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Chapter 10: Problem 46

Determine whether each number is rational, irrational, or not a real number. If a number is rational, give its exact value. If a number is irrational, give a decimal approximation to the nearest thousandth. Use a calculator as necessary. See Examples 4 and 10. $$ \sqrt{33} $$

Short Answer

Expert verified
Irrational, approximately 5.745

Step by step solution

01

- Define Rational and Irrational Numbers

Rational numbers can be expressed as a fraction of two integers (e.g., \(\frac{a}{b}\) where b 鈮� 0). Irrational numbers cannot be expressed as a simple fraction and their decimal form is non-repeating and non-terminating.
02

- Analyze the Number

The number given is \(\sqrt{33}\). To determine if it is rational or irrational, check if 33 is a perfect square. If 33 were a perfect square, its square root would be an integer.
03

- Check if 33 is a Perfect Square

The integer part of the square root of 33 lies between 5 and 6 (since \(\sqrt{25} = 5\) and \(\sqrt{36} = 6\)). Since 33 is not a perfect square, \(\sqrt{33}\) is irrational.
04

- Calculate the Decimal Approximation

Use a calculator to find a decimal approximation of \(\sqrt{33}\). The approximation to the nearest thousandth is 5.745.

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Key Concepts

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

square root
When we talk about the square root of a number, we are looking for a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Mathematically, this is expressed as \( \sqrt{9} = 3 \). Calculating square roots can become tricky if the number isn鈥檛 a perfect square. In our exercise, we dealt with the square root of 33. Since 33 isn't a perfect square, we need to find its decimal approximation instead.
perfect square
A perfect square is a number that is the product of an integer multiplied by itself. Examples include 1 (1 * 1), 4 (2 * 2), 9 (3 * 3), 16 (4 * 4), and so forth. It becomes easier to work with these numbers because their square roots are always whole numbers. In our exercise, \( \sqrt{33} \) isn't a perfect square because no integer multiplied by itself gives 33. This fact points us to the conclusion that \( \sqrt{33} \) is irrational.
decimal approximation
When we can't find the exact value of a square root (since it isn't a perfect square), we use a calculator to find a decimal approximation. A decimal approximation gives us an idea of how close we can get to the actual value. To approximate \( \sqrt{33} \), a calculator yields 5.745. This means that \( \sqrt{33} \) is very close to 5.745 but is not exactly that number. For practical purposes, this approximation is useful and helps us understand the size of \( \sqrt{33} \).
fraction
Fractions represent rational numbers, which can be written as the ratio of two integers. For instance, \( \frac{1}{2} = 0.5 \), \( \frac{3}{4} = 0.75 \), and even larger integers like \( \frac{22}{7} \). These fractions can either terminate or repeat in their decimal form. In our exercise, we determined that \( \sqrt{33} \) couldn't be expressed as a fraction due to its non-repeating, non-terminating decimal form, branding it as an irrational number.

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Most popular questions from this chapter

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