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Real Numbers – Definition With Examples

Jo-ann Caballes

Updated on January 11, 2024

Welcome to Brighterly, your trusted source for making mathematics a delightful learning experience for children of all ages! Real numbers might sound complex at first, but at Brighterly, we break down even the most intricate concepts into digestible lessons. In this comprehensive guide, we will dive into the intriguing world of real numbers. Whether you are a curious student eager to explore the mathematical universe or a parent seeking to provide your child with quality education, you will find our guide on real numbers to be both informative and engaging. From understanding what real numbers are to exploring their properties, differences, and representations, we’ve got it all covered. So fasten your seatbelts as we embark on this mathematical adventure with Brighterly!

What Are Real Numbers?

Real numbers encompass both rational numbers and irrational numbers. In simple terms, any number that can be located on the number line is considered a real number. This includes all positive numbers, negative numbers, and zero. For example, numbers like 2, -5, and 22​ are all real numbers. Unlike imaginary numbers, real numbers are tangible and can be related to quantities in the physical world.

Definition of Rational Numbers

Rational numbers are those that can be expressed as a ratio or fraction of two integers, where the denominator is not zero. They can be either positive or negative. Examples include 1221​, −34−43​, and 5 (since 5 can be written as 5115​).

Definition of Irrational Numbers

Irrational numbers, on the other hand, cannot be expressed as a simple fraction or ratio. They continue indefinitely without repeating. Examples include $\pi$ and √2​. These numbers often appear in geometrical and trigonometrical contexts. 

Properties of Rational Numbers

The rational numbers have their unique set of properties, including:

• Density Property: Between any two rational numbers, there exists another rational number.
• Closure Property: The sum, difference, and product of any two rational numbers are also rational.

Properties of Irrational Numbers

Irrational numbers have properties such as:

• Non-repeating and Non-terminating: Irrational numbers never repeat or terminate.
• Not Expressible as a Ratio: No irrational number can be expressed as a simple ratio of two integers.

Difference Between Rational and Irrational Numbers

The difference between rational and irrational numbers can be summarized as follows:

• Rational Numbers: Can be expressed as a ratio; they may terminate or repeat.
• Irrational Numbers: Cannot be expressed as a ratio; they do not terminate or repeat.

Representation of Real Numbers on the Number Line

Real numbers can be represented on a number line, with rational numbers having specific points and irrational numbers filling the gaps. For example, 1221​ is halfway between 0 and 1, while √2​ is an irrational point on the line.

Writing Rational Numbers in Decimal and Fraction Forms

Rational numbers can be converted between decimal and fraction forms. A repeating decimal like 0.333… can be expressed as 1331​, and a terminating decimal like 0.5 can be written as 1221​.

Writing Irrational Numbers in Decimal Form

Irrational numbers can be approximated in decimal form but will not terminate or repeat. For example, $\pi$ is approximately 3.14159.

Practice Problems on Real Numbers

1. Identify whether the number 2.875 is rational or irrational.
2. Express the repeating decimal 0.666… as a fraction.
3. Represent 33​ on the number line.

Conclusion

As we wrap up this enlightening journey through the land of real numbers, we hope that you’ve found the answers to all your questions and more. At Brighterly, our mission is to light up the world of mathematics for children, making complex ideas accessible and fun. We believe that understanding real numbers isn’t just about solving equations but about nurturing a love for logical thinking and problem-solving. We encourage you to explore our other resources and practice problems to solidify your grasp of real numbers. And remember, Brighterly is here for you, making mathematics brighter, one concept at a time!

Frequently Asked Questions on Real Numbers

What are complex numbers?

Complex numbers consist of two parts: a real part and an imaginary part. Unlike real numbers, complex numbers include an imaginary component, usually denoted by $i$, which is the square root of -1. At Brighterly, we offer further lessons on complex numbers, so don’t hesitate to explore this fascinating subject further!

Is zero a rational number?

Yes, zero is considered a rational number. A rational number is defined as any number that can be written as the ratio of two integers, with the denominator not being zero. In the case of zero, it can be expressed as 0/1. This is a great example of how mathematics can often surprise us, and at Brighterly, we aim to make these surprises exciting learning opportunities.

How do irrational numbers differ from rational numbers?

Rational numbers can be expressed as a simple ratio or fraction of two integers, while irrational numbers cannot. Rational numbers either terminate or have a repeating decimal pattern. In contrast, irrational numbers continue indefinitely without any repeating pattern. The exploration of rational and irrational numbers is a fascinating aspect of mathematics, and Brighterly’s resources are designed to guide you through these concepts in an engaging way.

Can every real number be represented on a number line?

Absolutely! Every real number, whether rational or irrational, can find its place on the number line. Rational numbers can be pinpointed exactly, while irrational numbers fill in the gaps. Understanding how numbers are represented on a number line can greatly enhance spatial awareness and conceptual understanding of numbers. At Brighterly, we provide interactive tools and lessons to help visualize these concepts.

Jo-ann Caballes

6 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.