Irrational Number vs. Rational Number

Attribute	Irrational Number	Rational Number
Type	Irrational numbers cannot be expressed as a ratio of two integers.	Rational numbers can be expressed as a ratio of two integers.
Representation	Irrational numbers are represented by non-repeating and non-terminating decimals.	Rational numbers are represented by either terminating or repeating decimals.
Examples	√2, π, e	1/2, 3/4, 0.25
Cardinality	The set of irrational numbers is uncountably infinite.	The set of rational numbers is countably infinite.

Definition

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. This means that rational numbers can be written in the form a/b, where a and b are integers. On the other hand, irrational numbers are numbers that cannot be expressed as a simple fraction. They are non-repeating and non-terminating decimals. Examples of irrational numbers include the square root of 2 and pi.

Representation

Rational numbers can be represented in various forms, such as fractions, decimals, and percentages. They can also be plotted on a number line. Irrational numbers, on the other hand, cannot be represented as fractions or ratios of integers. They are typically represented as decimals that go on forever without repeating. For example, the decimal representation of the square root of 2 is 1.41421356...

Properties

Rational numbers have some unique properties that distinguish them from irrational numbers. One key property of rational numbers is that they can be written as a terminating or repeating decimal. This means that the decimal representation of a rational number will either end or repeat after a certain point. Irrational numbers, on the other hand, have decimal representations that go on forever without repeating.

Operations

When performing operations on rational numbers, such as addition, subtraction, multiplication, and division, the result will always be a rational number. This is because rational numbers closed under these operations. However, when irrational numbers are involved in operations with rational numbers, the result will be an irrational number. For example, the sum of √2 and 1 is an irrational number.

Examples

Some examples of rational numbers include 1/2, 3/4, and -5. These numbers can be expressed as fractions and have finite decimal representations. On the other hand, examples of irrational numbers include √2, √3, and π. These numbers cannot be expressed as fractions and have non-repeating decimal representations.

Real Numbers

Both rational and irrational numbers are considered real numbers. Real numbers include all rational and irrational numbers, as well as integers and whole numbers. Real numbers can be plotted on a number line, with rational numbers evenly spaced between integers. Irrational numbers are located in between rational numbers on the number line.

Applications

Rational numbers are commonly used in everyday life for measurements, calculations, and comparisons. For example, when measuring ingredients for a recipe or calculating the total cost of items at a store, rational numbers are used. Irrational numbers are often used in mathematics and science to represent quantities that cannot be expressed as simple fractions, such as the circumference of a circle or the diagonal of a square.

Conclusion

In conclusion, rational and irrational numbers have distinct attributes that set them apart from each other. While rational numbers can be expressed as fractions and have finite decimal representations, irrational numbers cannot be written as simple fractions and have non-repeating decimal representations. Both types of numbers are essential in mathematics and have various applications in real-world scenarios.