Questions and Answers
Which of the following is an example of an irrational number?
All decimal numbers are either rational or irrational.
True
What is the approximate value of π to the hundredths?
3.14
The square root of 3 is approximately ______ when rounded to the tenths.
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Match the following expressions with their approximate values:
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The decimal expansion of an irrational number is non-repeating and non-terminating.
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The square root of 4 is an example of an irrational number.
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It is possible to compare the sizes of irrational numbers using rational approximations.
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An example of a rational approximation for π would be 3.14.
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Cube roots cannot be approximated to the tenths.
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Study Notes
Rational Approximations and Irrational Numbers
- Rational approximations help in estimating the size and location of irrational numbers on a number line.
- Irrational numbers cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions.
Estimating Values
- Square roots and cube roots can be estimated to the tenths for accuracy in comparisons and calculations.
- The value of pi (𝜋) should be approximated to the hundredths for better precision in mathematical contexts.
Characteristics of Numbers
- Every number has a decimal expansion; rational numbers have repeating or terminating decimals.
- Understanding the distinction between rational and irrational numbers is crucial for mathematical reasoning and problem-solving.
Understanding Rational and Irrational Numbers
- Every number can be expressed as a decimal expansion, which may either terminate (rational) or continue infinitely without repeating (irrational).
- An irrational number is defined as a non-repeating, non-terminating decimal.
Estimating Irrational Numbers
- Rational approximations can be used to compare and locate irrational numbers on a number line.
- Key irrational numbers to approximate include:
- Square roots and cube roots: Estimate their values to the tenths.
- Pi (𝜋): Approximate its value to the hundredths.
Number Line Placement
- By estimating irrational numbers, one can effectively place them on a number line between two rational numbers.
- For instance, the approximate value of √2 is about 1.4, and it can be placed between 1 and 2.
Examples of Approximations
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Square roots:
- √3 ≈ 1.7
- √5 ≈ 2.2
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Cube roots:
- ∛5 ≈ 1.7
- ∛8 = 2
- Pi: 𝜋 ≈ 3.14
Application in Problem Solving
- Using these approximations allows for easier comparisons and calculations in various mathematical contexts, especially when exact values are not necessary.
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Description
This quiz focuses on understanding rational and irrational numbers, specifically how to approximate irrational numbers like square roots, cube roots, and pi. Students will practice comparing the size of these numbers and locating them on a number line, while also learning about their decimal expansions.
