Why does any nonzero number to the zeroth power = 1?
Numbers to the Power of Zero
What causes the expansion of space? generic function with a "has property X" constraint? Set normal orientation of origin I want to change the orientation of my object to a specified direction.  Say I have an axle of a train whose normal orientation is set off by 90° from the global orientation:    The green arrow (Y-axis) points to the back of the train (along the global X-axis). But I want it to point to the sides of the train, so all I want is to rotate the orientation by 90° without rotating the object itself.  In other words, I want the normal and global orientation of the axle to be the identical:    Is this possible and if so, how?  Background: The game engine to use this model will move the bogies and the axles according to their orientations. After importing this train into the game the axles rotated 90° off. Why is my iron ore disappearing when mined with a wooden pickaxe? I am on Peaceful mode, and I am using a Wooden Pickaxe. Whenever I mine Iron Ore, it just disappears. Am I doing something wrong? What does the term "BODMAS" mean? What is BODMAS and why is it useful in programming? If I take a sponge and place it in a shallow dish of water (i.e. water level is lower than height of sponge), it absorbs water until the sponge is wet, including a portion of the sponge above the water level. In other words, it seems the sponge pulls some water from the bath up into itself, doing work, and the water gains some gravitational potential energy.  Where does the energy required to do this work come from? My suspicion is that the answer involves physical and/or chemical bonds between the water and the sponge, or possibly the change in the surface area to volume ratio of the water. Examples and further results about the order of the product of two elements in a group Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$?   says "not much":     There is no general formula relating the order of a product $ab$ to the orders of $a$ and $b$. In fact, it is possible that both $a$ and $b$ have finite order while $ab$ has infinite order, or that both $a$ and $b$ have infinite order while $ab$ has finite order.   On the other hand no examples are provided. $(\mathbb{Z},+), 1$ and $-1$ give an example of elements of infinite order with product of finite order. I can't think of any example of the other kind! So:   What's an example of a group $G$ and two elements $a,b$ both of finite order such that their product has infinite order?     Wikipedia then states:     If $ab = ba$, we can at least say that $\mathrm{ord}(ab)$ divides $\mathrm{lcm}(\mathrm{ord}(a), \mathrm{ord}(b))$   which is easy to prove, but not very effective. So:   What are some similar results about the order of a product, perhaps with some additional hypotheses? Turning a cipher into a hashing function What happens when the Runner stack is empty? When the corporation R&amp;D is empty the corporation loses.  What happens when the Runner's stack is empty?    I think that: He must finish the game with only the cards on the table.  or  I doubt that: He reshuffles the heap into a new stack. iOS 11-13 - unable to paste into comments Factorizing polynomials over fields other than $\mathbb{C}$ Numbering figures and tables by their section I am having trouble formatting my list of figures and list of tables.  Currently, my tex document is printing the list in the following manner:     1 Foreign Key Constraint Problem Due   to Improper Table Design. . . . 3      2 Application Flow of Territory   Maintenance Tool. . . . . . . . . . .   . . 4   But I want to number figures by their sections:     3.1.2 Foreign Key Constraint Problem Due to Improper Table Design Installed Teamviewer using a 64-bit system, but I get a dependency error Suppose that we have a vector space $V$ and the subspaces $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$. Then how comes $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}?$$ I followed this  to install Ubuntu as dual-boot with Windows 8.1 pre-installed as(UEFI),but I am not able to dual-boot and tried    I first burned an Ubuntu ISO on a blank DVD-R to create the LiveCD. Even with secure boot disabled, when I tried restarting my computer, it never booted to the installation screen, and it just skipped to Windows.  After a lot of skimming, I thought I had to put my computer in Legacy Mode in order to get give my CD/ROM drive priority. I installed Ubuntu and used my existing EFI partition. Ubuntu works, but Windows doesn't, probably because my laptop is stuck in Legacy Mode. I tried switching back to UEFI using boot-repair on the Linux Terminal, and going into Advanced Properties -> Grub Location -> Ticking the checkbox for EFI, but it says I need to be in an EFI session, and now I'm just really lost and confused at what I've all done. I can see my Windows8.1 OS on a partition but I want to dual-boot Windows 8.1 along with Ubuntu?   Did I just brick my computer? Command not found $ (dollar sign) Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. Probability of selecting an even natural number from the set $\Bbb N$. I confirmed on  thread that there are as many as even natural numbers as there are natural numbers.   Question : Suppose I have selected a number $n \in \mathbb N$; what is the probability that $n$ is even?  My Thought : $\text{Probability} = \dfrac{\text{n(E)}}{\text{n(S)}}$ Here $\text{n(S)}$ is the set of all natural numbers i.e. $\mathbb N$, and $\text{n(E)}$ is set of all even natural numbers. Since it is proved that number of elements is the set $\mathbb N$ is exactly the same as the number of elements in the set of natural numbers (it’s very easy to put the set of natural numbers, $\Bbb N=\{0,1,2,3,\dots\}$, into one-to-one correspondence with the set $\text{E}=\{0,2,4,6,\dots\}$ of even natural numbers; the map $\Bbb N\to \text{E}:n\mapsto 2n$ is clearly a bijection.) ; Thus, Probability $= \boxed 1$ I know this is definitely wrong.Probability must be $0.5$. But where am I wrong? Can anyone explain ? Thanks! How do I make sure that a socket bound to a port is properly released on process exit such that the port can be reused without bind() failing with EADDRINUSE? I've written a tiny program which just creates a socket, binds it to a fixed port, waits for a connection and then immediately terminates. When I rerun the program, the bind() call fails with EADDRINUSE, but if I wait a few minutes, it succeeds.  Is there a way I can explicitly "unbind" the socket, thereby freeing the port number? Create longer arrows It didn't fixed the error. So, how do I fix my GUI update?  For every software I have to install it from command line because I got same error when install software from Ubuntu Software Center.