Solution: Step 1: Express the number 1 as a fraction with the same denominator as frac1e. This gives us fracee - frac1e. Step 2: Merge the numerators while keeping the denominator the same, resulting in frace - 1e. Step 3: Present the final answer in its various acceptable forms. In its exact form, the answer is frace - 1e. If we convert it to decimal form, it is approximately 0.63212055 ldots. Knowledge Notes: To solve the problem of evaluating 1 - frac1e, we need to understand the following concepts: 1. Common Denominator: When subtracting fractions, it's necessary to have a common denominator. This allows us to combine the fractions by simply subtracting their numerators. 2. The Number e: The number e is an important mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is used in many areas of mathematics, including calculus and complex analysis. 3. Fraction Subtraction: When subtracting fractions with common denominators, we subtract the numerators and keep the denominator the same. 4. Exact vs Decimal Form: An exact form of a number is the form that represents the number without any approximation, often as a fraction or algebraic expression. The decimal form is an approximate value of the number, usually rounded to a certain number of decimal places. 5. LaTeX Formatting: To properly display mathematical expressions, LaTeX formatting is used. For instance, to write a fraction, we use the fracnumeratordenominatorsyntax. By applying these concepts, we can evaluate the expression 1 - frac1e by first expressing 1 with a common denominator of e, then subtracting the numerators, and finally presenting the result in both exact and decimal forms.

Evaluate 1-1/e

Based on the given problem, the question "Evaluate 1-1/e" is asking for a calculation to be performed where the number one (1) is subtracted by the reciprocal of the mathematical constant e. The mathematical constant e (approximated as 2.71828) is the base of the natural logarithm and is a fundamental constant in calculus, particularly in the context of exponential functions and differential equations. The question requires the use of this constant to find the difference between 1 and 1/e, resulting in a numerical value.

$1 - \frac{1}{e}$

Answer

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Solution:

Step 1:

Express the number $1$ as a fraction with the same denominator as $\frac{1}{e}$. This gives us $\frac{e}{e} - \frac{1}{e}$.

Step 2:

Merge the numerators while keeping the denominator the same, resulting in $\frac{e - 1}{e}$.

Step 3:

Present the final answer in its various acceptable forms. In its exact form, the answer is $\frac{e - 1}{e}$. If we convert it to decimal form, it is approximately $0.63212055 \ldots$.

Knowledge Notes:

To solve the problem of evaluating $1 - \frac{1}{e}$, we need to understand the following concepts:

Common Denominator: When subtracting fractions, it's necessary to have a common denominator. This allows us to combine the fractions by simply subtracting their numerators.
The Number $e$: The number $e$ is an important mathematical constant approximately equal to $2.71828$. It is the base of the natural logarithm and is used in many areas of mathematics, including calculus and complex analysis.
Fraction Subtraction: When subtracting fractions with common denominators, we subtract the numerators and keep the denominator the same.
Exact vs Decimal Form: An exact form of a number is the form that represents the number without any approximation, often as a fraction or algebraic expression. The decimal form is an approximate value of the number, usually rounded to a certain number of decimal places.
LaTeX Formatting: To properly display mathematical expressions, LaTeX formatting is used. For instance, to write a fraction, we use the $\frac{numerator}{denominator}$syntax.

By applying these concepts, we can evaluate the expression $1 - \frac{1}{e}$ by first expressing 1 with a common denominator of $e$, then subtracting the numerators, and finally presenting the result in both exact and decimal forms.