Linked Questions

22
votes
9answers
4k views

Why is $i^3$ (the complex number “$i$”) equal to $-i$ instead of $i$? [duplicate]

$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i $$ Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?
41
votes
9answers
6k views

Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
28
votes
9answers
3k views

$1/i=i$. I must be wrong but why? [duplicate]

$$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{-1} = i$$ I know this is wrong, but why? I often see people making simplifications such as $\frac{\sqrt{...
1
vote
3answers
1k views

Why does $2+2=5$ in this example? [duplicate]

I stumbled across the following computation proving $2+2=5$ Clearly it doesn't, but where is the mistake? I expect that it's a simple one, but I'm even simpler and don't really understand the ...
6
votes
5answers
372 views

Confused with imaginary numbers [duplicate]

In 9th grade I had an argument with my teacher that ${i}^{3}=i$ where $i=\sqrt{-1}$ But my teacher insisted (as is the accepted case) that: ${i}^{3}=-i$ My Solution: ${i}^3=(\sqrt{-1})^3$ ${i}^...
4
votes
3answers
546 views

$1 +1$ is $0$ ?​ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? $i^2$ why is it $-1$ when you can show it is $1$? So: $$ \begin{align} 1+1 &= 1 + \sqrt{1} \\ &= 1 + \sqrt{1 \times 1} ...
0
votes
2answers
2k views

What did I do wrong? 1 = √1 = √(-1)(-1) = √(-1) √(-1) = i.i = i² = -1 [duplicate]

I'm a simple man living my life and enjoying mathematics now and then. Today during lunch my friend asked me about complex numbers and $i$. I told him what I knew and we went back to work. After work ...
6
votes
1answer
2k views

Simple Complex Number Problem: $1 = -1$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? I'm trying to understand the exact point of failure in the following reasoning: \begin{equation*} 1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{...
2
votes
4answers
270 views

imaginary number $i$ equals $-6/3.4641$? [duplicate]

$$-4^3 = -64$$ so the third root of $-64$ should be $-4$ than. $$\sqrt[3]{-64} = -4$$ But if you calculate the third root of -64 with WolframAlpha( http://www.wolframalpha.com/input/?i=third+root+of+-...
-2
votes
3answers
202 views

What is the square root of $(-5)^2$? [duplicate]

Is this statement true? $$\sqrt{(-5)^2} = -5$$
-1
votes
3answers
973 views

Square root multiplication [duplicate]

$\sqrt {-3}$ multiplied by $\sqrt{-3}$ is $-3$. But this can also be written as $\sqrt {-3} \cdot \sqrt{-3} = \sqrt {(-3).(-3)}= \sqrt{9} =3$ So my question is why is this not possible?
2
votes
4answers
185 views

Is $i$ equal to $-i$? [duplicate]

When I was in high school, I learned about $i$ in math class and I remember asking my teacher back then if $i$ was equal to $-i$ according to the simple following development : \begin{equation} i=\...
0
votes
2answers
2k views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ \...
1
vote
1answer
647 views

Possible fake proof of $1= -1$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? Simple Complex Number Problem: 1 = -1 Well, I remembered this after having Algebra II a year ago, is it possible that this is a valid ...
2
votes
3answers
237 views

What's wrong with this demonstration? (1 = -1) [duplicate]

What's wrong with this demonstration?: $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{...

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