Real numbers satisfy a(bc) = (ab)c as well as ab = ba. They are also comparable. Generalising to complex numbers, everything stays the same, except the numbers lose their comparibility. Generalising to quaternions, the ab = ba no longer holds Generalising to octonions, a(bc) = (ab)c no longer holds. My question is, can you keep generalising indefinitely, and is there ever a point where the resulting number system has lost so many useful relations/identities that it just becomes a useless mush? (As a speculative example, perhaps it may become so generalised that something as simple as &quot;a + b&quot; becomes meaningless or uncomputable)
61817
In the construction of types of numbers, we have the following sequence:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$  or:  $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$  or:  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"  With the following "properties":   From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering). From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity. From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity. From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.   The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?
27649
It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.  Why isn't this an axiom of arithmetic, like the commutative law or associative law?  Or is it a consequence of some other axiom of arithmetic?  Thanks!  Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number?   In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a &lt; b$ then $a + c &lt; b + c$. In my view $&lt;$ and $=$ are similar so the absence of a definition for equality is strange.
11590
What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think  $$ \mathbb R^+ $$  but I believe that that is usually used to denote "all real numbers including infinity".  So is there a standard way to denote the set  $$ \{x \in \mathbb R : x \geq 0\} \; ?$$
15760
Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").   Yet no one uses these. Why is this?
103839
This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary operators defined on that set, perhaps over some other algebraic structure. Not counting categories here, just "low-level"-ish stuff.  I'm an undergraduate so my knowledge is of course very limited but I can't help to wonder why I never stumble upon algebraic structures with $n$-ary operators? Is there a good reason for this, something along the lines of $n$-ary operators behave "badly" when $n&gt;2$ or is it just my ignorance because I'm a beginner in the field?
76918
Complex numbers are usually formally defined as pairs of real numbers. Although there are operations on $\mathbb{C}$, such as complex multiplication, which are not found in operations usually applied to $\mathbb{R}^2$, the sets themselves seem to be the same. Each consists of pairs of real numbers.  So is it okay to say that $\mathbb{C} = \mathbb{R}^2$? It seems formally correct, but something doesn't feel quite right about it.
67370
I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive integer, therefore there are the same number of rationals as integers. I've ignored sign-related issues, but these are easily handled.  To count the rationals, consider sets of rationals where the denominator and numerator are positive and sum to some constant. If the constant is 2 there's 1/1. If the constant is 3, there's 1/2 and 2/1. If the constant is 4 there's 1/3, 2/2 and 3/1. So far we have counted out 6 rationals, and if we continue long enough, we will eventually count to any specific rational you care to mention.  The trouble is, I find this very hard to accept. I have two reasons. First, this logic seems to assume that infinity is a finite number. You can count to and number any rational, but you cannot number all rationals. You can't even count all positive integers. Infinity is code for "no matter how far you count, you have never counted enough". If it were possible to count to infinity, it would be possible to count one step less and stop at count infinity-1 which must be different to infinity.  The second reason is that it's very easy to construct alternative mappings. Between zero and one there are infinitely many rational numbers, between one and two there are infinitely many rational numbers, and so on. To me, this seems a much more reasonable approach, implying that there are infinite rational numbers for every integer.  But even then, this is just one of many alternative ways to map between ranges of rationals and ranges of integers. Since you can count the rationals, you can equally count stepping by any amount for each rational. You can use 1..10 for the first rational and 11..20 for the second etc. Or 1..100 and 101..200 etc, or 1..1000 and 1001..2000 etc. You can map finite range of integers of any size to each rational this way and, since there is no finite upper bound to the stepping amount, you could argue there are potentially infinite integers for every single rational.  So... can anyone convince me that there is a single unambiguous correct answer to this question? Are there more rational numbers than integers, or not?  EDIT  Although I've already accepted an answer, I'll just add some extra context.  My reason for questioning this relates to the Hilbert space-filling curve. I find this interesting because of applications to multi-dimensional indexing data structures in software. However, I found Hilberts claim that the Hilbert curve literally filled a multi-dimensional space hard to accept.  As mentioned in a comment below, a one meter line segment and a two meter line segment can both be seen as sets of points and, but (by the logic in answers below), those two sets are both the same size (cardinality). Yet we would not claim the two line segments are both the same size. The lengths are finite and different. Going beyond this, we most certainly wouldn't claim that the size of any finite straight line segment is equal to the size of a one-meter-by-one-meter square.  The Hilbert curve reasoning makes sense now - the set of points in the curve is equal to the set of points in the space it fills. Previously, I was thinking too much about basic geometry, and couldn't accept the size of a curve as being equal to the size of a space. However, this isn't based on a fallacious counting-to-infinity argument - it's a necessary consequence of an alternative line of reasoning. The two constructs are equal because they both represent the same set of points. The area/volume/etc of the curve follows from that.
