How to find the solutions to $z^4 = -16$
Complex numbers equation: $z^4 = -16$
Cannot find hard drives after installing Ubuntu 13.10 How can gravity affect light? I understand that a black hole bends the fabric of space time to a point that no object can escape.   I understand that light travels in a straight line along spacetime unless distorted by gravity. If spacetime is being curved by gravity then light should follow that bend in spacetime.  In Newton's Law of Universal Gravitation, the mass of both objects must be entered, but photon has no mass, why should a massless photon  be affected by gravity in by Newton's equations?  What am I missing? Pull down resistor on logic gate input I would like to use a pull-down resistor on the input of a NOT logic gate. I am using a 5V microcontroller, and I want to ensure that the input of the gate will never be floating (eg. microcontroller damage).  General information:  The logic gate I am using is a HD74LS04 inverter whose datasheet can be found . I am testing the output by placing my positive multi-meter lead on R4 and the negative lead on GND. With the digital pin floating, the output of 1Y is 4.33V. This is the expected behavior.  The logic gate requires a minimum of 2.0V for Vih. If I understand this circuit correctly, I expect the voltage at pin 1A to be 3.76V since R2 and R3 will form a voltage divider. This value seems plenty to signify a 'high' in the logic gate.  R2 exists to protect the digital pin. Ideally, I would have liked R2 to be 1KΩ. Increasing the value of R2 means I have to increase the value of R3 to maintain the same voltage divider. During testing I found that if R3 was any value over 3.3KΩ, the pin 1A would not get pulled down all the way if I left the digital pin floating.   Tests: If I disconnect everything except GND and Vcc and place the positive lead of my multi-meter on pin 1A (now floating) and the negative lead to GND, I read 1.49V. Placing a pull-down resistor (R3) on pin 1A to GND with a value of 100Ω changes the reading to 0.02V. Increasing the resistor's value to 330Ω increases the reading to 0.07V. At the value I need, 3.3KΩ, the reading increases to 0.60V. Finally, changing the resistor's value to 1MΩ increases the reading to 1.47V and actually changes the output of pin 1Y to low.  The test that uses the 3.3KΩ resistor is concerning. The reading I took (0.60V) seems to be approaching the Vil maximum of the logic gate which is 0.80V.   Questions:  This must be a trivial question, but why does an R3 value of 3.3KΩ not pull pin 1A all the way to GND? Is there a formula for determining the correct pull-down resistor value? Other answers I found on here simply state 10KΩ as a 'good' value for a pull-up resistor. I'm not sure where that value comes from or if it is applicable to a pull-down resistor.         &ndash; Schematic created using   Edit: Although Ignacio's answer technically answers the question, I thought I would share some information I found for others who may come across this question.  User 'Papabravo' on the  states that the calculation for a pull down resistor can be calculated by taking the Vil (0.8V) and dividing it by the "standard TTL load" of 1.6mA. As Ignacio pointed out, my specific logic gate can source up to 0.4mA, so the calculation would be 0.8/0.0004A which equals 2KΩ. For safety, I performed the calculation at 0.5V in order to give myself some headroom.  Other Resources: Control custom properties How can I add remove and create a callback/event for a custom property using bpy. I have a scene, called scene, and I want to add a custom property named x, defulat value 2 (as intereger), and a callback foo, to run when it is changed. I also need to be able to read the value from it, and delete it. How do I do this? Is $\exists x(x=a)$ a valid formula in first order logic with equality? When I ordered this TP-Link AC1300 Wireless Dual Band PCI-Express Adapter (Archer T6E), I looked into customers reviews and they said it worked fine with no drivers on Ubuntu. I didn't have a CD/DVD player on my desktop so I (on a laptop with a DVD drive) copied and pasted the DVD support disk content on to a USB drive and used that. But when I tried to run autorun.exe it puts up archive manager and says An error occurred while loading the archive Ubuntu verison- Ubuntu 16.04 LTS.  Output from: lspci -vnn  02:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 11) Subsystem: ASRock Incorporation Motherboard (one of many) [1849:8168] Kernel driver in use: r8169  Kernel modules: r8169   I am fairly new to Ubuntu (1 month) and have learned alot (sudo commands) and now know that windows is overpriced and that ubuntu is better in most fields.  Thanks for the help. As a new user on a site, what's the best way to ensure your question will be positively received? I understand that you have to have visas to enter some countries, but I was wondering whether you have to pay for each visa separately or you can get a visa to go everywhere. You see videos and lists on what you should pack to travel the world, but no lists on the documentation needed. Restart my graphics card without rebooting? Let $a_n=\sqrt[n]n-1$. Does $\sum_{n=1}^\infty a_n$ converge? How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? If $f'$ is bounded, then $f$ is uniformly continuous Let $f\colon(a,b) \to \mathbb R$ be differentiable and suppose that there exists an $M&gt;0$ such that $|f&#39;(x)| \leq M$ for all $x$ in $(a,b)$. Prove that $f$  is uniformly continuous. How do you judge the quality of a journal? Are there any versions of t-SNE for streaming data? My understanding of  and the Barnes-Hut approximation is that all data points are required so that all force interactions can be calculated at the same time and each point can be adjusted in the 2d (or lower dimensional) map.  Are there any versions of t-sne that can efficiently deal with streaming data? So if my observations are arriving one at a time, it will find the best location on the 2d map to place the new observation, or continuously update all points on the 2d map to account for ht new observation.  Would this even make sense or does it go against the setup of t-sne. What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases? How to keep processes running after ending ssh session? Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$ Any hint to prove this? I tried with properties of binomial coefficient but I can't get anything. $$\sum_{n=k}^{\infty}{{n}\choose{k}}x^n=\dfrac{x^k}{(1-x)^{k+1}}$$ I propose to let users vote on closing activities of moderators and other users with closevoting power.  Like this:       closed as exact duplicate by Michael Mrozek, waiwai933, Arjan, YOU, Yi Jiang 6 hours ago      This question covers exactly the same ground as earlier questions on this topic; its answers may be merged with another identical question.   Make it like this:     closed as exact duplicate by Michael Mrozek, waiwai933, Arjan, YOU, Yi Jiang 6 hours ago   fair/unfair close? [agree] [disagree]      This question covers exactly the same ground as earlier questions on this topic; its answers may be merged with another identical question.   Then the system will gather statistics and automatically grant more closing powers to those users who enjoy support of the community and revoke those powers from those mostly casting contradictory votes.  It would seem that granting the same powers to everybody has not resulted in balance but in public disorder. Perhaps it is reflective of American lifestyle (everybody has the right and means to shoot anybody), but it's not the best way to run a society by far.  What I'm trying to say is that while granting powers to everybody in advance is generous, the maturity to wheel these powers in a fair way must be proven. By introducing community feedback this assessment process can be automatized. Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. I use a 7805 for a project where the circuit needs a higher current (~2.8&nbsp;A) at 5&nbsp;V. So I assume that if I use both ICs in parallel I can increase the maximum current capacity. Would it work?