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In case anyone was curious, 1 billion miles at 70mph is 1631 years.

This is within a factor of ten to a mole of bits. That's an analogy science geeks can relate to.

I remembered incorrectly... Let me check. Actually, the first High Pressure Core Flooder System (for block 3) failed almost exactly 12 hours after the SCRAM. Decay heat generation at that point was was 7-8 MW. (Block 2 and 3 had a thermal power of 2.3 GW, by the way.)

Now, about 8 weeks after the SCRAM, decay heat generation is still approximately 1.6 MW - still a bit more than your average household kettle (actually equivalent to over 500 standard kettles).

A nice plot of the decay heat can be found at Wolfram Alpha (adjust the last bit in brackets if you want to change the time period - unit is in seconds)

If it was on all of the time, then yes. 100W on all the time would be a very significant fraction of my entire power usage. Wolfram Alpha can work out how much it costs for you, for me it's £1.03, if electricity costs 12p/kWh. If electricity is cheaper for you, then it may be less.

You have it backward with regard to the energy cost. You see you have 150 000 V * 0.001mA *25s and for a typical 1kg wood log (assuming total combustion) you have 18 MJ/kg * 1kg and according to wolframalpha 150000 V * 0.001mA * 25s 18 MJ/kg * 1kg = true

There is as yet insufficient data for a meaningful answer.

Your number is even higher than the one I got out of WolframAlpha the other day, which worked out to $1.379 billion (1.379x10^9 or in binary 1010010001100011101111011000000, ) per day. I am reminded of that Oreo video from a few years ago. Seems like if we'd bring some of our forces home to stand like an army aught, then maybe we could do something about the facts that schools are fail, infrastructure is crumbling, health care is a luxury, etc. This is an effing farce. We can't rub two dimes together to help our own people out, but we fall all over ourselves to make sure that the war machine is well oiled.

It's powers of a million, so 1000000^8 or A thousand trillion trillion, a billion billion billion: 1 followed by 27 zeros, 10^27.

One octillion.

The problem, of course, lies in the "too much" in this sentence :

Net neutrality doesn't cover oversubscription at all. You can oversubscribe all you want as long as you don't break neutrality rules - do it too much and your service will just be shitty.

Have you given any thought at all to how much bandwidth this actually *is*, that you demand here ? Mathematically speaking I mean. Let me help you a little bit. You want to guarantee good service. "In the limit" that means that every upstream connection (and *every* interconnect) must have as much bandwidth as ALL downstream connections added together. So let's do this exercise for DSL. You have a million DSL customers, 4 megabit each. To satisfy your absurd requirements the ISP needs 4 TERABIT connections to each and every interconnection point (question : do ISP's get to refuse to connect their network with anyone who cannot delive the required net-neutral interconnect ? Because other than AT&T, Google and Comcast, I seriously doubt anyone could even do it, never mind economically).

Have you noticed the assumption I made in the previous paragraph ? The assumption is that there is NO peer to peer traffic. If you allow peer-to-peer traffic there is NO amount of bandwidth that will ensure smooth operation, since every endpoint would need to be able to serve the entire internet simultaneously. If this is not true, mathematically speaking, *somewhere* you will have congestion and horrible performance.

So let's introduce a few variables : T1 = bandwidth of some user, T2 = bandwidth of some other user, T3 = yet another user. So what do we need to provide internet service without congestion in this situation ?

T1 > T2 + T3, since it must be able to serve them at full speed
T2 > T1 + T3, idem
T3 > T2 + T3, idem
and obviously T1 > 0, T2 > 0, T3 > 0

Why don't we ask wolframalpha what the solution is ?. This is for 3 users, in reality there's a few hundred million internet users.

EVEN disregarding those arguments, that you can never have a congestion-free network that does not violate network neutrality ... there are other problems.

Let's have you use a network *without* qos. Try it :

iptables -A INPUT -m statistic --mode random --probability 0.01 -j DROP

Tell me, this would be the normal tcp way of throttling (without QoS). There would be a consistent low-level of packet loss spread out over all connections, because that's just how tcp works, how it detects the required speed of the link (and with a lot of tcp connections, like every modem has since browser tabs were invented). This would occur mostly on the downlink from the ISP to you, so this would be a relatively accurate simulation.

