Which Of The Following Is Rational Number

Which of the Following is a Rational Number? A Comprehensive Guide

Meta Description: Learn to easily identify rational numbers! This guide explains what rational numbers are, provides examples, and helps you distinguish them from irrational numbers. Master this fundamental math concept today.

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. Understanding this definition is key to identifying rational numbers amongst a set of numbers. This article will break down this concept, providing clear examples and explanations to help you confidently determine which numbers are rational.

Understanding Rational Numbers

The core concept revolves around the ability to represent a number as a fraction of two whole numbers. This means the number can be precisely expressed as a ratio. Let's look at some characteristics:

• Integers are Rational: All whole numbers (positive, negative, and zero) are rational. For example, 5 can be written as 5/1, -3 as -3/1, and 0 as 0/1.
• Fractions are Rational: Any number that can be expressed as a fraction, where both the numerator and denominator are integers (and the denominator is not zero), is rational. Examples include 1/2, 3/4, -2/5.
• Terminating Decimals are Rational: Decimals that end after a finite number of digits are rational. For example, 0.75 can be written as 3/4, and 0.2 as 1/5.
• Repeating Decimals are Rational: Decimals that have a repeating pattern of digits are also rational. For example, 0.333... (0.3 recurring) is equivalent to 1/3, and 0.142857142857... (142857 recurring) is 1/7.

Identifying Irrational Numbers (The Counterpart)

To better understand rational numbers, it's helpful to understand what they are not. Irrational numbers cannot be expressed as a fraction of two integers. They have decimal representations that neither terminate nor repeat.

• Examples of Irrational Numbers: The most famous irrational number is π (pi), approximately 3.14159..., which continues infinitely without repeating. Another well-known example is the square root of 2 (√2), approximately 1.414..., which also continues infinitely without repeating. The Euler's number, e, is another example.

Examples: Which are Rational?

Let's examine some examples to solidify our understanding. Determine which of the following numbers are rational:

1. 5: Rational (can be written as 5/1)
2. -2/3: Rational (already in fraction form)
3. √9: Rational (√9 = 3, which can be written as 3/1)
4. π: Irrational
5. 0.666... (0.6 recurring): Rational (equals 2/3)
6. 0.125: Rational (equals 1/8)
7. √2: Irrational
8. -7: Rational (can be written as -7/1)

Practical Applications

Recognizing rational numbers is a fundamental skill in various mathematical fields, including algebra, calculus, and number theory. It's also crucial in many scientific and engineering applications where precise calculations are needed.

Conclusion

Identifying rational numbers comes down to understanding the definition: a number that can be expressed as a fraction of two integers. By remembering this definition and the characteristics discussed above (integers, fractions, terminating and repeating decimals), you can confidently determine whether a number is rational or irrational. Practice identifying rational and irrational numbers to strengthen your understanding of this fundamental mathematical concept.