Αποτελέσματα Αναζήτησης
Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$ $$(p/q)^2 = 2 = p^2 / q^2 $$ $$\text{ (What was the purpose of squaring √2 and p/q?)}$$
17 Απρ 2022 · (a) Prove that for each reach number \(x\), \((x + \sqrt 2)\) is irrational or \((-x + \sqrt 2)\) is irrational. (b) Generalize the proposition in Part(a) for any irrational number (instead of just \(\sqrt 2\)) and then prove the new proposition.
However, by finding that in 1 place on the number line, there exists a irrational between 2 rationals, we know that the set of rationals is insufficient to cover all distance on the number line.
Sal proves that the square root of 2 is an irrational number, i.e. it cannot be given as the ratio of two integers. Created by Sal Khan.
The sum of any rational number and any irrational number will always be an irrational number. This allows us to quickly conclude that ½+√2 is irrational. Created by Sal Khan.
By scaling and translating you only need to show there is an irrational number between, say, $0$ an $1$. Just show that. If $\alpha$ is irrational then $r \alpha$ is irrational and $n + \alpha$ is irrational, for all rational numbers $r$ and integers $n$.
17 Απρ 2022 · In this section we will prove one of the oldest and most important theorems in mathematics: 2–√ 2 is irrational (see Theorem 6.19). First, we need to know what this means. Definition 6.18. Let r ∈ R r ∈ R. We say that r r is rational if r = m n r = m n, where m, n ∈ Z m, n ∈ Z and n ≠ 0 n ≠ 0 .
