Hello. This is β (Beta). This time, I'd like to write about proof by contradiction. I'll be quoting from "Proving the world's most beautiful mathematical formula, e^(iπ)=-1".
"√2 is not an irrational number."
√2 = m/n ...① (m and n have no common divisors other than 1). Multiply both sides by n and square.
2n² = m² ...②. m² is a multiple of 2. If m is not a multiple of 2, then m² cannot be a multiple of 2.
m is a multiple of 2.
m = 2k (where k is an integer) ...③. Substitute this into ②.
2n² = 4k². n² = 2k², and n² is a multiple of 2.
n is also a multiple of 2...④. From ③ and ④.
m and n have 2 as a common divisor. This contradicts statement ①.
√2 is an irrational number. (Quoted from the above book)
From here on, these are my ideas. In conclusion, the point that m and n have no common divisors other than 1 is that m and n are relatively prime, that is, m and n are prime numbers or relatively prime composite numbers. In other words, from here on, we will consider cases other than when m and n are prime numbers or other cases where they have no common divisors other than 1.
Furthermore, if √2 can be a rational number, then √2 = (uk)/(2t²). In other words, if √2 can be a rational number, then √2 = composite number / (2 x squared expression).
The assumption that m and n have no common divisors other than 1 is based on the assumption that m and n are prime numbers, isn't it? Now, if we replace n and m with N and M, and set N=n/t and M=m/u, then by proof by contradiction we can see that √2=(uk)/(2t²) is a rational number equal to √2.
The equation √2=m/n assumes that both the numerator and denominator are prime numbers. However, if √2 is a rational number, then we can see that uk is a composite number and 2t² is a squared expression. The method for finding √2=(uk)/(2t²) will be explained in detail later. (Life Lessons).
Find the value of √2 as a continued fraction close to (uk)/(2t²). (Visual confirmation)
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