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Help: Proof that 1 = -1?

I came across something the other day involving complex numbers.

It is obvious that: \( \frac {1}{ -1} = -1 = \frac {-1}{1} \).

If we take square roots on both sides, we get: \( \frac {1}{i} = \frac {i} {1} \).

Multiplying gives \( 1 = 1 \times 1 = i \times i = -1 \).

So, \(1 = -1\)?

I don't understand what is wrong. Can someone help?

Note by Yumin Wang
3 years, 9 months ago

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I believe the problem here is that \(\sqrt{-1}\) is not an ordinary square root, as it is not real and thus it does not follow the same square-rooting rules as real square roots..

\( \sqrt{\frac{x}{y}} \) is not necessarily equal to \( \frac{\sqrt{x}}{\sqrt{y}} \)

Which goes back to the basic precalc premise that \( \sqrt{x}\sqrt{y} \neq \sqrt{xy} \) Tim Ye · 3 years, 9 months ago

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@Tim Ye Why not ? \(\sqrt{\dfrac{x}{y}}\) can be written as \((\dfrac{x}{y})^{\dfrac{1}{2}}\) which is \(\dfrac{x^{\dfrac{1}{2}}}{y^{\dfrac{1}{2}}}\)which is \({\dfrac{\sqrt{x}}{\sqrt{y}}.}\) Sai Ram · 1 year, 1 month ago

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Yes, Tim is right. Notice that \(\sqrt{-4}\sqrt{-9}=(2i)(3i)=-6\), while \(\sqrt{(-4)(-9)}=\sqrt{36}=6\). This is why the premise below works for non-negative real numbers only. Raoul Danniel Manuel · 3 years, 9 months ago

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Yes, as Krishna pointed out, the inverse of the square function is NOT a function, since it can take on 2 values. As such, when we talk about \( \sqrt{x} \), we have to be very certain about which value we are actually referring to. We cannot immediately conclude that \( \sqrt{1} = 1\), since there is the possibility that it is actually \(-1\). It can be impossible to tell from the context, and we have to be aware that only one of the values might make sense.

For exponents, if the exponent is not an integer, there can be ambiguity about what the value actually refers to. However, the laws of indices will still hold. For example, regardless of what value we choose for \( (-1)^{0.5} \), then \( (-1)^ {0.5} \times (-1)^{0.5} = -1\) holds. Calvin Lin Staff · 3 years, 9 months ago

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That is a great observation.

Do you know why we have to make such an exception when it comes to dealing with complex numbers, but not with the non-negative reals?

Hint: \( \sqrt{x} : \mathbb{C} \rightarrow \mathbb{C}\) is interpreted as the inverse of the square function \( x: \mathbb{C} \rightarrow \mathbb{C}, x \rightarrow x^2\). Same is true of \( \sqrt{x} : \mathbb{R} \rightarrow \mathbb{R}\) Calvin Lin Staff · 3 years, 9 months ago

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complex numbers are imaginary but natural numbers are real we cannot deal the imaginary with real no Keshav Kumar · 2 years, 6 months ago

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when the laws of indices were first proved they were proved for postive radicands so as tim says one cannot directly apply them for the negative radicands.also in the books that i have read it is mostly given that a+ib is a complex number where i is the square root of -1 and i square is -1 .so its mostly treated as an axiom Pranav Chakravarthy · 3 years, 9 months ago

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Also, the square roots of 1 are 1, and -1, so it's ambiguous Krishna Kinnal · 3 years, 9 months ago

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