Problem 83 Is $\pi$ greater than or less ... [FREE SOLUTION]

91影视

Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 8: Problem 83

Is $\pi$ greater than or less than $\sqrt{6}$ ?

Short Answer

Expert verified

$\pi$ is greater than $\sqrt{6}$.

Step by step solution

Understand the Comparison

To determine whether $\pi$ is greater than or less than $\sqrt{6}$, we need to compare their numerical values.

Determine the Value of \pi\

The value of $\pi$ is approximately 3.14159.

Calculate the Value of \sqrt{6}\

Calculate $\sqrt{6}$. Using a calculator, we find that $\sqrt{6}$ is approximately 2.44949.

Compare the Values

Now that we have the approximate values: $\pi \approx 3.14159$ and $\sqrt{6} \approx 2.44949$, we can compare them. Since 3.14159 > 2.44949, we conclude that $\pi$ is greater than $\sqrt{6}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

pi (蟺)

Pi, denoted as $ \pi $, is a special irrational number representing the ratio of a circle's circumference to its diameter. This ratio is constant for all circles. The value of $ \pi $ is approximately 3.14159 but extends infinitely without repeating. Because $ \pi $ is irrational, its decimal representation is non-terminating and non-repeating, making it unique. In various mathematical contexts, $ \pi $ is crucial, such as in geometry, trigonometry, and calculus.
When comparing $ \pi $ to other numbers, we often use its approximate value, 3.14159, to make numerical comparisons simpler.

square root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented with the radical symbol (鈭�). In our exercise, we are concerned with $ \sqrt{6} $, which is the square root of 6. To find this, we look for a number that, when squared, equals 6.
The value of $ \sqrt{6} $ is approximately 2.44949. Just like $ \pi $, $ \sqrt{6} $ is also an irrational number, meaning its decimal representation is endless and non-repeating. Despite being irrational, knowing the first few decimal places of $ \sqrt{6} $ provides a good approximation for many practical purposes.

numerical comparison

Comparing numerical values is a fundamental skill in mathematics. To compare $ \pi $ and $ \sqrt{6} $, we converted them to their approximate decimal forms: $ \pi $ being about 3.14159 and $ \sqrt{6} $ about 2.44949.
The comparison becomes straightforward when both numbers are in their decimal forms. We can directly observe that 3.14159 is greater than 2.44949. Therefore, $ \pi $ is greater than $ \sqrt{6} $. This step-by-step approach is useful in making clear and accurate comparisons, especially when dealing with irrational numbers.

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91影视

Is \(\pi\) greater than or less than \(\sqrt{6}\) ?

Short Answer

Step by step solution

Understand the Comparison

Determine the Value of \pi\

Calculate the Value of \sqrt{6}\

Compare the Values

Key Concepts

pi (蟺)

square root

numerical comparison

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