Irrational numbers ( a ) , ( b ) exist that make ( a^b ) rational.
Prove with law of excluded middle and without fixing what ( a ) and ( b ) are.
Proof
Supporse ( x = sqrt{2} ^ sqrt{2} ) .
Case: ( x ) is Rational
( sqrt{2} ) is irrational, so ( a = b = sqrt{2} ) make ( a^b ) rational. ( ( sqrt{2} ) が無理数であることの証明は 証明: ( sqrt{2} ) は無理数 をご覧ください。)
Case: ( x ) is Irrational
begin{eqnarray*}
sqrt{2} & = & (sqrt{2}^sqrt{2})^sqrt{2}
& = & sqrt{2}^{sqrt{2} cdot sqrt{2}}
& = & sqrt{2}^2
& = & 2
end{eqnarray*}
Then, ( a = x = sqrt{2}^sqrt{2} ) and ( b = sqrt{2} ) are irrational and make ( a^b ) rational.
Above all, there are irrational numbers ( a ) and ( b ) that make ( a^b ) rational.
Law of Excluded Middle
形式論理学の用語で、ある命題についてその肯定と否定とがある場合、一方が真ならば他方は偽、他方が真ならば一方は偽であり、その両方のどちらでもない状態は存在しないというもの。 命題 ( P ) について ( P lor lnot P ) が必ず真となることをいう。
How much is the summation from 1 to 100?
ガウスがこの計算を即座にやってのけたという話はあまりに有名で、整数を順次足す、等差数列を順次足す方法はご存じの方も多いと思います。
よく使われる説明の方法として、石を三角の形に置き、それと点対象な三角の形に石を置くというのがあります。
Continue reading Summation of \( k ^ n \) Lead the Formula →
Sorry, this entry is only available in 日本語 .
I prove that infinite decimal of rational number is recurring decimal. It is decimal of rational number but is not finite decimal.
While the process of division may seem intuitive, I will provide a more concrete explanation as a formal proof.
Feeling
For example, when 89 is sequentially divided by 13, including decimal places, the remainders are as follows:
\begin{eqnarray*}
89 \div 13 & = & 6 \; \textrm{remainder is} 11 \\
110 \div 13 & = & 8 \; \textrm{remainder is} 6 \\
60 \div 13 & = & 4 \; \textrm{remainder is} 8 \\
80 \div 13 & = & 6 \; \textrm{remainder is} 2 \\
20 \div 13 & = & 1 \; \textrm{remainder is} 7 \\
70 \div 13 & = & 6 \; \textrm{remainder is} 5 \\
50 \div 13 & = & 3 \; \textrm{remainder is} 11 \\
110 \div 13 & = & 8 \; \textrm{remainder is} 6
\end{eqnarray*}
On the 7th division, the same remainder as in the 1st division appeared again. Since there are only 12 possible remainers when divided by 13, ranging from 1 to 12, if the division doesn’t result in a perfect quotient after 13 divisions, it becomes evident that at some point, the same remainder would recur, leading to a repeating decimal.
Continue reading Proof: Infinite Decimal of Rational Number is Recurring Decimal →
Let’s assume the definition of rational numbers is known. Real numbers that are not rational are called irrational numbers.
Here, we will prove that \( \sqrt{2} \) is an irrational number.
Continue reading Proof: \( \sqrt{2} \) is an irrational number →
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