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TopNewestMy text book named additional mathematics quoted that a and b must be real numbers for all convenience you mentioned. Basically, imaginary numbers or complex numbers posses many possibilities while staying in a function but ought to be an only value when they are not. Example: x^2 = -1, x can be + j or - j but Sqrt (-1) or j itself cannot choose to own both values. Another example: e^x = -1, x can be -3 Pi j, - Pi j, Pi j, 3 Pi j, 5 Pi j and etc. Here, we define principal value for Ln -1 which is j Pi.

Complex number undoubtedly have been studied by many people including mathematicians of course and it is found to be something important that unable to be denied by people. Without complex number, there are many logical outcomes that cannot be explained and topics remained incomplete. The most significant example is solution to x^3 + p x + q = 0. We may refuse quadratic formula when j appears, but we cannot do the same for cubic formula.

When we realize that something ought to survive, we shall have thousands of reasons to prove them right; on the other hand, we could have ten thousands of reasons instead just to reject them. Towards the surviving path, we ought to study and understand the feature of complex number rather than to go for ways that may find them some flaws.

When we do in such a way that j suddenly becomes -j, we ought to realize its nature and also the meaning of a principal value. In other words, we shall find NO TROUBLE at all when we don't do in a way that purposely try to reveal their nature of many values. Sincerely, I personally find that complex numbers are as true as they can be.

Just try our best to cope with their features. Then, we shall find that Complex Number is the way towards all truth.

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Indeed. A lot of times, we memorize rules without recalling what are the conditions under which it holds. In this case, as you pointed out, the rule only applies for real numbers, and cannot be applied to complex numbers.

This is why in Rules of Exponents, we added in a warning about when these rules hold:

Other common instance of forgetting conditions is applying AM-GM to negative numbers.

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I meant not all convenience are applicable if not real number, not meant for not applicable to complex number. I concluded from analysis that when there comes with fractional index with simplest co-prime, example (-1)^(5/ 3) or (-1)^(7/ 2), always take the effect of its denominator before numerator. Without list of black and white one by one, we just need to emphasize that we conclude according to whatever reasonable. I think 0^0 = 1 should be included in the list. As its limit tells, 0^0 = 0^(1 - 1) = 0/ 0 doesn't deny the fact for it to be 1. 0/ 0 includes 1 but generally something instead of invalid. Indeterminate is just something but we cannot know one of them in general to satisfy particular need to be sole definable value. Thanks anyway for providing the list for me to do some revision and thinking.

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thank you sir Lu Chee Ket for helping me to clear my doubt and also thanks to calvin lin.

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Proof by contradiction

I can write -

\( i^2 = \sqrt{-1}\times\sqrt{-1}\)

According to you ,

\( i^2 = \sqrt{-1}\times\sqrt{-1} = \sqrt{-1\times-1} = 1\) - this is a contradiction as \(i^2 = -1\)

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yes exactly i've the confusion with the contradiction.

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Good for doubting. This is the spirit that we should request to ourselves. If you truly prove it incorrect, then I personally feel happy to admit the fact. This should be our correct attitude.

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