Is -5/7 A Rational Number

Is -5/7 a Rational Number? A Deep Dive into Rational Numbers

Is -5/7 a rational number? The short answer is: yes. But understanding why requires a deeper exploration of what constitutes a rational number. This article will not only definitively answer this question but will also provide a comprehensive understanding of rational numbers, their properties, and how to identify them. We'll cover the definition, examples, and even delve into some related mathematical concepts. By the end, you'll be able to confidently identify rational numbers and understand their significance in mathematics.

Understanding Rational Numbers

Before we determine if -5/7 is a rational number, let's solidify our understanding of what a rational number actually is. 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. The key here is that both the numerator (p) and the denominator (q) must be whole numbers (positive or negative, including zero for the numerator), and the denominator cannot be zero. The reason for excluding a zero denominator is that division by zero is undefined in mathematics.

Let's break this down further:

• Integers: Integers are whole numbers, including positive numbers (like 1, 2, 3...), negative numbers (like -1, -2, -3...), and zero (0).

• Fraction: A fraction represents a part of a whole. It's expressed as a ratio of two numbers, the numerator and the denominator.

Therefore, any number that can be written in the form of a fraction where both the numerator and denominator are integers (and the denominator isn't zero) is a rational number.

Examples of Rational Numbers

To further illustrate the concept, let's examine some examples of rational numbers:

• 1/2: This is a classic example. Both 1 and 2 are integers, and the denominator is not zero.

• -3/4: Negative numbers are allowed in rational numbers. Both -3 and 4 are integers.

• 5: The integer 5 can be expressed as a fraction: 5/1. Both 5 and 1 are integers. All integers are rational numbers.

• 0: Zero can also be expressed as a fraction: 0/1. This fits the definition of a rational number.

• 0.75: This decimal can be written as the fraction 3/4, satisfying the definition. All terminating and repeating decimals are rational numbers.

• -2.666...: This repeating decimal represents -8/3. While it appears complex, it can still be expressed as a fraction of two integers.

Why -5/7 is a Rational Number

Now, let's get back to our original question: Is -5/7 a rational number? The answer is a resounding yes. Here's why:

• -5 is an integer: It's a negative whole number.
• 7 is an integer: It's a positive whole number.
• The denominator (7) is not zero: This is crucial for the fraction to be defined.

Since -5/7 satisfies all the conditions of the definition of a rational number, it is, without a doubt, a rational number.

Irrational Numbers: The Counterpoint

To fully appreciate rational numbers, it's helpful to understand their counterpart: irrational numbers. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They have decimal representations that neither terminate nor repeat. Famous examples include:

• π (pi): The ratio of a circle's circumference to its diameter, approximately 3.14159... The digits continue infinitely without repeating.

• e (Euler's number): The base of natural logarithms, approximately 2.71828... Similar to pi, its decimal representation is non-terminating and non-repeating.

• √2 (the square root of 2): This number cannot be expressed as a simple fraction. Its decimal approximation is approximately 1.41421...

Distinguishing Between Rational and Irrational Numbers

The key difference between rational and irrational numbers lies in their ability to be expressed as a fraction of two integers. If a number can be written in the form p/q (where p and q are integers, and q ≠ 0), it's rational. If it cannot, it's irrational. This distinction is fundamental in many areas of mathematics.

Real Numbers: The Bigger Picture

Both rational and irrational numbers together form the set of real numbers. Real numbers encompass all the numbers we typically use in everyday life and in most mathematical applications. The real number line includes all rational and irrational numbers, stretching infinitely in both positive and negative directions.

Decimal Representations: A Practical Approach

While the fraction definition is mathematically precise, understanding decimal representations can be helpful in identifying rational numbers in practice.

• Terminating Decimals: These decimals have a finite number of digits after the decimal point (e.g., 0.25, -3.7). All terminating decimals can be expressed as fractions and are therefore rational.

• Repeating Decimals: These decimals have a sequence of digits that repeats infinitely (e.g., 0.333..., 1.272727...). These repeating decimals can also be converted into fractions and are rational.

• Non-Terminating, Non-Repeating Decimals: These are the hallmarks of irrational numbers. The digits continue infinitely without any repeating pattern.

Converting Repeating Decimals to Fractions

Converting repeating decimals to fractions may seem challenging, but there's a systematic approach. Let's illustrate with an example:

Convert 0.333... to a fraction:

1. Let x = 0.333...
2. Multiply both sides by 10: 10x = 3.333...
3. Subtract the first equation from the second: 10x - x = 3.333... - 0.333...
4. Simplify: 9x = 3
5. Solve for x: x = 3/9 = 1/3

This demonstrates that the repeating decimal 0.333... is equivalent to the fraction 1/3, confirming its rational nature. A similar method can be used for other repeating decimals.

Frequently Asked Questions (FAQ)

Q1: Can a rational number be negative?

A1: Yes, absolutely. Both the numerator and the denominator can be negative integers, resulting in a negative rational number. For example, -5/7, -2/3, and -1 are all rational numbers.

Q2: Is every fraction a rational number?

A2: Yes, provided that the numerator and the denominator are integers, and the denominator is not zero.

Q3: Are all integers rational numbers?

A3: Yes. Every integer can be expressed as a fraction with a denominator of 1 (e.g., 5 = 5/1, -2 = -2/1).

Q4: Can a rational number be expressed as a decimal?

A4: Yes. All rational numbers can be represented as either terminating or repeating decimals.

Q5: How can I tell if a decimal is rational or irrational?

A5: If the decimal terminates or repeats, it's rational. If it continues infinitely without repeating, it's irrational.

Q6: What is the significance of rational numbers?

A6: Rational numbers are fundamental to many areas of mathematics and science. They are used extensively in calculations, measurements, and problem-solving across various fields.

Conclusion

In conclusion, -5/7 is definitively a rational number because it fulfills all the criteria: it's a fraction, its numerator (-5) and denominator (7) are both integers, and its denominator is not zero. This article not only answered the initial question but also provided a comprehensive exploration of rational numbers, their properties, and their relationship to irrational and real numbers. By understanding the fundamental principles of rational numbers, you can confidently identify them and appreciate their significance in the broader landscape of mathematics. Remember the key: a rational number is any number expressible as a fraction p/q, where p and q are integers, and q ≠ 0.