Problem 5
Question
Select the appropriate word to complete each of the following. If \(a\) is a whole number that is not a perfect square, then \(\sqrt{a}\) is a(n) ______ number.
Step-by-Step Solution
Verified Answer
Irrational
1Step 1: Understand whole numbers and perfect squares
A whole number is a number without fractions; an integer that is non-negative. A perfect square is a whole number that can be expressed as the square of an integer. For example, 1, 4, 9, and 16 are perfect squares because they are 1², 2², 3², and 4² respectively.
2Step 2: Recognize non-perfect squares
If a whole number is not a perfect square, it means there is no integer that when squared equals that number. For example, 2, 3, 5, and 6 are not perfect squares.
3Step 3: Determine the nature of \( \sqrt{a} \ \)
For a number \( a \ \) that is not a perfect square, \( \sqrt{a} \ \) cannot be expressed as a simple fraction, thus it is not a rational number. Therefore, \( \sqrt{a} \ \) is an irrational number.
4Step 4: Fill in the blank
Based on the above steps, the appropriate word to complete the sentence is 'irrational'.
Key Concepts
Whole NumbersPerfect SquaresRational NumbersIrrational Numbers
Whole Numbers
Whole numbers are a basic concept in mathematics. They include all the non-negative integers: 0, 1, 2, 3, and so on.
Whole numbers do not include fractions, decimals, or negative numbers.
For example, the numbers 4, 15, and 100 are whole numbers, but 4.5, -1, and 3/4 are not.
Whole numbers do not include fractions, decimals, or negative numbers.
For example, the numbers 4, 15, and 100 are whole numbers, but 4.5, -1, and 3/4 are not.
Perfect Squares
A perfect square is a whole number that can be represented as the product of an integer with itself.
For example, 1 (1²), 4 (2²), 9 (3²), and 16 (4²) are perfect squares.
This concept is important because it differentiates numbers whose square roots are integers from those whose square roots are not.
For example, 1 (1²), 4 (2²), 9 (3²), and 16 (4²) are perfect squares.
This concept is important because it differentiates numbers whose square roots are integers from those whose square roots are not.
Rational Numbers
Rational numbers can be expressed as fractions or ratios of two integers, where the denominator is not zero.
For example, 1/2, 3, and 4.75 are rational numbers because they can be written as 1/2, 3/1, and 19/4 respectively.
In decimal form, rational numbers either terminate or repeat.
For example, 1/2, 3, and 4.75 are rational numbers because they can be written as 1/2, 3/1, and 19/4 respectively.
In decimal form, rational numbers either terminate or repeat.
Irrational Numbers
Irrational numbers cannot be written as simple fractions – their decimal expansions are non-terminating and non-repeating.
Examples include the numbers π (pi) and e, as well as the square roots of non-perfect squares like √2 and √3.
When a whole number is not a perfect square, its square root falls into this category, making it an irrational number.
Examples include the numbers π (pi) and e, as well as the square roots of non-perfect squares like √2 and √3.
When a whole number is not a perfect square, its square root falls into this category, making it an irrational number.
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