HalfWit

The mathematical abstraction that describes distances is called a metric space. In a metric space, distance is required to be a nonnegative real number. There's no mathematical theory of negative lengths. That's not to say that someday someone won't come up with something like that, but it's not currently available. The field's wide open to you. http://en.wikipedia.org/wiki/Metric_space

I hear the blacksmiths have been put out of business by these newfangled automobiles.

Pi is finite in every base. It's between 3 and 4 no matter how you represent it. It can't possibly have a terminating or repeating expansion in any integer or rational base (however you define rational bases, it's tricky) because pi is an irrational number.

Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann http://en.wikiquote.org/wiki/John_von_Neumann

The reason that a negative times a negative equals a positive is that it is a logical consequence of the basic properties of the integers, which are known as the ring axioms. You can read (way too much) about them here ... http://en.wikipedia.org/wiki/Ring_(mathematics) I have to apologize in advance for that link because it's way too technical and fails to provide a simplified introduction to the subject. But basically we define a "ring" as any mathematical system in which we can add, subtract, and multiply; and that addition and multiplication are related by the distributive law a(b + c) = ab + ac valid for all objects a, b, and c, as long as they are members of the system in which we're interested. [From now on I'll use the word "numbers" to refer to the members of our system; but keep in mind that these are not necessarily the familiar integers, but rather the members of any system that obeys these rules.] The ring axioms say that there's a magic number called 0 that has the property that 0 + n = n for all numbers n and that for each number n there is an "additive inverse" called -n, whose defining property is that = n + (-n) = 0. There's also a magic number called 1 with the property that 1 * n = n for all numbers n. All these things are true about the everyday integers, so we can use these properties to prove things about the integers. And it turns out to be the distributive property that is crucial. Here's a formal derivation. We want to find out what is (-1)(-1). The way I'm going to do this is to evaluate the quantity (-1)(-1) - 1 and show that it must be zero. This will then prove that (-1)(-1) = 1. I'll use C++ style comments ('//') to provide hopefully helpful commentary. (-1)(-1) - 1 // Expression we want to evaluate. = (-1)(-1) + (-1) // Because "subtraction" is actually adding the additive inverse. = (-1)(-1) + (-1)(1) // Multiplying something by 1 doesn't change it. = (-1)(-1 + 1) // This is the distributive law, which says that a(b + c) = ab + ac. = (-1)(0) // Because -1 + 1 = 0 by definition. -1 is the additive inverse of 1. = 0 // Anything times zero is zero. That's actually a consequence // of the ring axioms and requires proof, which which we'll assume. We just derived a logically certain chain of equality between (-1)(-1) - 1 and 0. If we add 1 to both sides of the equation, we get (-1)(-1) = 1. Conclusion: The ring axioms logically imply that (-1)(-1) = 1. One is still free to impute metaphysical significance to all this, or to try to "figure it out." Mathematicians prefer abstraction. We write down the properties of the thing being studied; and then we derive logical consequences. This methodology of abstraction often provides structural insight. We just discovered that the reason a negative times a negative is positive is simply that it's a logical consequence of the distributive law. Once you accept the distributive law, you have no other choice. That's interesting! By the way we should complete the job by proving the general case (-a)(-b) = ab. That's left to the reader of course

Jeez I'm not going to read about this in the newspapers I hope!

Where do you live? If you are in the SF Bay area you can get a job easily. There's a huge tech boom right now. If you live in a small town in a place with no tech, you will find it much more difficult. Perhaps you might consider moving to where the jobs are.

How do you figure that? If a/i = ai and you multiply both sides by i you get a = -a. Surely you don't believe that every number is equal to its negative.

Somebody make it stop.

Yes. 1, x, x^2, ... are a basis for P(x). There are infinitely many vectors in the basis. And each element of P(x) can be written as a finite linear combination of basis vectors. It doesn't make a whole lot of sense to point out that x^2 = 0 + 0*x + 1*x^2 + 0*x^3 ... It's true, but so what? In any vector space V, any vector v whatsoever can be written v = v + 0*x1 + 0*x2 + ... where the xi's are all the other vectors in the entire vector space. But so what? What's the significance of this to you? That would be true about any basis vector in any vector space. In the Cartesian plan with standard basis {(1,0), (0,1)} you could certainly make the point that (1,0) = (1,0) + 0 * (0,1) but what would be the point of saying that?

