考慮 a^b = c。
若 a 為無理數， b 為有理數，那麼 c 為有理數的例子多的是。
可是，若 a 為有理數， b 為無理數，那麼有 c 為有理數的例子嗎？
更新1:
真抱歉，我要改改題： 若 a 為‘大於 1’的有理數， b 為無理數，那麼有 c 為有理數的例子嗎？
～～無理數次方～～
2006-12-17 7:24 am
一條比較簡單的問題：
考慮 a^b = c。
若 a 為無理數， b 為有理數，那麼 c 為有理數的例子多的是。
可是，若 a 為有理數， b 為無理數，那麼有 c 為有理數的例子嗎？
考慮 a^b = c。
若 a 為無理數， b 為有理數，那麼 c 為有理數的例子多的是。
可是，若 a 為有理數， b 為無理數，那麼有 c 為有理數的例子嗎？
回答 (4)
2006-12-17 3:23 pm
✔ 最佳答案
Take 2^b =3 .b is irrational otherwise there exits non-zero m and n such that 2^m = 3^n.
b is irrational and hence b and 1 are linearly independent over rational.
According to the (Theorem 2.4) below, b is not algebraic irrational otherwise the number 3 is transcendental !!!
This means that b is transcendental and hence b is not rational number.
2006-12-17 07:34:06 補充：
SORRY{{ Take 2^b =3 .b is irrational otherwise there exits non-zero m and n such that 2^m = 3^n. }}The above statement is enough. Just forget the nonsense following it and please forgive me about that. I have been confusing it with another question.
2006-12-17 07:36:35 補充：
Please forgive me about not deleting this answer.I just want to keep some record for my work even this is very foolish work.
2006-12-23 20:32:20 補充：
It is very lucky to be selected because I answered the question in a very dreadful way.
參考: Transcendental Number Theory, by Alan Baker, p11, (Theorem 2.4)
2007-03-12 6:01 am
有Transcendental Number Theory
的soft copy嗎?
的soft copy嗎?
2006-12-17 7:38 am
若a=1
b是任何數
c都會是1
2006-12-17 00:52:19 補充：
3^0.630929753571457437....=20.630929753571457437....=log2/log3但log2和log3不是有理數
b是任何數
c都會是1
2006-12-17 00:52:19 補充：
3^0.630929753571457437....=20.630929753571457437....=log2/log3但log2和log3不是有理數
收錄日期: 2021-04-23 16:28:55
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