Question: does every prime divide some fibonacci number?

Answer: yes


Question: 5 coins are tossed how many outcomes?

Answer: There are $2^5$ equally likely strings of length $5$ made up of the letters H and/or T

Answer: there are $\;2^5 = 32$ possible outcomes


Question: how to make electromagnetic waves visible?

Answer: by using them to make something else emit energy in a frequency we can see

Answer: liquid crystals are applied on an RF absorbing film and change color when RF waves heat the film


Question: are all cosets the same size?

Answer: the size of every coset is equal to the size of $H$


Question: do mathematicians, in the end, always agree?

Answer: no


Question: does the square root of a negative number exist?

Answer: When we say that the square root of a negative number "doesn't exist

Answer: we mean that there is no real number solution


Question: is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?

Answer: No

Answer: with all distances strictly greater than 5cm it is not possible to place 26 points


Question: how does the mass of an object affect its velocity?

Answer: It determines how quickly an object can change speed (accelerate) under the action of a given force

Answer: mass determines how strong a force has to be to accelerate an object at a given rate


Question: 2 is the only prime even number because?

Answer: Because every even number other than 2 is obviously divisible by 2


Question: why is integration so much harder than differentiation?

Answer: differentiation, you can mechanically apply the rules. For integration, you need to recognize patterns


Question: what is the basic difference between compton effect and photoelectric effect?

Answer: Photo electric effect occur in bound electron, while compton effect occur in free electron


Question: is there a math expression equivalent to the conditional ternary operator?

Answer: Kronecker Delta


Question: every nonzero element in a finite ring is either a unit or a zero divisor

Answer: every nonzero element is either a zero-divisor or a unit


Question: in classical logic, why is $(p\rightarrow q)$ true if both $p$ and $q$ are false?

Answer: since the empty set is a subset of any set, a false statement implies any statement


Question: why isnt a virus alive?

Answer: they always rely on the host for the machinery with which to replicate


Question: how can one prove that $e<\pi$?

Answer: Use $e<3$ and $\pi>3$


Question: how do i tell if matrices are similar?

Answer: if and only if they have the same characteristic polynomial, and neither of them is a scalar matrix

Answer: if and only if identical lists of invariant factors are found


Question: prove that kinetic energy is not conserved in inelastic collision?

Answer: kinetic energy is turned into heat, so although energy is conserved, kinetic energy is not


Question: what is the physical significance of hysteresis loop of ferromagnetic material?

Answer: The area inside the hysteresis loop is the energy dissipated in the material per cycle


Question: are ripples in water circular?

Answer: ripples are not circular at all

Answer: the ripples in water are always circular


Question: is potential energy a state function?

Answer: potential energy is a state function


Question: what does find the magnitude of acceleration mean?

Answer: how quickly your speed changes in a given unit of time

Answer: how quickly velocity changes


Question: why atmosphere is the bad conductor of heat?

Answer: the very low density of gases that make them bad conductors of heat

Answer: The particles in air are far apart

Answer: Heat does not travel easily by conduction through air


Question: is dough a solid or liquid?

Answer: Dough, and Play-Doh, are best described as non-Newtonian fluids


Question: why does nasa use gold foil on equipment and gold-coated visors?

Answer: reflects IR to keep it from getting “hot


Question: are all quadratic equations functions?

Answer: No


Question: what is the main difference between continuous mapping at point and the uniform continuous?

Answer: continuity is defined at a point $c$, whereas uniform continuity is defined on a set $A$


Question: are objects always in motion?

Answer: all objects are in motion with respect to some other object


Question: what structure does the alternating group preserve?

Answer: orientation


Question: what is the relationship of sound intensity to sound pressure?

Answer: The intensity of the sound wave, however, is proportional to the square of the sound pressure


Question: what is the physical relationship between mass and speed?

Answer: mass determines how strong a force has to be to accelerate an object


Question: if two events are mutually exclusive are they independent or dependent?

Answer: mutually exclusive events are not independent, and independent events cannot be mutually exclusive

Answer: if the two events are mutually exclusive, they can not be independent


Question: are sat scores discrete or continuous?

Answer: discrete


Question: is square root of negative 1 a rational number?

Answer: it cannot be rational

Answer: it is not rational

Answer: it is neither rational nor irrational

Answer: not a rational number


Question: difference between supremum and maximum

Answer: A maximum of a set must be an element of the set. A supremum need not be

Answer: maximum is the largest member of the set, while the supremum is the smallest upper bound


Question: why do ballpoint pens write better on pages that have pages below them?

Answer: the steel ball of the pen makes a better deeper contact by indentation with side cushioning


Question: natural selection occurs at which level?

Answer: individual

Answer: population


Question: are there any turing-undecidable problems whose undecidability is independent of the halting problem?

Answer: Yes


Question: can we theoretically balance a perfectly symmetrical pencil on its one-atom tip?

Answer: NO

Answer: No


Question: what is a good book for learning math, from middle school level?

Answer: How to solve it


Question: is it possible to write a sum as an integral to solve it?

Answer: Yes you can use the Euler Maclaruin formula to write the sum as an integral


Question: are mutually exclusive events also independent?

Answer: mutually exclusive events are not independent, and independent events cannot be mutually exclusive

Answer: if the two events are mutually exclusive, they can not be independent


Question: why does rainwater form moving waves on the ground? is there a name for this effect?

Answer: These waves are called "roll waves." They are due to an instability in shallow shear flows


Question: why do the french count so strangely?