111356
Given:$ A_{n\times n} , B_{n\times n} .$  Is it necessary true that there is similarity between $AB$ and $BA$?  I'm not quite sure how to check it is true for every non-singular M matrix. $M^{-1}ABM = BA$  Anyone have any ideas what am I missing to understand in similarity? If it's not necessary true then give a counter example.
24154
Base-b Representation of an Integer: Why can I make the assumption about number of terms in expansions in uniqueness part?   Reading the book of Koshy I have come across the below theorem. It states that I can write any integer $N$ in a base $b\ge2$ representation $N=a_kb^k+...+a_1b+a_0$ such that $0\le a_i&lt;b$, $a_k \neq 0$ for some $k\ge0$.  In the uniqueness part, why can I assume the expansions contains the same number of terms ? I must prove that all expansions of $N$ equal the one in the theorem, but if I add terms to the expansion by zero coefficients, well, then the expansion is not in the proper form anymore ? (the greatest term in the expansion is $0$ if I add zero coefficients) .  Can someone help clear this out ?     Why can I assume the expansions contain the same number of terms ? If I add terms to the expansion, well then I've violated the form of the expansion ?
54040
I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this.  In the domain of natural numbers, addition and multiplication always generate natural numbers, staying in the same domain.  However subtraction of a large number from a smaller one needs to "escape" into the domain of integers, and division may result in escape to the real domain (like 3 / 5 -&gt; 0.6).  It was a simple step from there to taking the square root of a negative number, hence requiring the escape into the complex domain, such as 4+7i.  He quite easily picked up that each of these domains was a superset of another, natural -&gt; integer -&gt; real -&gt; complex.  However, he then asked if an operation on a complex number would require yet another escape, a question I had to investigate. Now, it turns out that the square root of a complex number is simply another complex number along the lines of mathematical distribution: (a+bi)2 -> a2 + 2abi - b2, from memory.  But I'm wondering if there are other mathematical operations performed on complex numbers (or any of its subset domains) that can't be represented within the complex domain.  Apologies if I've used the wrong terms, it's been about 30 years since I did University level math.
80972
I've ended up in a debate on a proof for .999˜ being equal to 1 (in the real number system.)  I think the arguments have boiled down to whether or not we can preform variable substitution on one side of the equation, while keeping that variable rather than substituting it on the other side of the equation.  What we are looking at:  Given: x = 0.999˜ (Using the ˜ symbol to represent repeating)     .. A     10x - x     = 9.999˜ - x     9x          = 9.999˜ - x     9(.999˜)    = 9.999˜ - (.999˜)     8.999˜      = 9     ..     OR     .. B     10x - x     = 9.999˜ - x     9x          = 9.999˜ - (0.999˜) // Preform a substitution on ONLY the RHS     9x          = 9     x           = 1     ..  What's so confusing is that both should be right...since 8.999˜ IS equal to 9.  But, since this is a proof you should expect the same representation of the number at the most simplified step (I believe?)  So for simplicity sake my questions are...:   Is the algebra valid in both examples A and B? Does example A act as proof regardless of giving different representations of the same number at the most simplified step?
12721
We've just been learning about complex numbers in class, and I don't really see why they're called numbers.  Originally, a number used to be a means of counting (natural numbers).  Then we extend these numbers to instances of owing other people money for instance (integers).  After that, we consider fractions when we need to split things like 2 pizzas between three people.  We then (skipping the algebraic numbers for the purposes of practicality), we use the real numbers to describe any length; e.g. the length of a diagonal of a unit square.  But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition of 'number'?  Wikipedia claims that "A number is a mathematical object used to count, label, and measure", but this definition fails to make sense after we extend $\mathbb{R}$.     Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?
38990
That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by.
89315
Since the  makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this:  If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics.  In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right?  Does this mean that the mathematical foundation of physics is flawed? (I hope this question is not too provocative!)
88366
Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help.
123313
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$. Also... $a,b$ and $n$ are natural numbers.  I feel I should begin with EEA to multiply out the gcd's, but I don't know where to go from there...
57240
I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? How can everything be based on something that is not real. Sorry if what I am saying don't make sense, it hard to put it into words. I feel like I've been known numbers all my life yet I don't know what they are. Also why don't complex numbers have an order? Real numbers are an invention just as complex numbers are, so how are imaginary numbers any more imaginary than real numbers? I mean an order was given to the real numbers, so why can't complex numbers be given a definite order?  Thanks