In this  at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why.   Why is this so? What is the intuition behind it?
70189
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?  This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.
59805
According to the  :  If the points are labeled sequentially in the counterclockwise direction, then the area is positive, if they are labeled in the clockwise direction, the area will be negative.  Can anyone please explain why the area is negative, with the change in the order of the points? How does this work? I want to know the logic behind this.
91168
What would the graph of $(-1)^{x}$ look like?   I know that the value of the function alternates between $1$ and $-1$ when it is defined so I think it would just be points spread over the lines $x=1$ and $x=-1$.   Is this correct? Also, will there be any definite pattern of the points.
34341
What's wrong with this : $$e^{i\pi} = -1$$ $$\therefore e^{2i\pi} = 1$$ $$\therefore \log \left( e^{2i\pi} \right ) = \log(1) = 0$$ $$\therefore 2i\pi = 0$$
122424
This is an equation given in my book.     The question is why have they used a negative sign on the LHS?   Now, if you try to derive the mirror equation with simple geometry, you get  1/v +1/u =1/f . I got that equation without applying any signs there!  So what is the need to apply the signs ‘here'?   If i try to derive this equation ( the one in the photo ) i get the exact answer, but with the positive sign in the LHS. WHY we use the negative sign in the derivation, and then  again we use conventions while solving the questions?
63730
"" (number $5$).  I'm looking at this, and i'm not sure what the $\equiv$ means in this instance? I know the rest of the values, but what does $\equiv$ mean right now?  Thanks in advance!
61068
Say a function's domain is $\{-\pi/2, \pi/2\}$. How should I interpret this interval?  It starts from where? To where? In what direction?  And if it starts from $3\pi/2$, would the next one be $-5\pi/3$.(because it starts from negative, $-\pi/2$).
27406
What is the $\sqrt{i^4}$?   $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?  Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?  When I plug it into my TI-89 Titanium, I get $1$.  Edit: I now see why $i^4 \neq -1$. $$\sqrt{x^2} = \left|{x}\right| $$ So$$\sqrt{(i^2)^2} = \left|i^2\right| = \left|-1\right| = 1$$
5833
I simply have no idea how this result is found $$I=\hbar  \sqrt{\ell(\ell+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that $I_z=\hbar m_j$. Please explain as clearly as possible how this is derived.  Also, any pointers towards any books/videos that may help me learn quantum mechanics would be appreciated.
18259
Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as $x = x + 1$, or $x = 1/0$?  Edit: OK, a lot of people have said that a number $x$ such that $x = x + 1$ would break the rule that $0 \neq 1$. However, let's look at the extension from whole numbers to include negative numbers (yes, I said that I wasn't going to include this) by defining $-1$ to be the number such that $-1 + 1 = 0$. Note that this breaks the "rule" that "if $x \leq y$, then $ax \leq ay$", which was true for all $a, x, y$ before the introduction of negative numbers. So I'm not convinced that "That would break some obvious truth about all numbers" is necessarily an argument against this sort of thing.
114176
I know that  $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$  But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
54133
Find the value of $\large i^{i^{.^{.^.}}}$ ?  How should we start to solve it ?  Also you can see  if it helps.  Thanks
43217
Where is the mistake? $$-1=(-1)^{2/2}=\left(\left (-1\right)\displaystyle ^2\right)^{1/2}=1^{1/2}=\sqrt 1=1$$
92838
The common argument against the existence of a total order on complex numbers that observes the known axioms on real numbers is that 0 &lt; i or 0 &lt; -i. But why? 0 is not even an imaginary number.  There's a possibility of treating ni &lt; -oo for all n in the set of natural numbers, such that we have the real line on the right and the imaginary line on the left, where a comparison operation compares the imaginary part before the real part.  The thing is, both negative numbers and imaginary numbers are not scalars that satisfy the usual axioms of multiplication. Why must imaginary numbers satisfy what even negative numbers cannot satisfy? In order to establish the 2-D complex plane, we just have to see the set of complex numbers as a cross product of 2 sets of real numbers, where the imaginary part comes before the real part, the opposite way of writing a complex number conventionally, namely, ai + b. Well, we've been using cross products to establish multiple dimensions for a long time. I don't see a problem here.  ------------ Appendix ------------  Recall that imaginary numbers are derived to be more negative than negative numbers in the first place, namely, i i = -1 and (-1)(-1) = 1. So, it makes intuitive sense to treat them as numbers less than negative infinity. However, I forgot that they also cancel out like real numbers, namely, i + -i = 0 i = 0. The key is to not treat 0 i = 0. The problem is that they are designed to be like real numbers, except for the i sign, which is similar to the - sign.  In any case, I was trying to define an INumber interface, but not being able to compare numbers is really inconvenient. It means that they cannot participate in many known formal verification and optimization techniques. The reason why I ask this question here is that if mathematicians can agree on this definition, it will be easy to adopt it in ISO, IEC and IEEE standards. Of course, I can define anything I want, but who will use my definition? If the new definition works, we will have safer and faster computer programs. It makes perfect economic sense.  I know I can simply try not to use complex numbers at all, but why invent them in the first place? Plus, it's really convenient to express wave functions in terms of complex numbers. Nowadays, what is not a wave in physics and engineering? By the way, we mostly deal with equations and not inequalities or any sort. So, it's not as meaningful as I pitch it here. Anyways.  I will work out the addition requirements before posting another question. Thank you all for your participation.
114348
Problem Statement:- The number of zeroes at the end of the integer   $$100!-101!+\ldots-109!+110!$$   I am having a bit of a trouble in thinking how do I proceed. A little push in the right direction would be appreciated.   And if you are posting a full solution do use the spoiler tag, as sometimes I cant stop myself from seeing the whole solution and lose the chance of thinking through it by myself with a push in the right direction from you guys.  Also, I dont know whether I am using the right tag feel free to correct it if its wrong.
22469
Context: Rings.     Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?   Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
85874
Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it simply, you don't understand it well enough” so how would I explain it.