Answer: 80 is also not invented, it is 4 20s, quatre-vingt


Question: if a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

Answer: LLN doesn't tell you anything about the probability of the next (or any) trial


Question: do ac circuits need ground?

Answer: You can absolutely operate an AC circuit without a ground wire


Question: what is the evolutionary advantage of red-green color blindness?

Answer: differentiate between much more shades of khaki than unaffected people

Answer: help detecting camouflaged food in a green environment

Answer: better at detecting color-camouflaged objects

Answer: spotting hidden dangers (predators) or finding camouflaged foods


Question: thought experiment - would you notice if you fell into a black hole?

Answer: A falling observer does not experience passing through an event horizon


Question: how to calculate manhattan distance between two vectors?

Answer: Take the sum of the absolute values of the differences of the coordinates

Answer: subtract the values of a dimensiin then abs them and add all the results


Question: are glass a good conductor of heat?

Answer: Glass is a very poor heat conductor


Question: can scientific notation be negative?

Answer: no problem having a negative number shown in scientific notation

Answer: Yes, you can use negative numbers too


Question: how come large herbivores have such thin legs?

Answer: Less mass in the lower leg means it swings easier


Question: the square root of a negative number does not exist?

Answer: negative numbers can't have square roots


Question: what is the difference between singular value and eigenvalue?

Answer: singular values are always real, while eigenvalues need not be real


Question: what is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

Answer: ${5\over27}$


Question: is the inverse of a symmetric matrix also symmetric?

Answer: Yes


Question: what is spin as it relates to subatomic particles?

Answer: Spin is a technical term specifically referring to intrinsic angular momentum of particles

Answer: Spin is angular momentum of particles


Question: are bugs attracted to infrared light?

Answer: insects do see, and are attracted to, IR light


Question: have electrons ever been observed?

Answer: only indirectly observed


Question: the last digit of $2^{2006}$

Answer: $2^{2006}$ ends in a $4$

Answer: $4$

Answer: 4$


Question: what exactly are eigen-things?

Answer: vectors that exhibit especially simple behaviour under a linear transformation


Question: why doesnt the moon fall onto the earth?

Answer: the pull on the Moon is just enough to keep it in a circular orbit around Earth


Question: how can anything ever fall into a black hole as seen from an outside observer?

Answer: from an outside perspective, nothing can ever pass the event horizon


Question: compact sets are closed?

Answer: not actually true that a compact subset of an arbitrary topological space is closed

Answer: Compact subsets of metric spaces are closed


Question: what is the difference between discrete and finite?

Answer: Discrete just have to be separated from one another, finite means the total number must be $<\infty$


Question: why does stephen hawking say black holes dont exist?

Answer: Stephen Hawking doesn't say that black holes don't exist


Question: is an irrational number odd or even?

Answer: only integers are even or odd

Answer: an irrational number, or in fact any number which is not an integer, is neither odd nor even


Question: how do moving charges produce magnetic fields?

Answer: charged particle moves through the "pervasive" EM field (space), it perturbs the EM field and generates a magnetic field


Question: an easy example of a non-constructive proof without an obvious fix?

Answer: Uncountability of the real numbers


Question: does associativity imply commutativity?

Answer: does not


Question: how many 7-note musical scales are possible within the 12-note system?

Answer: $38$ possible heptatonic scales

Answer: 60


Question: is fatty acid a polymer?

Answer: not considered polymers


Question: is pi an imaginary number?

Answer: not an imaginary number


Question: 6 options how many combinations?

Answer: 2^6, which is 64


Question: is pde harder than ode?

Answer: PDEs are generally more difficult to understand the solutions to than ODEs


Question: intuitive interpretation of the laplacian

Answer: behaves like a local averaging operator

Answer: Laplacian as behaving like an 'average rate of change

Answer: local averaging operator


Question: does normal force affect the coefficient of friction?

Answer: the coefficient does not change


Question: sum of two closed sets in $\mathbb r$ is closed?

Answer: The sum $E +F$ may fail to be closed even if $E$ and $F$ are closed


Question: can octopus go on land?

Answer: for short periods


Question: is it possible to plot a graph of any shape?

Answer: Definitely not


Question: why are adjoint functor important?

Answer: they show up everywhere and are an extremely useful organizing principle


Question: what are the 2 types of current flow in a semiconductor?

Answer: the electron current and the hole current


Question: what is the size of each side of the square?

Answer: $18$

Answer: 18$


Question: if you are vacuuming your carpet and you wrap the cord around your body do you become a magnet?

Answer: No


Question: every real number has a decimal expansion?

Answer: yes

Answer: Yes

Answer: Every real number $x$ has a decimal expression $$x = a_0.a_1a_2a_3\ldots


Question: are imaginary numbers and complex numbers the same?

Answer: imaginaries are a subset of the complex numbers


Question: are neutrinos affected by gravity?

Answer: Neutrinos are certainly affected by gravity

Answer: All particles, even massless ones, are affected by gravity


Question: will a hole cut into a metal disk expand or shrink when the disc is heated?

Answer: the hole would still get larger

Answer: the hole will expand

Answer: Heat a hole, and it expands


Question: reference request: introduction to commutative algebra

Answer: A Singular Introduction to Commutative Algebra

Answer: "Commutative Algebra with a view towards algebraic geometry


Question: what is lost when we move from reals to complex numbers?

Answer: the notion of an order

Answer: ordering

Answer: order


Question: are all bounded sequences convergent?

Answer: No


Question: are pivot columns linearly independent?

