Math Problem Statement

Place the numbers below in order from least to greatest. Which number will go in the yellow box? The numbers are: -√2, √3, 1.5, -1.6, -√3.

Solution

Let's solve the problem step-by-step.

The task is to order the numbers from least to greatest and determine which number goes into the yellow box. The numbers given are:

2,3,1.5,1.6,3-\sqrt{2}, \quad \sqrt{3}, \quad 1.5, \quad -1.6, \quad -\sqrt{3}

Step 1: Approximate the square roots.

  • 21.414\sqrt{2} \approx 1.414
  • 31.732\sqrt{3} \approx 1.732

Thus:

  • 21.414-\sqrt{2} \approx -1.414
  • 31.732\sqrt{3} \approx 1.732
  • 31.732-\sqrt{3} \approx -1.732

Step 2: Rewrite the numbers using their approximate values.

  • 21.414-\sqrt{2} \approx -1.414
  • 31.732\sqrt{3} \approx 1.732
  • 1.51.5 stays as it is.
  • 1.6-1.6 stays as it is.
  • 31.732-\sqrt{3} \approx -1.732

Step 3: Order the numbers from least to greatest.

From least to greatest: 1.732(-3),1.6,1.414(-2),1.5,1.732(3)-1.732 \, (\text{-}\sqrt{3}), \quad -1.6, \quad -1.414 \, (\text{-}\sqrt{2}), \quad 1.5, \quad 1.732 \, (\sqrt{3})

Step 4: Assign the numbers to the colored boxes.

  • Blue: 31.732-\sqrt{3} \approx -1.732 (Least)
  • Red: 1.6-1.6
  • Yellow: 21.414-\sqrt{2} \approx -1.414 (Answer for the yellow box)
  • Green: 1.51.5
  • Orange: 31.732\sqrt{3} \approx 1.732 (Greatest)

Final Answer:

The number that goes in the yellow box is:

2-\sqrt{2}


Do you want further details or have any other questions?
Here are 5 related questions to deepen your understanding:

  1. How do you find exact square roots for non-perfect squares like 2\sqrt{2}?
  2. What are the properties of irrational numbers like 2\sqrt{2} and 3\sqrt{3}?
  3. How do negative numbers compare with positive ones in terms of size?
  4. What are other real-life applications of ordering numbers?
  5. How can you visualize irrational numbers on a number line?

Tip: When comparing square roots, it's helpful to memorize common approximations such as 21.414\sqrt{2} \approx 1.414 and 31.732\sqrt{3} \approx 1.732.

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Math Problem Analysis

Mathematical Concepts

Irrational Numbers
Ordering Real Numbers
Square Roots

Formulas

√2 ≈ 1.414
√3 ≈ 1.732

Theorems

Properties of Irrational Numbers
Ordering of Negative and Positive Numbers

Suitable Grade Level

Grades 7-10