Kummer said:[tex]\log_2 5 = a/b[/tex] so [tex]2^{a/b} = 5 \implies 2^a = 5^b[/tex]. Now use unique factorization.
Kummer said:[tex]\log_2 5 = a/b[/tex] so [tex]2^{a/b} = 5 \implies 2^a = 5^b[/tex]. Now use unique factorization.
The proof involves using a proof by contradiction method. Assume that Log2 of 5 is rational and can be expressed as a fraction of two integers. Then use the properties of logarithms to show that this leads to a contradiction.
Proving that Log2 of 5 is irrational is important because it helps to establish the irrationality of the number 2, which is a fundamental number in mathematics. It also helps to show that not all numbers can be expressed as fractions, which expands our understanding of numbers and their properties.
Yes, the same proof by contradiction method can be used to show that Logb of a is irrational for any base b and positive integer a that is not a perfect power of b. This is known as the Gelfond-Schneider theorem.
While the proof itself may not have direct practical applications, the concept of irrational numbers is used extensively in various fields of mathematics, science, and engineering. Understanding the irrationality of Log2 of 5 helps to build a strong foundation for more complex mathematical concepts.
Yes, there are other proofs for the irrationality of Log2 of 5, such as using continued fractions or the prime factorization of numbers. However, the proof by contradiction method is the most commonly used and easiest to understand.