Answer: Pivot columns are linearly independent

Answer: Yes of course pivot columns are linearly independent


Question: what is a non-conservative field?

Answer: one where the integral along some path is not zero


Question: what is the difference between cell line and cell strain?

Answer: A cell strain is a subpopulation of a cell line that has been positively selected from the culture


Question: what is the probability of getting a pair in a deck of cards?

Answer: 3/51 = 1/17


Question: can hiv be transmitted via mosquitos?

Answer: No


Question: is it possible to represent every huge number in abbreviated form?

Answer: No


Question: average distance between random points on a line segment

Answer: $\frac L 3$

Answer: D=L/3


Question: what is the difference between analytical and numerical methods?

Answer: Analytical is exact; numerical is approximate


Question: a body moving with a constant velocity will have?

Answer: the total of all the forces applied to it is zero


Question: in uniform circular motion acceleration is uniform or variable?

Answer: variable


Question: how to know if a point is inside a circle?

Answer: if the distance from it to the center is less than $r$


Question: is zero odd or even?

Answer: even


Question: can mutually exclusive events be independent and dependent?

Answer: mutually exclusive events are not independent, and independent events cannot be mutually exclusive

Answer: if the two events are mutually exclusive, they can not be independent

Answer: If two events are independent, they cannot be mutually exclusive


Question: what is the relationship between ir rays and heat?

Answer: energy carried away by the infrared radiation reduces the heat content of the radiating body

Answer: most bodies radiate most of their heat in the infrared spectrum


Question: can 2 events be independent and mutually exclusive?

Answer: This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive

Answer: If two events are independent, they cannot be mutually exclusive

Answer: a mutually exclusive pair of nontrivial events are also necessarily dependent events


Question: what is the jacobian matrix?

Answer: a matrix of partial derivatives


Question: what is the difference between analytical and empirical?

Answer: empirical: based on, concerned with, or verified by observation or experience. analytical: based on analysis or logic


Question: why are hermitian matrices important?

Answer: because we have the spectral theorem for them, i.e. they admit an orthonormal eigenbasis


Question: are prime numbers still prime in other bases?

Answer: A prime number is a prime by defination, irrespective of base

Answer: It is prime, no matter how you represent it


Question: explicit norm on $\mathcal{c}^0(\mathbb{r},\mathbb{r})$

Answer: There is no explicit norm on $\mathcal{C}^0(\mathbb{R}, \mathbb{R})$


Question: is there a 0-1 law for the theory of groups?

Answer: a FO 0-1 law for group theory as defined by Dominik doesn't exist and may never exist

Answer: such a law cannot exist for the whole group theory or even for finite group theory


Question: why are so few foods blue?

Answer: Blue color is rare in both the animal and plant Kingdoms in general


Question: what is the effect of non-vaccinated people on vaccinated people?

Answer: vaccinated people are still at risk of disease following exposure to unvaccinated people


Question: there exists a power of 2 such that the last five digits are all 3s or 6s. find the last 5 digits of this number

Answer: 66336


Question: probability of 3 people in a room of 30 having the same birthday

Answer: $$P(365,30,3)\simeq 2,85\%$$


Question: is there a symmetry associated to the conservation of information?

Answer: Liouville theorem, which can be interpreted in terms of time-translation symmetries

Answer: there is a symmetry associated with information conservation


Question: is sin x one to one?

Answer: it is not one-to-one


Question: help find hard integrals that evaluate to $59$?

Answer: $$ \frac{64}{\pi^3} \int_0^\infty \frac{ (\ln x)^2 (15-2x)}{(x^4+1)(x^2+1)}\ dx $$


Question: does juggling balls reduce the total weight of the juggler and balls?

Answer: no


Question: are sinx and cosx orthogonal?

Answer: $\sin(x),\cos(x)$ are orthogonal


Question: whats 4 times more likely than 80%?

Answer: 95\%


Question: is sin x an onto function?

Answer: The sine is not onto

Answer: not onto


Question: number of onto functions

Answer: $n!S(m,n)$


Question: intuitive explanation of a definition of the fisher information

Answer: a measure of the "curvature" of the support curve near the maximum likelihood estimate of θ


Question: is it possible for a function to be smooth everywhere, analytic nowhere, yet taylor series at any point converges in a nonzero radius?

Answer: No


Question: how to calculate the difference between two negative numbers?

Answer: subtract the smaller absolute value from the larger absolute value


Question: why is radian so common in maths?

Answer: Radians are simply the natural units for measuring angles

Answer: radians are the natural unit of measure for angles


Question: are all central forces conservative?

Answer: central forces are conservative


Question: matrices commute if and only if they share a common basis of eigenvectors?

Answer: This is false


Question: how to find solutions of linear diophantine ax + by = c?

Answer: use the extended Euclidean algorithm

Answer: extended Euclidean algorithm


Question: why is there a scarcity of lithium?

Answer: there is quite a lot of lithium in the earth that is not so easy to obtain

Answer: it is found in many rocks and some brines, but always in very low concentrations

Answer: it does not tend to accumulate in rich geological deposits


Question: am i attracting pluto?

Answer: The mass that makes up your body and everything inside your body is attracting Pluto


Question: are two disjoint events independent?

Answer: Two disjoint events can never be independent, except in the case that one of the events is null

Answer: Disjoint events aren't independent, unless one event is impossible


Question: can you make a sphere out of a plane?

Answer: impossible


Question: how many meal combinations are possible with the current menu?

