Linked Questions
154 questions linked to/from Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
21
votes
9
answers
6k
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 $-i$?
- 345
44
votes
9
answers
10k
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 ...
- 575
27
votes
9
answers
4k
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{...
- 481
2
votes
3
answers
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 ...
- 458
6
votes
5
answers
439
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}^...
- 352
6
votes
3
answers
11k
views
What is $\sqrt{-4}\sqrt{-9}$? [duplicate]
I assumed that since $a^c \cdot b^c = (ab)^{c}$, then something like $\sqrt{-4} \cdot \sqrt{-9}$ would be $\sqrt{-4 \cdot -9} = \sqrt{36} = \pm 6$ but according to Wolfram Alpha, it's $-6$?
- 2,023
3
votes
2
answers
2k
views
What's wrong with this proof $1=i^2=-1$ [duplicate]
I just thinking what $i^{i}$ should be, arrived at a quiet awkward thing.
So this was that awkward thinking :
Let $ i^{i} = a$
$(i^{i})^{2i} = a^{2i}$
$i^{-2}=a^{2i}$
$-1=a^{2i}$
Now if we take power ...
- 353
4
votes
3
answers
590
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:
$$
\begin{align}
1+1 &= 1 + \sqrt{1} \\
&= 1 + \sqrt{1 \times 1} ...
- 683
0
votes
2
answers
4k
views
What did I do wrong? 1 = √1 = √(-1)(-1) = √(-1) √(-1) = i.i = i² = -1 [duplicate]
I'm a simple man living my life and enjoying mathematics now and then. Today during lunch my friend asked me about complex numbers and $i$. I told him what I knew and we went back to work.
After work ...
- 3,842
-2
votes
3
answers
541
views
6
votes
1
answer
3k
views
Simple Complex Number Problem: $1 = -1$ [duplicate]
Possible Duplicate:
-1 is not 1, so where is the mistake?
I'm trying to understand the exact point of failure in the following reasoning:
\begin{equation*}
1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{...
- 717
-1
votes
3
answers
2k
views
Square root multiplication [duplicate]
$\sqrt {-3}$ multiplied by $\sqrt{-3}$ is $-3$. But this can also be written as $\sqrt {-3} \cdot \sqrt{-3} = \sqrt {(-3).(-3)}= \sqrt{9} =3$
So my question is why is this not possible?
- 91
0
votes
2
answers
3k
views
Paradox - minus one equals one using square roots [duplicate]
I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows:
$$ \sqrt{-1} = \sqrt{-1} $$
$$ \...
2
votes
4
answers
214
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=\...
- 125
2
votes
4
answers
323
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+-...
- 79