Go on, enter that rule, and tell me if you like the result. But odds are you're not a fan of hanging web browsers, sites that respond perfectly well one time then refuse to answer for 2 minutes the next, without any fault on the part of either you, the site, or the ISP (but, of course, you're going to blame the ISP, hell I've seen customers complain to high heaven for 0.01% packet loss due to bursty traffic).

So can you please wake up to the technical reality, the necessity of traffic prioritization (including, obviously, prioritizing the ISP's own traffic, prioritizing business and voice users*, ...)

* obviously any isp can only prioritize it's OWN voice users. Because it can make sure no customer disguises bittorrent traffic as voice traffic. It is impossible to do the same for voip traffic of other providers in a secure manner. No QoS (bittorrent throttling, kazaa throttling, ...) means it is impossible for anyone to deliver reliable voice service.

So what are you net neutrality advocates suggesting ? Destroying all internet telephony just because the scorched earth of a nuclear blast is the only truly level playing field ? No QoS means no (reliable) voice service. QoS policies mean no net neutrality. It is that simple.

http://www.wolframalpha.com/input/?i=pi+base+3
the sample size is a little small, but it seems to win for pi, e, pi^e, but not by much and e^pi tied but its not like i play many games

Common misconception!

It is true that the average male life expectancy is 75.5 at birth. But that number includes all the things that kill children, people in their teens, people in their twenties, etc.

If, like Martin, you survive to age 62, then your remaining life expectancy is 81.65 years of age.

So we're safe!

(PS: Understanding this concept about life expectancies is pretty important for retirement planning too! If you live to 65, then your American male life expectancy is 82 years.)

Wouldn't have helped much with the Police here in the UK, you can get thrown in jail for not handing over your encryption password/keys.

For 2 years, rather than 5. Do the maths.

Okay.

Apparently, 5 years is 3 years more than 2 years.

Besides, infinity is not a number nor a size, but a concept.

Here's the mistake which I addressed

Infinity divided on infinity is still theoretically speaking 1.

You're treating infinity as a number such as 1 or 200. If you divide 200 by 200, of course it's 1. Your range of x (1 to 200) is not infinity. Doing mathematical operations on infinity has its own set of rules (i.e. infinity times infinity is infinity, but not dividing them), since infinity is not a real/normal number.

A/A=1, yes?

If A is a real number, excluding 0. But since there are variables out there that range from the negative numbers to positive numbers (charge, for instance, may be -1 C, 0 C, or +1 C), you can see where the problem may lay.

And I might still be learning lim at the moment, so I do not understand the evidence you bring against it either.

Two functions, x-e (changed for clarity) and ln x are both infinity if you let x be infinity. If x is a number that's roughly 4.1386519464791 (irrational number), then x-e /(ln x) = 1 (or nearly 1). But if x is infinity, then x-e /(ln x) is indeterminate because x-e gets larger faster than ln x in the long run. That's why you have to take the limit, to see what happens when x approaches infinity (and you have to perform L'Hospital's Rule).

Try infinity/infinity on a graphing calculator and see what you get, or visit Wolfram Alpha.

Yea except a gram of it will weigh a few million pounds.

I'm almost positive that a gram will always weigh a gram. Did you mean a cubic centimeter?

Pounds are really a measure of force, though. So if the acceleration due to gravity is 9.8m/s^2, a gram will weigh about 0.0098N.

On the surface of a neutron star that was 1.5 solar masses and whose radius is 8km, a gram would weigh 6.99 * 10^8 pounds, so the original poster is in the ballpark.

Yeah, it was kind of ironic the previous poster's linked page having a link to:

http://www.wolframalpha.com/input/?i=population%20growth%20of%20China&lk=2

which shows China has a 0.63% population growth --- 154th in the world.

William

Why on earth would anyone want to remove yet another limit to human growth?

Where do you see a correlation between access to energy and population growth?

The countries with greater population countries are Liberia, Burundi, Afghanistan, Western Sahara, East Timor, Niger, Eritrea, Uganda, Democratic Republic of the Congo and the Palestinian territories. Clearly they have too much access to energy.

What we really need is a Chinese-style one child policy, or better yet incentives for no children at all.

Because, not only that doesn't have any moral implications, as it clearly worked in reducing their population.

Don't get me wrong, I agree that having many children with our current population is completely immoral, but I think that approach to dealing with the problem is misguided.