Answer: 96


Question: why do units (from physics) behave like numbers?

Answer: units are not separate from numbers but rather extensions of numbers


Question: what is the difference between wave and pulse?

Answer: A single vibration is called pulse and continued vibration of a particle in medium is called wave


Question: why is negative times negative = positive?

Answer: doing the opposite twice means doing the usual


Question: are top floor apartments hotter?

Answer: The top floor can be very hot in a cement apartments flat without an attic


Question: how many numbers from 1 to 100 are not divisible by 2 3 and 5?

Answer: 26


Question: is zero positive or negative?

Answer: neither positive nor negative

Answer: neither a negative or a positive


Question: subgroup of abelian group is abelian?

Answer: Yes, subgroups of abelian groups are indeed abelian


Question: what concept does an open set axiomatise?

Answer: semidecidable property


Question: is arrow notation for vectors not mathematically mature?

Answer: No


Question: why is the vibration in my wire acting so oddly?

Answer: the stiffness and elasticity of it is not homogenous lengthwise or even along cross-section


Question: how to find a basis for the intersection of two vector spaces in $\mathbb{r}^n$?

Answer: Parameterize both vector spaces (using different variables!) and set them equal to each other


Question: what does surface area of a sphere actually mean (in terms of elementary school mathematics)?

Answer: painting the surface area, and working out how much paint was required to paint the sphere's complete surface


Question: what is the difference between binomial and bernoulli distribution?

Answer: All Bernoulli distributions are binomial distributions, but most binomial distributions are not Bernoulli distributions


Question: in linear algebra, what is a vector?

Answer: a quantity with magnitude and direction

Answer: an element of a vector space


Question: what is the highest frequency of electromagnetic radiation?

Answer: Gamma-rays


Question: can a woman give birth to twins with different fathers?

Answer: there have been documented cases where twins had different fathers


Question: what is 5' and 3' dna?

Answer: 5' carbon has a phosphate group attached to it and the 3' carbon a hydroxyl (-OH) group

Answer: five prime" and "three prime", which indicate the carbon numbers in the DNA's sugar backbone


Question: in russian roulette, is it best to go first?

Answer: Your best bet is actually to go last

Answer: it's a disadvantage to go first


Question: is a 2x3 matrix invertible?

Answer: No


Question: are all mathematicians human calculators?

Answer: no

Answer: No


Question: death because of distilled water consumption

Answer: related to hyperhydration/water intoxication

Answer: to drink nothing but distilled water, and eat no food, you probably would die of hyponatremia


Question: where does the word torsion in algebra come from?

Answer: Its origin lies in algebraic (or combinatorial) topology


Question: is there a known well ordering of the reals?

Answer: No


Question: what exactly is a number?

Answer: a mathematical notion which represents quantity


Question: is gas a good conductor of electricity?

Answer: At atmospheric pressure, air and other gases are poor conductors


Question: naturally occurring non-hausdorff spaces?

Answer: Sierpiński space


Question: why does an integral change signs when flipping the boundaries?

Answer: $\int_a^b f(x) dx + \int_b^a f(x) dx = \int_a^a f(x) dx$. But $\int_a^a f(x) dx = 0$, so if we want this formula to hold, we need $\int_a^b f(x) dx = -\int_b^a f(x) dx $

Answer: The change in $x$, $dx$ becomes negative


Question: why do we not have spin greater than 2?

Answer: a consequence of little group invariance and charge conservation


Question: function that is the sum of all of its derivatives

Answer: $f(x)=\exp(\frac{1}{2}x)$


Question: intuitively, what is the difference between eigendecomposition and singular value decomposition?

Answer: SVD is for bilinear forms and the eigendecomposition is for quadratic forms


Question: is the law of large numbers empirically proven?

Answer: we cannot empirically verify LLN

Answer: not empirically proven


Question: what is the difference between regression and classification?

Answer: Regression: the output variable takes continuous values. Classification: the output variable takes class labels

Answer: Regression involves estimating or predicting a response. Classification is identifying group membership

Answer: regression predicts a value from a continuous set, whereas classification predicts the 'belonging' to the class


Question: prawn is not a fish but an?

Answer: Crustacean


Question: what does := mean?

Answer: item on the left hand side is being defined to be what is on the right hand side


Question: are rational points dense on every circle in the coordinate plane?

Answer: The set of rational points on a circle are dense in the circle


Question: what does express in terms of x mean?

Answer: to write an algebraic expression using only that variable or variables


Question: does someone falling into a black hole see the end of the universe?

Answer: they will not see the end of the universe


Question: what is the magnitude of acceleration called?

Answer: magnitude of the acceleration vector


Question: can you answer my sons fourth-grade homework question: which numbers are prime, have digits adding to ten and have a three in the tens place?

Answer: 37

Answer: 433

Answer: 631


Question: can mutually exclusive events also be independent of one another?

Answer: mutually exclusive events are not independent, and independent events cannot be mutually exclusive

Answer: if the two events are mutually exclusive, they can not be independent

Answer: If two events are independent, they cannot be mutually exclusive


Question: what does ∈ mean?

Answer: an element of

Answer: element in the set of

Answer: Element of


Question: how many connected components does $\mathrm{gl}_n(\mathbb r)$ have?

Answer: two components


Question: fourier transform of derivative

Answer: $ \mathcal{F}(f')(\xi)=2\pi i\xi\cdot\mathcal{F}(f)(\xi)


Question: is the power set of the natural numbers countable?

Answer: uncountable


Question: if we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?

