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The number $w$ is irrational. $e$ is the base of the natural logarithm. Which statement about $w + e$ is true? $\newline$ Choices: $\newline$ (A) $w + e$ is rational. $\newline$ (B) $w + e$ is irrational. $\newline$ (C) $w + e$ can be rational or irrational, depending on the value of $w$ .

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Properties of operations on rational and irrational numbers

Full solution

Q. The number

w

is irrational.

e

is the base of the natural logarithm. Which statement about

w + e

is true?

\newline

Choices:

\newline

(A)

w + e

is rational.

\newline

(B)

w + e

is irrational.

\newline

(C)

w + e

can be rational or irrational, depending on the value of

w

.

Identify Type of $e$ : Identify whether $e$ is a rational or irrational number. $\newline$ $e$ is the base of the natural logarithm and is known to be an irrational number.
Properties of Irrational Numbers: Consider the properties of irrational numbers. The sum of an irrational number and a rational number is irrational. The sum of two irrational numbers can be either rational or irrational, depending on the specific numbers.
Analysis of Expressions: Analyze the given expressions. $\newline$ $w$ is an irrational number. $\newline$ $e$ is an irrational number. $\newline$ We need to determine the nature of $w + e$ .
Consider Specific Cases: Consider specific cases. $\newline$ If $w = -e$ , then $w + e = 0$ , which is rational. $\newline$ If $w$ is any irrational number other than $-e$ , then $w + e$ is irrational because the sum of two non-identical irrational numbers is irrational.
Conclusion: Conclude the statement about $w + e$ . $w + e$ can be rational or irrational, depending on the value of $w$ .

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