No, because "root" is ambiguous. Every nonzero complex number has two distinct square roots. And there is no canonical way to distinguish them. So you have to say which of the two roots you mean in the equation above. All the problems in this thread come down to choosing one square root on the left side and the other square root on the right. This is different from defining sqrt(2) as the positive of the two real numbers whose square is 2. In the real numbers, we can define a privileged subset of positive numbers. In the real numbers, 2 can be algebraically distinguished from -2. But in the complex numbers, there are no positive numbers so that we can not distinguish between the two square roots of a number without explicitly saying which square root we're choosing. There's a somewhat heavygoing discussion of this subject here ... http://en.wikipedia.org/wiki/Branch_point Basically they teach you this stuff in a math major class on complex analysis. The key point is that in the reals, sqrt can be defined unambiguously as the positive one. In the complex numbers, sqrt can not be defined unambiguously except by explicit saying which square root you mean.

No, -1^3 = 1(1^3) since exponentiation has precedence over negation. One of the many persistent errors in this thread. Hello, people, two things please: 1) (-2)^2 = 2^2 does not imply that -2 = 2. 2) -1^48 = -1. (-1)^48 = 1.

I haven't been following this (silly IMO) thread in any detail. But I'm not aware of the above either. Please explain.

If you find a polynomial with rational coefficients that has e as a zero, then e would not be transcendental. That's the definition.

How do you figure that? Multiplying both sides by i gives a = -a, which is your mistake.

If it's the history you're curious about, look up Napier. He's the guy who discovered logarithms ... and having done so, he spent the next 20 years of his life calculating log tables. Talk about dedication. No mechanical calculators were in existence at that time. There are a lot of links about the history. Just google "Napier" and you'll find lots of info.

That's a research project at the professional level, to count the number of pieces of fruit in an arbitrary arrangement. What level is your paper supposed to be at? High school, college, grad school, professional research?

[insert obligatory Up Up and Awaaaaay joke] The first dimension consists of the real numbers, along with the usual addition and multiplication of real numbers. Geometrically, it's often called the real line. The second dimension consists of all the possible pairs (x1, x2) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically, it's known as the plane. The third dimension consists of all possible triples (x1, 2, x3) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically it's generally called 3-space. (Or *real *3-space, in contexts where we are considering n-tuples of complex numbers, etc.) The fourth dimension consists of all possible 4-tuples (x1, x2, x3, x4) of real numbers, along with the operations of component-wise addition and scalar multiplication. Generally it's called 4-space. The fifth dimension consists of all possible 5-tuples (x1, x2, x3, x4, x5) of real numbers, along with the operations of component-wise addition and scalar multiplication. Geometrically it's called 5-space. Dot dot dot. In each case we define the distance of two points as the square root of the sum of the squares of the componentwise differences. We then use the distance function to define a metric; and we use the metric to give us a topology. Then we can have continuous functions, limits, derivatives and all that other good stuff in as many dimensions as we like. Exercise to verify that you've understood this: What is the 475th dimension, mathematically? What's it called geometrically? Write down an explicit expression for the distance between two arbitrary points in real 475-space. Oops I just gave away one of the answers. Moral of the story: Math is not physics. Mathematics is much simpler than physics!

OF COURSE!!! But if you can do that, then you have already no doubts that .999... = 1. Of course you CAN multiply a convergent series term by term. But you have to PROVE that you can do it, since infinitary operations are not mentioned in the field axioms. By the time you prove the theorem on term-by-term multiplication, you've already developed the theory of real numbers, limits, and infinite series. And if the sum of a series is the limit of the sequence of partial sums, then what is a sequence, and what is a limit? A sequence is a function whose domain is the natural numbers. And what's a function? It's a subset of the Cartesian product of two sets, and so on down the chain. In order to prove term-by-term multiplication, you need to start from the axioms of set theory and work your way up through half a semester of Real Analysis. It takes undergrad math majors three years to get to this point. By the time you've done all that, you have no doubt that .999... = 1. The 10x proof is circular. It assumes facts far more powerful than .999... = 1. It's not that the 10x proof is wrong. It's that it already assumes a huge amount of technical machinery far more powerful than the mere fact that .999... = 1. That's why the 10x argument is only an informal justification, and not a proof.

If you already know the series is convergent; and you already know what convergence means; then there is nothing to prove, you're already done. So you see, that step is not valid. Because it assumes what you are trying to prove. By the way, you don't need absolute convergence to multiply a series term by term. Plain old convergence suffices. But you have to prove the theorem on term-by-term convergence of an infinite series, since it's not directly allowed by the field axioms. But if you can prove that theorem, you certainly have no confusion about .999... = 1.