Answer: \begin{align} E[X] &= E[X_1] + E[X_2] + \cdots +E[X_{99}]\\ &= 99E[X_1]\\ &= 99\frac{25}{100}\frac{24}{99} = 6 \end{align}

Answer: $$ 24 {99\choose 24} \times {100\choose 25}^{-1} = 6.$$


Question: how does a knife cut things at the atomic level?

Answer: the atoms amorphise in the immediate vicinity due to the high pressure and then deform around the blade

Answer: material either fractures along faults in the lattice, or you separate molecules (as when you cut bread


Question: how to show a curve has no turning points?

Answer: The curve has no turning point when its derivative takes constant sign

Answer: A differentiable function $f(x)$ has no turning points if its derivative $f'(x)$ has no real roots


Question: light is electromagnetic wave or not?

Answer: yes


Question: every subsequence of $x_n$ has a further subsequence which converges to $x$. then the sequence $x_n$ converges to $x$.

Answer: TRUE


Question: is $\pi^2 \approx g$ a coincidence?

Answer: pure coincidence


Question: is time orthogonal to space?

Answer: Time is orthogonal to space


Question: why is it important for a matrix to be square?

Answer: powers of a matrix only make sense if the matrix is square


Question: why does a mirror split my laser beam?

Answer: You are getting reflections from the front (glass surface) and back (mirrored) surface, including (multiple) internal reflections


Question: is torque measured in joules?

Answer: The units for torque, as you stated, are Newton-meters

Answer: Joules are generally not appropriate units for torque


Question: is $0! = 1$ because there is only one way to do nothing?

Answer: Yes


Question: why does a full moon seem uniformly bright from earth, shouldnt it be dimmer at the border?

Answer: Surface roughness is the main reason the edges are not as dim as the sphere model predicts


Question: what is the difference between an electron neutrino and a muon neutrino?

Answer: muon neutrinos scattering on protons created muons, whereas electron neutrinos created electrons


Question: apparently sometimes $1/2 < 1/4$?

Answer: $1/2$ is NOT smaller than $1/4$


Question: is there any evidence that sexual selection may lead to extinction of species?

Answer: sexual selection increased risks of local extinction


Question: does a particle exert force on itself?

Answer: in quantum mechanics, we find that particles do indeed interact with themselves

Answer: a real self-interaction is excluded by the principles of Newtonian mechanics


Question: do sieve tubes have mitochondria?

Answer: retain mitochondria


Question: natural selection occurs at which level individual or population?

Answer: population level


Question: a is 25 percent more efficient than b?

Answer: A is 25% more efficient than B


Question: do any gases conduct electricity?

Answer: Gases do conduct electricity


Question: what is the largest eigenvalue of the following matrix?

Answer: 21


Question: what is category theory useful for?

Answer: to talk about structures recurring throughout other disciplines of mathematics


Question: why does a remote car key work when held to your head/body?

Answer: your body is reasonably conductive

Answer: it can couple to RF sources capacitively


Question: given a die, what is the probability that the second roll of a die will be less than the first roll?

Answer: 5/12


Question: how discontinuous can a derivative be?

Answer: $D$ can be dense in $\mathbb R$. $D$ can have cardinality $c$ in every interval

Answer: $D$ can have positive measure in every interval

Answer: $D$ can have full measure in every interval


Question: understanding gödels incompleteness theorem

Answer: any effectively generated formal system which is sufficiently powerful is either inconsistent or incomplete


Question: are all algebraic integers with absolute value 1 roots of unity?

Answer: No

Answer: no


Question: are sinx and cosx linearly independent?

Answer: $\sin x$ and $\cos x$ are linearly independent on any interval $I$


Question: development of the idea of the determinant

Answer: as a property which tests for the existence of unique solutions for a system of linear equations


Question: what instrument is used to measure optical density?

Answer: spectrophotometer


Question: is there a function with a removable discontinuity at every point?

Answer: No


Question: how far can one see over the ocean?

Answer: 5 nautical miles of visibility (1nm = 1852m)


Question: did the big bang happen at a point?

Answer: there was a "beginning point singularity

Answer: representing universe in 4D then big bang had happened at a point

Answer: in 3D the big bang should have happened in every point of the universe


Question: is a cell wall an organelle?

Answer: isn't an organelle


Question: is gcd an injective function?

Answer: it is not injective


Question: a body is moving with a constant velocity?

Answer: the total of all the forces applied to it is zero


Question: how were old-school mathematics graphics created?

Answer: drawn by hand


Question: is it bad form to write mysterious proofs without explaining what one intends to do?

Answer: If your proof is obscure, then you have failed to communicate


Question: is it possible that a counter-example exists but it cannot be found

Answer: Yes


Question: is there such a thing as proof by example (not counter example)

Answer: there is such a thing as a proof by example

Answer: any proofs of existential claims will qualify as a proof by example


Question: what is the standard basis for fields of complex numbers?

Answer: $\left\lbrace(1,0),\, (0,1)\right\rbrace$


Question: is $6.12345678910111213141516171819202122\ldots$ transcendental?

Answer: This is a transcendental number


Question: i roll a die repeatedly until i get 6, and then count the number of 3s i got. whats my expected number of 3s?

Answer: 1


Question: are both eyes exactly the same?

Answer: No


Question: can disjoint events be independent?

Answer: Disjoint events aren't independent, unless one event is impossible, which makes the two events trivially independent


Question: why do humans not produce vitamin c like other mammals?

