6.7 DETERMINING RATIONAL OR IRRATIONAL

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What is a rational number?

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A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). Examples include \( \frac{1}{2}, 3, -4.5 \).

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What is an irrational number?

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An irrational number is a number that cannot be expressed as a simple fraction. It cannot be written as \( \frac{a}{b} \) where \( a \) and \( b \) are integers. Examples include \( \sqrt{2}, \pi, e \).

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Is the sum of a rational number and an irrational number rational or irrational?

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The sum of a rational number and an irrational number is always irrational.

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Is the product of two irrational numbers always irrational?

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No, the product of two irrational numbers can be rational. For example, \( \sqrt{2} \times \sqrt{2} = 2 \), which is rational.

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What is the result of adding two irrational numbers?

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The sum of two irrational numbers can be either rational or irrational. For example, \( \sqrt{2} + (-\sqrt{2}) = 0 \) (rational), but \( \sqrt{2} + \sqrt{3} \) is irrational.

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What is the definition of a rational expression?

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A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, \( \frac{x^2 + 1}{x - 3} \) is a rational expression.

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Can a rational number be expressed as a decimal?

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Yes, a rational number can be expressed as a decimal that either terminates (e.g., 0.75) or repeats (e.g., 0.333...).

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What is the significance of the square root of a non-perfect square?

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The square root of a non-perfect square is an irrational number. For example, \( \sqrt{3} \) is irrational.

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Is the number 0 a rational number?

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Yes, 0 is a rational number because it can be expressed as \( \frac{0}{1} \).

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What is the product of a rational number and an irrational number?

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The product of a rational number and an irrational number is always irrational. For example, \( 2 \times \sqrt{3} = 2\sqrt{3} \) is irrational.