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Chapter 2: Problem 4

Is every rational number a real number?

Short Answer

Expert verified
Explain briefly. Answer: Yes, every rational number is a real number. This is because rational numbers can be expressed as a fraction and have a specific location on the number line, which is the main characteristic of real numbers.

Step by step solution

01

Understand Rational Numbers

Rational numbers are numbers that can be expressed as a fraction (thus, a ratio) of two integers, where the numerator is an integer and the denominator is a non-zero integer. In other words, a rational number can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q\) is not equal to 0. Examples of rational numbers include 1, -5, \(\frac{1}{2}\) and \(\frac{-7}{3}\).
02

Understand Real Numbers

Real numbers are the numbers that can be found on the number line, which includes both rational and irrational numbers. All integers, fractions, and decimals (both repeating and non-repeating) are real numbers. In simple terms, real numbers include all numbers which can be assigned a precise position on the number line. Irrational numbers are those that cannot be expressed as a simple fraction, like the square root of a non-perfect square or transcendental numbers such as \(\pi\) or \(e\). These numbers are also real numbers.
03

Determine the Relationship Between Rational and Real Numbers

As stated earlier, real numbers include both rational and irrational numbers. This means that all rational numbers must fall within the set of real numbers; they have a precise location on the number line.
04

Conclusion

Yes, every rational number is a real number. This is because rational numbers can be expressed as a fraction, having a specific location on the number line, which is the main characteristic of real numbers.

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