Problem 81 - Calculus Early Transcendentals: Pearson New International Edition

Question

Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) \(0.375\) (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)

Step-by-Step Solution

Answer

(a) Rational, (b) Rational, (c) Rational, (d) Irrational.

Step 1: Determine if (a) is Rational or Irrational

The square root of 9 is 3, so \(-\sqrt{9} = -3\). A rational number is any number that can be expressed as a quotient of two integers. Since -3 can be written as \(-\frac{3}{1}\), it is a rational number.

Step 2: Determine if (b) is Rational or Irrational

The number 0.375 is a terminating decimal. Terminating decimals can be expressed as a fraction, \(\frac{375}{1000}\), which can be simplified to \(\frac{3}{8}\). Therefore, 0.375 is a rational number.

Step 3: Determine if (c) is Rational or Irrational

Multiply the terms: \(3 \sqrt{2} \times 5 \sqrt{2} = 15 \times 2 = 30\). Since 30 can be written as \(\frac{30}{1}\), it is a rational number.

Step 4: Determine if (d) is Rational or Irrational

Expand the expression \(1+\sqrt{3})^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\): \(= (1)^2 + 2(1)(\sqrt{3}) + (\sqrt{3})^2\), giving \(1 + 2\sqrt{3} + 3\), which equals \(4 + 2\sqrt{3}\). Because the term \(2\sqrt{3}\) is irrational, the entire expression is irrational.

Key Concepts

Square Roots

Square roots play a fundamental role in mathematics. They ask the simple question: "Which number, when multiplied by itself, gives the original number?" For example, the square root of 9 is 3, because 3 × 3 = 9. This concept is essential to determine if numbers like the \(-\sqrt{9}\) are rational or irrational.
When working with square roots, we often encounter perfect squares. Those are numbers like 1, 4, 9, 16, etc., whose square roots are integers.

• If the square root results in an integer, the original number is usually a perfect square.
• Square roots of non-perfect squares are usually irrational numbers because they don't have a simple fraction equivalent.

Understanding square roots helps in correctly identifying the nature of a number in any calculation.

Terminating Decimals

Terminating decimals are decimals that have a finite number of digits after the decimal point. An example found in the exercise is 0.375. It is easy to understand why they are considered rational numbers.

• These decimals can always be expressed as fractions. For example, 0.375 converts to the fraction \(\frac{375}{1000}\), which further simplifies to \(\frac{3}{8}\).
• Because they end after a certain number of decimal places, it is straightforward to convert them into a fraction using powers of 10.

When you know a decimal is terminating, you can quickly determine it as a rational number because it can be precisely expressed as a fraction.

Irrational Numbers

Irrational numbers cannot be written as simple fractions. They are numbers that have non-repeating, non-terminating decimal expansions.
An example from the exercise is the expression \(1+\sqrt{3})^2\). Expanding this results in a mix of whole numbers and irrational components like 2\(\sqrt{3}\), ensuring the entire expression stays irrational.

• Irrational numbers include quantities like \(\pi\) and \(e\), as well as square roots of non-perfect squares.
• They often appear in expressions involving roots of numbers that aren't square integers, resulting in a decimal that goes on forever without forming a repeatable pattern.

Understanding when a number cannot be expressed as a fraction is crucial for identifying irrational numbers.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers. This means they can either be a whole number, a terminating decimal, or a repeating decimal.

• For example, \(-3\) from \(-\sqrt{9}\) and 30 are rational because they can be written as \(-\frac{3}{1}\) and \(\frac{30}{1}\).
• Any decimal that ends or repeats is also rational, as it can be converted to a fractional form.

Rational numbers are everywhere in daily life, from simple fractions like halves and quarters to prices and measurements.