What is irrational numbers?
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Irrational numbers are numbers that cannot be expressed as a fraction of two integers. In other words, they are non-repeating and non-terminating decimal numbers. Examples of irrational numbers include √2 (approximately 1.41421356), π (approximately 3.14159), and e (approximately 2.71828).
To understand how to identify irrational numbers, it is important to know that rational numbers are numbers that can be expressed as a fraction of two integers. For example, 1/2, 5/7, and -3/4 are all rational numbers because they can be written as a ratio of two integers.
To determine if a number is irrational, there are a few methods you can use:
1. Square Root Method: Take the square root of the number. If the square root is a non-repeating, non-terminating decimal, then the original number is irrational. For example, √2 is an irrational number because its square root is approximately 1.41421356.
2. Decimal Representation: If a number's decimal representation goes on forever without repeating, it is likely an irrational number. For example, π has a decimal representation of 3.14159 and it never repeats or terminates.
3. Rational Approximation: Another approach is to approximate a number using rational numbers. If a number cannot be expressed as a fraction with a finite number of digits, it is irrational. For example, e is approximately 2.71828, and you cannot find two integers that can represent it exactly.
These methods can help you identify irrational numbers, but it's important to note that there are infinitely many irrational numbers between any two rational numbers.