11. In general, everyone felt a bit safer in every day life. Killings, roberies, etc...not nearly as prevalent.

I'm not sure exactly when or where you're referring to, but the rate of violent crime in the USA has not been lower since the early 70s.

http://www.wolframalpha.com/input/?i=9.5+millimeters+to+pancakes

Wolfram Alpha can't even do that. But the thickness is 9.5mm for those wondering how much "flatter than a pancake" is.

This has to be the most efficient aircraft ever made: those paper airplanes got a glide ratio of 200:1! That's insane! Comparatively, most aircraft get glide ratios around 15:1.

According to the article they launched from Wolfsburg, Germany and some landed in Bangalore, India. Wolfram alpha says that is >7000km. The 122,00 feet is about 37km.

Now witness the power of our fully armed and operational knowledge engine. Integrates at will!

That it's not a search engine.

&#183 (and I think even &middot) has worked fine in HTML for years (albeit not on /. for some reason). I use parentheses, dots, and carets myself, (or enter it in http://www.wolframalpha.com/ and copy the pretty mathematically-correct image), but I'm not New Scientist.

It looks like you ran your calculation with UK barrels. Honestly, I had no idea there was a difference until I checked this out.

http://www.wolframalpha.com/input/?i=one+year+of+crude+oil+consumption+by+one+acre+per+eight-hundred+barrels

You can calculate against the size of Texas (land area of texas), but it doesn't seem to know the size of an American football field.

Your latency would be ridiculous almost 40 minutes but it would be easy enough to get internet there. Although most sites would obviously time out waiting for ACK.

I'm pretty sure the answer is 28... but my calculus is a bit rusty. One of use just failed that captcha (and basic math)!

I cheated

Also, it makes sense. Sin(theta - Pi/2) is the same as Cos(theta). The derivative of Cos(theta) is Sin(theta). And substituting x=0 will give you zero for the problem. (I'm in O.D.E.'s right now so it's all rather fresh)

Not only that, but you could give about 7 IPs to every atom in the body of every human alive on Earth! Taking the number of stars in the observable Universe, each star could get about a quadrillion IP addresses. So yeah, there's plenty of IPs for your toaster :)

Not only that, but you could give about 7 IPs to every atom in the body of every human alive on Earth! Taking the number of stars in the observable Universe, each star could get about a quadrillion IP addresses. So yeah, there's plenty of IPs for your toaster :)

The troubling thing is that's not so far off.

http://www.wolframalpha.com/input/?i=5+MB%2Fhr+in+GB%2Fmonth

And don't forget:

Sweden - 22.4
Norway - 15.7
Canada - 3.66
Australia - 2.77

And don't forget:

Sweden - 22.4
Norway - 15.7
Canada - 3.66
Australia - 2.77

And don't forget:

Sweden - 22.4
Norway - 15.7
Canada - 3.66
Australia - 2.77

And don't forget:

Sweden - 22.4
Norway - 15.7
Canada - 3.66
Australia - 2.77

Sweden - 58

Don't forget about Canada http://www.wolframalpha.com/input/?i=people+per+square+mile+in+Canada

Or Australia http://www.wolframalpha.com/input/?i=people+per+square+mile+in+australia

But back on topic, I've know no internet WITHOUT download caps...

Don't forget about Canada http://www.wolframalpha.com/input/?i=people+per+square+mile+in+Canada

Or Australia http://www.wolframalpha.com/input/?i=people+per+square+mile+in+australia

But back on topic, I've know no internet WITHOUT download caps...

Here's what I think about when people mention universal health care.

Population density in people per square mile:
France - 304
UK - 656
Germany - 611
US - 88

Now, as someone who isn't a coal miner and doesn't have a lifetime of poverty ahead of me, where would I rather live? Because, let me tell you, healthcare costs, they don't bother a lot of people.

Here's what I think about when people mention universal health care.

Population density in people per square mile:
France - 304
UK - 656
Germany - 611
US - 88

Now, as someone who isn't a coal miner and doesn't have a lifetime of poverty ahead of me, where would I rather live? Because, let me tell you, healthcare costs, they don't bother a lot of people.

Here's what I think about when people mention universal health care.

Population density in people per square mile:
France - 304
UK - 656
Germany - 611
US - 88

Now, as someone who isn't a coal miner and doesn't have a lifetime of poverty ahead of me, where would I rather live? Because, let me tell you, healthcare costs, they don't bother a lot of people.