Answer: due to a mutation in the GULO (gulonolactone oxidase) gene


Question: are all finite sets closed?

Answer: every finite set is closed in a metric space


Question: what is the physical significance of electric dipole moment?

Answer: it gives a measure of the polarity/polarization of a net neutral system


Question: do all prokaryotes have introns?

Answer: Prokaryotes can't have introns


Question: explanation on arg min

Answer: $\arg\min _{x} f(x)$ is the value of $x$ for which $f(x)$ attains its minimum

Answer: $arg min$ (or $arg max$) return the input for minimum (or maximum) output


Question: meaning of rays in polar plot of prime numbers

Answer: Each ray thus corresponds to an arithmetic progression $a\pmod{710}$


Question: are all isosceles triangles equilateral?

Answer: every equilateral triangle is isosceles, but not every isosceles triangle is equilateral


Question: is infinity a number?

Answer: no

Answer: not a number


Question: why are rings called rings?

Answer: derived from Hilbert's term "Zahlring" (number ring


Question: why do we still do symbolic math?

Answer: theory behind numerical estimates is based on 'symbolic' math


Question: are all fields vector spaces?

Answer: a field is always a vector space over itself


Question: why do electrons, according to my textbook, exist forever?

Answer: Electrons are the lightest negatively charged particle and therefore cannot decay into lighter particles


Question: there are 4 cups of liquid. three are water and one is poison. if you were to drink 3 of the 4 cups, what is the probability of being poisoned?

Answer: 3/4

Answer: $1-\frac{3!}{4!} = \frac34$

Answer: 25%

Answer: 75%

Answer: 1/2


Question: why does hot water clean better than cold water?

Answer: hot water melts oil/grease/etc while cold water would solidify it

Answer: hot water is a better solvent than cold water


Question: are sets closed under addition?

Answer: not closed under addition

Answer: not closed under this kind of addition


Question: why should we prove obvious things?

Answer: sometimes things that should be "obvious" turn out to be completely false

Answer: something that seems obvious may be wrong


Question: difference between bijection and isomorphism?

Answer: an isomorphism (in the context of algebra) needs to map between the operations involved, while a bijection doesn't


Question: a conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Answer: $$ \frac{1}{2\log^2{2}}\int_{0}^{\infty} \frac{du}{\cosh{(u/2)}} = \frac{\pi}{2 \log^2{2}} $$ as correctly conjectured


Question: what is the greatest common factor of all four-digit palindromes?

Answer: $11$


Question: how does a calculator calculate the sine, cosine, tangent using just a number?

Answer: Taylor Series for $\sin / \cos$ or the CORDIC algorithm


Question: what is the difference between a point and a vector?

Answer: Points describe locations while vectors describe directions

Answer: Points are locations in space. Vectors are displacements in space


Question: the generating function for the fibonacci numbers

Answer: $\dfrac{1}{1-(z+z^2)}=1+(z+z^2)+(z+z^2)^2....$

Answer: $$ \sum_{n=0}^\infty F_nz^n = \frac{1}{1-(z+z^2)} $$

Answer: $$F(x)=\frac{1}{1-x-x^2}$$


Question: probability that random moves in the game 2048 will win

Answer: essentially zero


Question: how to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?

Answer: It's not possible

Answer: The problem as stated is impossible under the usual interpretation


Question: is length an intensive property?

Answer: Length is an extensive property


Question: do irrational roots come in pairs?

Answer: The complex roots of a polynomial with real coefficients always come in conjugate pairs


Question: a and b are mutually exclusive are they independent?

Answer: mutually exclusive events are not independent

Answer: if the two events are mutually exclusive, they can not be independent

Answer: a mutually exclusive pair of nontrivial events are also necessarily dependent events


Question: if events a b and c are disjoint then they are independent?

Answer: Two disjoint events can never be independent

Answer: Disjoint events aren't independent


Question: is cdf the integral of pdf?

Answer: Yes

Answer: yes


Question: what’s the difference between analytical and numerical approaches to problems?

Answer: Analytical is exact; numerical is approximate

Answer: Analytical solutions can be obtained exactly with pencil and paper; Numerical solutions cannot be obtained exactly


Question: why does a rubber band become a lighter color when stretched?

Answer: the amount of absorption per unit area decreases, and the band become lighter in color


Question: are upper triangular matrices invertible?

Answer: only if it has no zeros on the main diagonal


Question: are all differentiable functions integrable?

Answer: If you are thinking Riemann integrable, Then every differentiable function is continuous and then integrable


Question: what was the major discovery on gravitational waves made march 17th, 2014, in the bicep2 experiment?

Answer: gravitational waves from inflation during the early universe are being indirectly observed

Answer: the detection of inflationary gravitational waves in the B-mode power spectrum

Answer: It is the first indirect observation of gravitational waves produced by inflation


Question: is the vector cross product only defined for 3d?

Answer: Yes


Question: number of monic irreducible polynomials of prime degree $p$ over finite fields

Answer: Gauss’s formula $$\frac{1}{n}\sum\limits_{d \mid n} \ \mu(n/d) \cdot q^{d}$$

Answer: $\displaystyle \frac{q^{p}-q}{p}$


Question: can cosh x be zero?

Answer: $\cosh x$ is always non-zero


Question: why can a venn diagram for $4+$ sets not be constructed using circles?

Answer: a graph created in such a way from a 4-set Venn diagram cannot be planar—a contradiction


Question: what remains in a students mind

Answer: a vague outline of the terminology and theorems


Question: amla is citrus fruit or not?

