places. Here's a quick guide on how to do it:

• Identify the digit in the spot you want to round to. For \( e \), this is the second decimal place, which is 1 in 2.71828.
• Look at the next digit to the right (the third decimal place). In the case of \( e \), the digit is 8.
• If this digit is 5 or greater, you round the digit up. Here, 8 is greater than 5, which means we increase the 1 to a 2.
• If this digit is less than 5, you keep the digit the same.

This procedure helps maintain accuracy in calculations while making numbers more manageable. As a result, the value of \( e \), when rounded to two decimal places, is 2.72.