SilentQ

Y'what? The first equation is not internally consistant. [math]m_1=2m_0[/math] So far so good. Now derive [math]m_2[/math] directly from [math]m_0[/math], whatever it may be. [math]m_2=3m_0[/math] But deriving it from [math]m_1[/math] above, we run into a contradiction. [math]m_2=2m_1=4m_0[/math] The second equation implies that [math]m_t=e^t+a[/math].

The concept of the supernatural makes no sense to me: how can you have something outside nature? If you find something which violates the laws of nature as you understand them, then you need to rethink your laws. Everything that happens is a part of nature by definition.

f18 As Sisyphus says, a fertilised ovum is not alive in its own right, whereas a newborn baby manifestly is. Where to draw the line is tricky. I would advocate the point at which the central nervous system has formed (as far as basic structure is concerned; the brain carries on developing long after birth, and in a sense throughout the individual's life). I don't believe that the circumstances of the pregnancy change the ethical status of an abortion. A rape victim can justifiably obtain abortion in any situation where any other pregnant woman can, and only in such situations. The perfect solution would perhaps be a way for doctors to remove the fœtus without killing it, and growing it in some kind of tank. Such technology is a way off yet, methinks. Hope this post isn't redundant.

Why would you do that?

Arguably that particular rule of real life doesn't apply to this problem, since the original wording of the problem has no problem with the existence of a hen and a half.

Rounding up to the next whole number of hens, yes. Of course, the posing of the problem seems to have no objection to fractions of hens leading a happy, independent existence, so perhaps it should be 20/7, or two and six sevenths of a hen.

This seems to me to be hopelessly naïve. Nobody, as the cliché says, is perfect. I broadly agree that it is a good idea to know someone a bit before starting a relationship with them, but you needn't get to be good friends first: that bit comes after. I know many former couples who are still perfectly good friends. It's a question of maturity. I have never personally been in a romantic relationship. I have certain issues I need to sort out first, and I don't really see it as something I need to be happy at the moment. I would automatically say no if someone came up to me out of the blue and said, 'Will you be my girlfriend?'

If a hen and a half lays an egg and a half in a day and a half, how many hens that are better by half lay a score and a half in a week and a half? I am an evil girl. The major part of the problem is translating it into mathematical notation.

May be generalised to base [math]k[/math] saying that a number is equal to [math]\sum_{i=0}^nk^ix_i[/math]. [math]\sum_{i=0}^nk^ix_i - \sum x = \sum_{i=0}^n(k^ix_i-x_i)[/math] [math]= \sum_{i=0}^n(k^i-1)x_i[/math] [math]= \sum_{i=0}^n(x_i\sum_{j=0}^{i-1}(k-1)k^j)[/math] We may bring the constant factor of [math]k-1[/math] out to the front of both sums, giving [math](k-1)\sum_{i=0}^n(x_i\sum_{j=0}^{i-1}k^j)[/math], which is obviously then a multiple of [math]k-1[/math].

It is circular reasoning. One newton is defined as the force required to produce an acceleration of one metre per second squared in a mass of one kilogramme. Changing Newton's second law would change the definition of a newton, in such a way as [math]F=ma[/math] still held.

There are almost certainly a vast number of species which we have not yet discovered. Most of these are bacterial, and one is probably not a giant primate. On the other hand, if such a creature did exist, the Himalaya / Rockies would be just where it could remain undiscovered for so long.

It's very clever twistage of the laws of mathematics, and I can see it's not meant seriously, but not to let out a small grrr would be a betrayal of my feminist principles. Hence, Grrrrrrrr.

Beg pardon. 180 degrees is [math]\pi[/math] radians, not [math]\frac{\pi}{2}[/math] as I said before. Sorry, my bad.

Hello. I'm Jo and I'm new. I'm an astronomer (currently amateur, but possibly professional eventually), a mathematician and a musician, among other things. I have a definite polymath-y streak to me. Did I mention languages? I am fascinated by languages. And building musical instruments.

How do you pronounce 'WiSci'? I've been alternately pronouncing it 'why-sye' and 'whisky'. Which (if any) of these is correct?

If anything, I've got more dependent on machines as I've progressed. I was told to acquire a graphical calculator upon entering the sixth form (last two years of high school in Britain). I have certainly never had to use infinite series to calculate trig functions. The Maclaurin expansion is used to prove de Moivre's Theorem, which lies at the basis of complex analysis, but using it to calculate values of functions is counterproductive when you can use a calculator. There are certain values you might want to commit to memory. In radians they are the following. [math]\sin 0=\cos\frac{\pi}{2}=0[/math] [math]\sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}[/math] [math]\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}[/math] [math]\sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}[/math] [math]\sin\frac{\pi}{2}=\cos 0=1[/math] [math]\tan 0=0[/math] [math]\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}[/math] [math]\tan\frac{\pi}{4}=1[/math] [math]\tan\frac{\pi}{3}=\sqrt{3}[/math] [math]\tan\frac{\pi}{2}=\infty[/math] Memorising these will certainly serve you well if you intend to take further mathematical study. If you prefer the angles in degrees, 180 degrees are equal to [math]\frac{\pi}{2}[/math] radians. If it helps, think of a right-angled isosceles triangle for [math]\frac{\pi}{4}[/math], and an equilateral triangle cut in half along a line of symmetry for the rest. It is the work of ten seconds to draw a triangle in the margin of an exam script if necessary.

Back in the olden times, you would use a book of tables containing sines, cosines, tangents, arcsines, arccosines, arctangents, natural logarithms and probably natural antilogarithms (exponentials). These were calculated by hand by a person called (confusingly) a computer. The computer would use some kind of infinite series (a Maclaurin or Taylor expansion or similar). It is also possible to build a mechanical device for calculating the sine. I'm not aware of a similar device for the cosine and tangent, but one could no doubt be invented with a little ingenuity.

This is the dark side of the Internet: any nitwit with a computer and an opinion can write garbage like that.

I think you're saying that [math]0.2^2=0.04[/math], or 4%. Remember, though, that it is a change of 20%, and thus the multiplication factor is not 0.2, but 1.2. Therefore the factor in the probability formula ([math]\frac{\pi r^2}{s^2}[/math]) is [math]1.2^2=1.44[/math], or a change of 44%, as the Tree says. The textbook is most probably wrong.