Answer: No


Question: why does it hurt the next day after doing significant exercise?

Answer: due to microtrauma or micro ruptures in the muscle fibers


Question: why do we bend a book to keep it straight?

Answer: curving the paper effectively multiplies it's stiffness by several orders


Question: is it possible to start fire using moonlight?

Answer: no


Question: is temperature a lorentz invariant in relativity?

Answer: There is no universal relativistic temperature transformation of the form $T' = T(v)$


Question: why, historically, do we multiply matrices as we do?

Answer: symbolic way of substituting one linear change of variables into another one


Question: what is the meaning of the third derivative of a function at a point

Answer: jerk


Question: is there a purely algebraic proof of the fundamental theorem of algebra?

Answer: No


Question: how to tell if a set of vectors spans a space?

Answer: set up a matrix and use Gaussian elimination


Question: can all nfa be converted to dfa?

Answer: every NFA can be converted to an equivalent DFA

Answer: DFAs, NFAs and regular expressions are all equivalent


Question: can oxaloacetate cross mitochondrial membrane?

Answer: cannot cross


Question: what is the difference between independent and mutually exclusive events?

Answer: mutually exclusive events are not independent, and independent events cannot be mutually exclusive


Question: is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

Answer: It is not a ratio


Question: sets are important in mathematics?

Answer: every field of mathematics uses or refers to sets


Question: why is information indestructible?

Answer: physical laws are reversible and therefore it stands to reason that information cannot be lost


Question: are commutative c*-algebras really dual to locally compact hausdorff spaces?

Answer: we simply don't have this duality as stated

Answer: the dual of this category is equivalent to the category of locally compact Hausdorff spaces


Question: how to prove that two vectors are perpendicular to each other?

Answer: the scalar (dot) product between the two considered vectors must be zero $(0)$


Question: why are quantum gates reversible?

Answer: because quantum mechanics is reversible


Question: is there a function whose antiderivative can be found but whose derivative cannot?

Answer: no


Question: why does lhopitals rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?

Answer: limit of $f'$ over $g'$ doesn't exist

Answer: l'Hopital requires existence of the limit of the ratio of the derivatives


Question: natural example of cosets

Answer: Hopf fibers


Question: can an irrational number raised to an irrational power be rational?

Answer: Yes


Question: what is the difference between transparent and white?

Answer: So while transparent materials transmit light, white materials reflect it


Question: why is googles quantum supremacy experiment impressive?

Answer: they have achieved sufficient experimental control over a circuit containing 53 qubits


Question: if vectors are linearly independent do they span?

Answer: Any set of linearly independent vectors can be said to span a space


Question: are all injective functions invertible?

Answer: only if it is bijective


Question: why is the laplacian important in riemannian geometry?

Answer: eigenvalues of the Laplacian provide invariants of the Riemannian manifold


Question: why must a physical theory be mathematically self-consistent?

Answer: because we expect nature to stick to laws that can be described mathematically


Question: what is the implicit function theorem?

Answer: a set defined by $$ S=\{z\in{\Bbb R}^d|f(z)=0\} $$ is locally a graph of some function


Question: why are rms values used?

Answer: using the RMS values makes the power easy to calculate

Answer: They help to find the effective value of AC (voltage or current

Answer: The rms value, not the peak value, is the equivalent DC value that gives the same average power


Question: division by $0$

Answer: would mean multiplying by the reciprocal of 0. But 0 has no reciprocal

Answer: division by 0 has no meaning in the set of real numbers


Question: what is the focal length of eye lens?

Answer: 18.5 mm


Question: is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Answer: transcendental


Question: can coefficient of restitution be negative?

Answer: It can never be negative


Question: are electrons affected by gravity?

Answer: everything is affected by gravity, just like everything is affected by inertial forces. This includes electrons


Question: what is difference between a ring and a field?

Answer: A field has multiplicative inverses, rings don't need to have that


Question: why do scientists think that all the laws of physics that apply in our galaxy apply in other galaxies?

Answer: Because what we can observe doesn't differ according to distance from us


Question: how often will a calendar repeat itself?

Answer: after $28$ years


Question: why do noble gases behave ideally?

Answer: because they are monatomic and interact only by van der Waals forces


Question: are there organisms with fewer than 1000 neurons?

Answer: roundworm C. elegens, a nematode with only 302 neurons

Answer: water snail with 8 distinct neurons


Question: what is the relationship between absolute pressure atmospheric pressure and gauge pressure?

Answer: Absolute pressure is zero-referenced against a perfect vacuum, so it is equal to gauge pressure plus atmospheric pressure

Answer: Gauge pressure is zero-referenced against ambient air pressure, so it is equal to absolute pressure minus atmospheric


Question: what is non conservative field?

Answer: the integral along some path is not zero


Question: what is bra and ket notation and how does it relate to hilbert spaces?

Answer: kets are vectors on your Hilbert space, while bras are linear functionals of the kets


Question: why do we not have to prove definitions?

Answer: because the act of defining is to give a meaning to a particular concept


Question: is it possible to have three real numbers that have both their sum and product equal to $1$?

Answer: $\left(-1,1-\sqrt{2},1+\sqrt{2}\right)$ and $\left(-1,1+\sqrt{2},1-\sqrt{2}\right)$ are valid answers


Question: what is the fastest/most efficient algorithm for estimating eulers constant $\gamma$?

Answer: Brent-McMillan algorithm


Question: how does bpg help at high altitudes?