Here's what I think about when people mention universal health care.

Population density in people per square mile:
France - 304
UK - 656
Germany - 611
US - 88

Now, as someone who isn't a coal miner and doesn't have a lifetime of poverty ahead of me, where would I rather live? Because, let me tell you, healthcare costs, they don't bother a lot of people.

They actually have gotten better, and they continue to get better.

Better according to who?"
Maybe the tax payers would like to have a word with you.

They actually have gotten better, and they continue to get better.

Better according to who?"
Maybe the tax payers would like to have a word with you.

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here and read. The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. Every Day Math B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught)

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here and read. The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. Every Day Math B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught)

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here and read. The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. Every Day Math B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught)

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here to read. Hint: The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught) Aughh, this properly formatted text doesn't display properly in the preview screen.

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here to read. Hint: The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught) Aughh, this properly formatted text doesn't display properly in the preview screen.

Ahhhh, an excellent, creative and paradoxical problem. Thank you. First I dissect and disprove U17, then present the reconciliation of the paradox. TO U17: The trouble with math is someone can write an excellent reasoning with advanced techniques and notation. However, if there's a single critical mistake, the whole thing is spoiled. Who would want to drink filtered purified, cold refreshing water, after someone put in a drop of used motor oil? It's perfectly acceptable to multiply by a non zero variable or expression. This spoils U17's otherwise beautiful reasoning. Multiplying by an expression which IS zero, (x = 0) can create extra solutions. Check x =-1 on the step P3, (-1) ^3 ?=? (-1)^2 - (-1), -1 ?=? 1 +1 doesn't work even after we multiplied by x. The extra solution must have come from somewhere else. SPOILER ALERT- THE RESOLUTION OF THE PARADOX IS BELOW. Consider working on the question a while, per Lockhart, then after you've worked on it for a while. Try to rediscover the challenge & creativity of math, then come back here to read. Hint: The truth is x^3 = -1 contains the solutions. We need to delve into complex numbers though. This particular equation has 3 solutions in the set of complex numbers. http://www.wolframalpha.com/input/?i=x^3%20%3D%20-1&t=ff3tb01 x = -1 is real, and reals are a subset of complex numbers. http://www.wolframalpha.com/input/?i={x%20%3D%3D%20%28-1%29^%281%2F3%29}%2C%20{x%20%3D%3D%20-%28-1%29^%282%2F3%29}&t=ff3tb01 x = (1 +- i * root 3) / 2 are two solutions. (should See DeMoivre's theorem (convert to polar coordinates or see the graph), multiply it out by hand or check with WolframAlpha.com to confirm x^3 =-1 And the two complex solutions are our answer. http://www.wolframalpha.com/input/?i=x%3D%3D+1%2F2+(1-i+sqrt(3))%2C+x^2+-+x++%2B+1%3D The original reasoning has two oversights. A. x = -1 is an extraneous solution akin to y = 5, y^2 = 25, y = +- 5 and assuming y can be -5 or +5. Apparently when taking a cube root, we need to check for all solutions if they are extraneous (extra) solutions which rules out x = -1. B. Failure to consider complex solutions when taking a cube root. (Not commonly taught, but hidden right next to the complex properties which are commonly taught) Aughh, this properly formatted text doesn't display properly in the preview screen.

If your car fines you every time you go over 65, then don't drive or 60 or so.

It is certainly politically correct to stare at the speedometer, but for some reason I mostly look at the road and the cars all around me. The speed is dialed as needed for safety. Sometimes I need to make room ahead or behind for other cars to merge; sometimes I need to change speed to move into an adjacent lane and get into the right spot there. Those things take precedence over the speedometer, as long as I am within a few mph around the speed limit.

Driving 60 is simply unsafe; the safest speed is the speed of traffic, and most of the traffic moves at about 5 mph above the limit. You only need to take care of the following distance, and that is under your control. If you have room ahead then you not only can stop your own car; you can also do that slowly enough so that the tailgater behind you has time to react. With rains moving into CA this becomes particularly important. So it's far safer to have 300' distance at 66 mph than 30' at 64 mph.

The first equation doesn't have a real number solution hence the substitution in step 5 is incorrect.

According to Wolfram Alpha, there are enough IPv6 addresses to have one for every 1.5 square picometres of the Earth's surface. If we're talking about /64 subnets, then you can only have one per 27 square millimetres.