Questions: Consider the following set of real numbers: [ left-sqrt2,-frac35, 1,1 . overline3, sqrt5, 2.9right ] Which of the following lists ALL of the rational numbers in the set? -sqrt2, 1 . overline3, sqrt5 -frac35, 1,2.9, sqrt5 -frac35, 1,-sqrt2 -frac35, 1,1 . overline3, 2.9

Consider the following set of real numbers: [ left-sqrt2,-frac35, 1,1 . overline3, sqrt5, 2.9right ] Which of the following lists ALL of the rational numbers in the set? -sqrt2, 1 . overline3, sqrt5 -frac35, 1,2.9, sqrt5 -frac35, 1,-sqrt2 -frac35, 1,1 . overline3, 2.9
Transcript text: Consider the following set of real numbers: \[ \left\{-\sqrt{2},-\frac{3}{5}, 1,1 . \overline{3}, \sqrt{5}, 2.9\right\} \] Which of the following lists ALL of the rational numbers in the set? $-\sqrt{2}, 1 . \overline{3}, \sqrt{5}$ $-\frac{3}{5}, 1,2.9, \sqrt{5}$ $-\frac{3}{5}, 1,-\sqrt{2}$ $-\frac{3}{5}, 1,1 . \overline{3}, 2.9$
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Solution

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Solution Steps

To determine which numbers in the given set are rational, we need to identify numbers that can be expressed as a fraction of two integers. Rational numbers include integers, fractions, and repeating decimals.

Solution Approach
  1. Identify each number in the set and determine if it is rational or irrational.
  2. List all the rational numbers.
Step 1: Identify the Set of Numbers

The given set of real numbers is: \[ \left\{-\sqrt{2}, -\frac{3}{5}, 1, 1.\overline{3}, \sqrt{5}, 2.9\right\} \]

Step 2: Classify Each Number

We classify each number in the set as either rational or irrational:

  • \( -\sqrt{2} \) is irrational.
  • \( -\frac{3}{5} \) is rational.
  • \( 1 \) is rational.
  • \( 1.\overline{3} \) (which is equal to \( \frac{4}{3} \)) is rational.
  • \( \sqrt{5} \) is irrational.
  • \( 2.9 \) is rational.
Step 3: List All Rational Numbers

From the classification, the rational numbers in the set are: \[ -\frac{3}{5}, 1, 1.\overline{3}, 2.9 \]

Final Answer

The list of all rational numbers in the set is: \[ \boxed{-\frac{3}{5}, 1, 1.\overline{3}, 2.9} \]

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