Answer: stabilizes the T- (taut; oxygen unbound) form of haemoglobin


Question: why do we use the word scalar and not number in linear algebra?

Answer: number is an individual entity in a ring while a scalar is an operation on a vector space

Answer: scalar" was a convenient name to use for the real multipliers of the vector elements

Answer: scalar conveys a notion of proportion and product, while number is more specialized, possibly involving ordering


Question: root 3 is irrational number?

Answer: The number $\sqrt{3}$ is irrational


Question: can $18$ consecutive integers be separated into two groups,such that their product is equal?

Answer: This is impossible


Question: are there any examples of non-computable real numbers?

Answer: Chaitin's constant


Question: what is gravity like in the center of the earth?

Answer: the gravitational force is zero, but the gravitational well is at its deepest


Question: what is the difference between null space and kernel?

Answer: nullspace when referring to a matrix and the word kernel when referring to an abstract linear transformation

Answer: kernel" is used more generally whereas "nullspace" is used essentially only in Linear Algebra


Question: what seemingly innocuous results in mathematics require advanced proofs?

Answer: Jordan Curve Theorem


Question: does friction always do negative work?

Answer: the work friction does is always negative


Question: why does cracking a joint make noise?

Answer: rapid collapsing of cavities inside the joint

Answer: breaking of intra-articular adhesions

Answer: caused by the formation of a gas cavity in synovial fluid of the joints


Question: can gravity waves travel faster than light?

Answer: No


Question: can a number have infinitely many digits before the decimal point?

Answer: there are numbers that have infinite digits before the decimal point

Answer: You can't have infinitely many digits before the decimal point


Question: the identity cannot be a commutator in a banach algebra?

Answer: Corollary 2.2.2 concludes that the identity is not a commutator


Question: discontinuous derivative.

Answer: Volterra’s Function is differentiable everywhere, but its derivative is discontinuous on a set of positive measure


Question: what does breaking stress depends on?

Answer: the nature of the material


Question: does a gun exert enough gravity on the bullet it fired to stop it?

Answer: The gun's gravity is never going to be enough to stop that bullet


Question: are the natural numbers finite?

Answer: not finite but rather infinite


Question: unexpected approximations which have led to important mathematical discoveries

Answer: the fast inverse square root


Question: do most mathematicians know most topics in mathematics?

Answer: I expect an average mathematician to have decent knowledge of between 5% and 15% of topics

Answer: No

Answer: no


Question: on average, how many friends would i need to have to have at least one friends birthday every day?

Answer: about $365\cdot(\ln(365)+0.577)=2364$ friends


Question: in a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

Answer: 1/3

Answer: 1/2

Answer: 33%


Question: intuitive explanation of a positive semidefinite matrix

Answer: a matrix that's "like" a single number $\geq 0$


Question: what is the arithmetic mean of no numbers?

Answer: should not exist

Answer: Error


Question: true or false? $x^2\ne x\implies x\ne 1$

Answer: TRUE


Question: what is the difference between a group and a ring?

Answer: A ring is a group with an additional operation, where the second operation is associative


Question: what is the tor functor?

Answer: the derived functors of the tensor product


Question: how do you explain to a 5th grader why division by zero is meaningless?

Answer: no number of times you can add zero to itself to get a non-zero sum

Answer: zero cookies and you split them evenly among zero friends


Question: proof that the trace of a matrix is the sum of its eigenvalues

Answer: matrix has its eigenvalues on the diagonal, its trace is the sum (with multiplicity) of its eigenvalues


Question: are leptons fermions or bosons?

Answer: fermions


Question: whats an intuitive way to think about the determinant?

Answer: think of a determinant as a volume


Question: what is the answer to an exponent called?

Answer: power of

Answer: power


Question: why does factoring eliminate a hole in the limit?

Answer: because one expression is equivalent to another except right at that hole


Question: why dont miners get boiled to death at $4$ km deep?

Answer: because the mines use suitable cooling and ventilation equipment

Answer: they cool the mine actively


Question: what is the result of $\infty - \infty$?

Answer: Indeterminate

Answer: indeterminate

Answer: there is no reasonable way to define $\infty - \infty$

Answer: it is indeterminate

Answer: The answer is undefined


Question: is every lebesgue measurable function on $\mathbb{r}$ the pointwise limit of continuous functions?

Answer: No


Question: fastest way to meet, without communication, on a sphere?

Answer: one of you stays still

Answer: stay put


Question: is atp synthase a channel protein?

Answer: ATP synthase is an enzyme, a molecular motor, and an ion channel


Question: what are some differences between insects and spiders?

Answer: spiders and insects have different forms of respiration, different mouthparts, and different exoskeletons


Question: what is exponentiation?

Answer: repeated multiplication


Question: more than 99% of groups of order less than 2000 are of order 1024?

Answer: assertion is true


Question: how to calculate difference between two dates in maths?

Answer: go through Julian day numbers and perform substraction


Question: are all triangular matrices invertible?

Answer: only if it has no zeros on the main diagonal


Question: why is the penrose triangle impossible?

Answer: You would end up with a triangle with three $90^{\circ}$ angles


Question: can you raise a number to an irrational exponent?

Answer: Yes


Question: are all diagonalizable matrices invertible?

Answer: No. For instance, the zero matrix is diagonalizable, but isn't invertible


Question: how does the brains energy consumption depend on mental activity?

Answer: The energy consumption doesn't vary that much between resting and performing tasks


Question: what does e mean in 9.0122222900391e-5?

Answer: exponent of 10