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Rational and irrational numbers are two different types of real numbers in mathematics.

Rational Numbers:

Rational numbers are numbers that can be expressed as a ratio or fraction of two integers, where the denominator is not zero.
They can be written in the form a/b, where a and b are integers, and b is not equal to zero.
Examples of rational numbers include 1/2, -3/4, 5, 0, and any integer.
Irrational Numbers:

Irrational numbers are numbers that cannot be expressed as a simple fraction or ratio of two integers.
They have non-repeating, non-terminating decimal expansions.
Examples of irrational numbers include the square root of 2 (√2), pi (π), and Euler's number (e).
Here are some key differences between rational and irrational numbers:

Representation: Rational numbers can be represented as fractions or ratios of integers, while irrational numbers cannot be represented in this way.

Decimal Expansion: Rational numbers always have finite or repeating decimal expansions, whereas irrational numbers have non-repeating, non-terminating decimal expansions.

Examples: Most integers and fractions are rational numbers, while famous mathematical constants like √2, π, and e are irrational numbers.

Density: There are an infinite number of both rational and irrational numbers, but between any two distinct rational numbers, there are infinitely many irrational numbers.

Operations: When you add, subtract, multiply, or divide two rational numbers, the result is always a rational number. However, performing these operations on irrational numbers may result in either rational or irrational numbers.

Approximations: Irrational numbers are often approximated as decimals or fractions for practical calculations because their exact values cannot be expressed. For instance, π is often approximated as 3.14159, which is a rational approximation.

Understanding the distinction between rational and irrational numbers is fundamental in mathematics, especially in real analysis, algebra, and calculus, where these types of numbers are used extensively.
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