You are correct that this commonly seen "proof" is a bit bogus. The field axioms for the real numbers only allow you to multiply a constant times a finite sum. In order to multiply 10 * .999... term by term, you need to develop the theory of infinite sequences and series ... by which time, the fact that .999... = 1 has already been proved. So this "10x" proof is really just a heuristic hand-wavy pseudo-proof for the benefit of high school students. However, it's still true that .999... = 1. For one thing, can you name a number that's strictly larger than .999... and strictly less than 1? No, you can't. Secondly, in freshman calculus they teach you about geometric series. .999... is a shorthand for 9/10 + 9/100 + 9/1000 + ..., which is a geometric series whose sum is 1. To sum up: .999... = 1, and that's a provable mathematical fact. However, the common "10x" proof is flawed, since it assumes facts that are already more mathematically sophisticated than the fact that .999... = 1. Namely, the "10x" proof assumes that you can do term-by-term multiplication of an infinite sum. However, that operation is not allowed by the axioms for real numbers; and can only be proven after you have developed a rigorous theory of the real numbers and then a rigorous theory of infinite sequences and series.

I'm a tau-ist. One-quarter of the way around the unit circle is ... pi/2. I have to stop and think about that every single time. It would be more natural to say that one-quarter of the way around is tau/4. One less mental translation to have to make every single time. You'd save a lot of cpu cycles in your head. I have a theory on how this came about. In ancient times, math was developed to keep track of the crops. Farmers needed to know how big their field was. So diameters were important. In our modern, abstract world, what's important is the unit circle in the plane. So much math and technology comes from the study of functions on the unit circle. From advanced abstract math to the digital signaling that underlies all our communications, the unit circle is where it all starts. It's the radius that's important, not the diameter. As in so many aspects of modern life, we are stuck with a system designed for a world we no longer live in.

It's very intuitive once you get it. It's hard to explain without diagrams. But draw your unit circle. i is an angle of pi/2 radians. Its square root is an angle of pi/4 radians. because **** Multiplying complex numbers is just adding their angles!! **** So when I have pi/4 and I square it, that's the same as rotating pi/4 and then rotating pi/4 again ... which would leave you at pi/2. So if you want the square root of pi/4, it must be pi/8 for the same reason. You just keep bisecting its angle on your unit circle. Now I must mention that I'm playing loose with terminology. What I really mean by pi/4 is "The complex number that lies on the unit circle and makes an angle of pi/4 between it, the origin, and the positive x-axis." If you just draw this I believe you will be enlightened about this whole business. It's just **adding angles**.

The way to understand i is geometrically. It's a counterclockwise rotation of the plane through an angle of pi/2 radians, or 90 degrees. As you can see, if you do it once and then do it a second time, you end up at the complex number -1, or the point on the plane (-1,0). So i^2 = -1 is a geometric triviality. Now, what is sqrt(i)? Well, what rotation can you do two times in a row to get to i? Answer: A rotation of pi/4 or 45 degrees. And what complex number corresponds to a rotation of pi/4? It's: sqrt(2) / 2 + i * sqrt(2) / 2 You can multiply it out to see that it works; and you can observe that if I rotate 45 degrees and then again 45 degrees, I have in effect rotated 90 degrees.

I seem to remember this essay from somewhere else. I see you haven't taken the trouble to improve it. But you should. The above statement is unsupportable. First, it only applies to finite sets. Secondly, if you are building up math from first principles, sets are logically prior to numbers. So I don't know what you mean by "number of elements." If you are going to take the trouble to lay out an overview of set theory for general readers, the least you could do is try for a bit of accuracy. You can't say that cardinality is a "number" without saying what you mean by number; and you can't then start waving your hands at the cardinality of infinite sets, having given a misleading and incorrect definition of cardinality in the first place. I remember being annoyed by these problems the first time I read this, and your repeating it doesn't make it any better. People who take the trouble to lay out general purpose overviews of technical subjects have a responsibility to try to be accurate within the limits of understandability. Making a false statement early on and then introducing additional confusion based on your faulty statement is poor pedagogy. Why not rewrite your essay based on the well-understood and accepted notion of one-to-one correspondence? That is both accurate math and good pedagogy. What you have here is neither.