what are irrational numbers?
Irrational numbers are numbers that cannot be written as a fraction. E.g. sqrt(2) cannot be a fraction p/q:
sqrt(2) = p/q --->
p^2/q^2 = 2 --->
p^2 = 2 q^2
Factor both sides in prime factors. The number p has a certain number of factors, p^2 has double that number, so the left hand side contains an even number of prime factors. By the same reasoning q^2 also contains an even number of prime factors, so 2 q^2 contains an odd number of prime factors as 2 is a prime number. But this means that p^2 can never equal 2 q^2!
HURRY PLEASE
Irrational numbers are a type of real number that cannot be expressed as a fraction or a ratio of two integers. They are numbers that go on infinitely without repeating and cannot be written as terminating or repeating decimals. Irrational numbers include famous examples such as the square root of 2 (√2), pi (π), and the golden ratio (φ).
To understand how to identify irrational numbers, you can follow these steps:
1. Rational numbers: Start by understanding rational numbers. These are numbers that can be expressed as the quotient of two integers (e.g., 1/2, 3/4, -5/6). They can be written as terminating decimals (e.g., 0.25) or repeating decimals (e.g., 0.333...).
2. Examples of irrational numbers: Familiarize yourself with some commonly known irrational numbers. Some well-known examples include √2, π (pi), e (the base of natural logarithms), and φ (the golden ratio). These numbers cannot be expressed as a fraction or a ratio of integers.
3. Non-repeating and non-terminating decimals: Another way to identify irrational numbers is by checking if a number has a non-repeating and non-terminating decimal representation. For example, the square root of 2 (√2) is approximately 1.41421356..., and pi (π) is approximately 3.14159265... These decimals continue indefinitely without repeating or terminating.
4. Proof or mathematical calculation: In some cases, proving that a number is irrational requires mathematical proofs or calculations. For example, it was proven that √2 is irrational using a proof by contradiction.
In summary, irrational numbers are real numbers that cannot be expressed as fractions or ratios of integers and have non-repeating and non-terminating decimal representations. By understanding the concept of rational numbers and the characteristics of irrational numbers, you can identify and grasp the concept of irrational numbers.