Problem 58 - Algebra A Combined Function

Question

Determine whether each statement is true or false.Every rational number is also a real number.

Step-by-Step Solution

Answer

True, every rational number is a real number.

Step 1: Understanding Rational Numbers

A rational number is any number that can be expressed as the quotient \( \frac{a}{b} \) of two integers, where \( a \) is the numerator and \( b \) is the non-zero denominator. This includes integers, fractions, and terminating or repeating decimals.

Step 2: Understanding Real Numbers

A real number is any number that can be found on the number line. This includes both rational numbers and irrational numbers. Irrational numbers are numbers that cannot be expressed as fractions of integers, such as \( \sqrt{2} \) and \( \pi \).

Step 3: Comparing Rational and Real Numbers

Rational numbers are a subset of real numbers since they can be located on the number line. As real numbers include all rational and irrational numbers, it follows that every rational number must also be a real number.

Step 4: Conclusion

Since rational numbers are a subset of real numbers, the statement 'Every rational number is also a real number' is true.

Key Concepts

Rational Numbers

Rational numbers are vital components of the real number system. A rational number is defined as any number that can be expressed as the fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This means:

• Integers like 3 and -2 are rational because they can be written as \( \frac{3}{1} \) and \( \frac{-2}{1} \) respectively.
• Fractions like \( \frac{1}{2} \) and \( \frac{-5}{3} \) are rational numbers as well.
• Terminating decimals such as 0.75 (which is \( \frac{3}{4} \)) and repeating decimals like 0.333... (which is \( \frac{1}{3} \)) are also rational.

Rational numbers play an essential role in mathematics because they fill the gaps between whole numbers, offering diverse values that can be used for precise calculations in everyday life and science.

Irrational Numbers

Irrational numbers, while less common than rational numbers, are equally significant. These numbers cannot be perfectly expressed as fractions of two integers. Some well-known examples include:

• \( \pi \), the ratio of a circle's circumference to its diameter, is an irrational number. It cannot be represented exactly by any fraction.
• \( \sqrt{2} \), the square root of 2, is also irrational. It results from trying to identify exactly two numbers that multiply to give 2, yet its decimal form is non-repeating and non-terminating.

Recognizing irrational numbers is important because they remind us that not all numbers can be neatly represented as fractions. These numbers often appear in geometry, calculus, and complex scientific formulas, reflecting nature's unpredictability.

Number Line

The concept of a number line is fundamental in understanding real numbers. The number line is a straight infinite line that visually represents numbers from \(-\infty\) to \(+\infty\). On it, we can plot:

• Integers such as -3, 0, and 7, which have fixed positions.
• Rational numbers, like fractions and decimals, positioned according to their value between integers.
• Irrational numbers, which can be located at specific points despite their complex non-repeating decimal forms.

The number line serves as a practical tool for visualizing and comparing different types of numbers; it shows their respective locations along a continuous scale. This makes it easier for students to see how rational and irrational numbers relate to one another within the broader category of real numbers.