Every Rational Number is a Real Number

Rational numbers and real numbers are two fundamental concepts in mathematics. While they may seem distinct, there is a deep connection between them. In this article, we will explore the relationship between rational and real numbers, and demonstrate why every rational number is indeed a real number.

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. They can be positive, negative, or zero. Examples of rational numbers include 1/2, -3/4, and 5/1.

Rational numbers can be represented on a number line, where each point corresponds to a specific value. For example, the rational number 1/2 would be located halfway between 0 and 1 on the number line.

Defining Real Numbers

Real numbers, on the other hand, encompass all rational and irrational numbers. Unlike rational numbers, real numbers cannot always be expressed as fractions. They include numbers such as √2, π (pi), and e (Euler’s number).

The set of real numbers is infinite and continuous. It spans the entire number line, from negative infinity to positive infinity. Real numbers can be represented as points on a number line, just like rational numbers.

The Inclusion of Rational Numbers in Real Numbers

Now that we have a basic understanding of rational and real numbers, let’s explore why every rational number is also a real number. This can be demonstrated through the concept of decimal representation.

Every rational number can be expressed as a terminating or repeating decimal. For example, the rational number 1/2 can be written as 0.5, which terminates. Similarly, the rational number 1/3 can be written as 0.333…, which repeats indefinitely.

Decimal representation is a way to express rational numbers as real numbers. Since real numbers encompass all rational numbers, it follows that every rational number is a real number.

Examples of Rational Numbers as Real Numbers

Let’s consider a few examples to further illustrate the concept:

  • The rational number 2/3 can be expressed as the repeating decimal 0.666…
  • The rational number -5/4 can be expressed as the terminating decimal -1.25
  • The rational number 7 can be expressed as the terminating decimal 7.0

In each of these examples, the rational number is represented as a real number through its decimal representation.

Q&A

Q: Are all real numbers rational?

A: No, not all real numbers are rational. Real numbers include both rational and irrational numbers. Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal representations.

Q: Can a rational number be negative?

A: Yes, rational numbers can be positive, negative, or zero. The sign of a rational number is determined by the sign of its numerator and denominator.

Q: Are integers rational numbers?

A: Yes, integers are rational numbers. Integers can be expressed as fractions with a denominator of 1. For example, the integer 5 can be written as 5/1.

Q: Can every real number be expressed as a fraction?

A: No, not every real number can be expressed as a fraction. Irrational numbers, such as √2 and π, cannot be written as fractions. They have non-repeating, non-terminating decimal representations.

Q: Are there more rational numbers or real numbers?

A: The set of real numbers is larger than the set of rational numbers. While rational numbers can be expressed as fractions, real numbers include both rational and irrational numbers, making the set of real numbers larger.

Summary

In conclusion, every rational number is indeed a real number. Rational numbers can be expressed as fractions and can also be represented as points on a number line. Real numbers encompass all rational numbers, as well as irrational numbers. By representing rational numbers as decimal representations, we can clearly see their inclusion in the set of real numbers. Understanding the relationship between rational and real numbers is essential in various mathematical applications and provides a foundation for further exploration in the field of mathematics.

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