Which property of real numbers justifies this?
360090
Everyone's right but w3rd, who fell for the standard teacher trick of making the commutative property LOOK like associative by using 3 numbers instead of just 2.  They key is to look closely at what changes, the placement of parentheses (associative) or the ORDER of the addition (commutative).
804983
did you even try to look this up?\n\ngot to www.yahoo.com and type\n"reflexive property"\n\nWhoa!!
19510
It is not true for all a and b. It is true if and only if a and b are positive real numbers.
758845
NO!\nThe cardinality of the integers the same as that for the rational numbers for instance.  Denoted aleph (א)-null, this "infinity" is "countable", i.e., it can be put in a 1-1 correspondence (bijection) with the natural numbers.\n\nThe cardinality of the real numbers is higher.( א-one *) the number of real numbers cannot be put into a 1-1 correspondence with the natural numbers.\n\nThe number of possible curves in a plane is even higher (א)-two\nETC.......(ad infinitum - no pun intended)\n\nWhat is strange here is that the number of odd natural numbers is the same as the number of all natural numbers, even though the former is a subset of the latter!  That is because a bijection exists between the two sets.\nFor example, let every natural number x be associated with the number 2x-1.   The pairings are:\n1 - 1\n2 - 3\n3 - 5\n4 - 7\n5 - 9\netc.\nEvery natural number is paired with every odd natural number.  Thus the sets have the same cardinality.\nNo bijection exists between these countable sets and say, the real numbers however.  In fact, if you pick ANY two numbers, no matter HOW CLOSE TOGETHER, there are still more real numbers between them than all rational numbers combined. The proof of this is rather cumbersome for here, I suggest you read up on Georg Cantor or infinite sets elsewhere\n\n*It is my "choice" to accept as convention (one side or another of) the continuum hypothesis - it doesnt really matter, but as one answerer pointed out the real numbers' cardinality usually represented as 'c' (Although I never liked that designation-too easily misinterpreted as something else.  I can try to sway supporters, cant I?)
348468
This hypothesis is false:\nSuppose 0/0 = 1. Since 1 is the multiplicative identity, this implies (0/0)*a = a for all real numbers a. But (0/0)*a = (0*a)/0 = 0/0 = 1. This contradicts our hypothesis, so the hypothesis must be false (proof by contradiction). QED\n0/0 is indeterminate as a limiting form. See http://mathworld.wolfram.com/Indeterminate.html . \nAs an algebraic expression over the real field, it is undefined: http://mathworld.wolfram.com/Undefined.html . \nThis is a Frequently Asked Question; the following is my previous answer:\n\nThis question may be asked so frequently (along with the infamous 0.999...) because no concrete axioms or definitions are ever given for the real number system until one studies higher algebra or real analysis; it is assumed that everyone knows what a real number is until then. The proliferation of these questions show that this assumption is no good.\nFor the sake of conciseness, I will not define what a real number is here, as only the algebraic structure of the real numbers is necessary to answer this question. The algebraic structure of the real numbers is what we call a field. A field is an algebraic structure that follows a finite set of axioms. These axioms define what statements are true and false in this system based on the laws of logic. You can find the field axioms here: http://mathworld.wolfram.com/Field.html .\nEach line is preceded by "For all real numbers a, b and c, the following is true...". Note that the proper form of the additive identity and multiplicative identity is that it prescribes the existence of some number 1 and some number 0 such that the equations hold. These same numbers are then used to prescribe the existence of additive and multiplicative inverses.\nDivision a/b is just shorthand for a*(b^(-1)). So in your question, you are looking for 0*0^(-1). It is easy to show that there is no such number as 0^(-1) in any field. Suppose 0^(-1) exists.\nThen 0*0^(-1) = 1. But multiplying both sides by 0^(-1) gives 0 = 0^(-1), which means 0*0 = 0 = 1. This is obviously false.\nSuppose 0 = 1. 0*a = a for all numbers a. But 0*a = (1 + (-1))*a = a + (-a) = 0. Thus 0 is not equivalent to 1 and our sole assumption that 0^(-1) exists must be false. Tus the expression 0*0^(-1) is undefined in a field.\nProof by contradiction is very powerful for many types of statements.\nAs to the limit argument, note that you are evaluating a limit over a plane of the function f(x,y) = x/y. As we approach (0,0) along the x-axis, this limit is 0, but as we approach (0,0) along the y-axis, the limit diverges. Thus, the limit of f(x,y) at (0,0) does not exist, so f(0,0) remains undefined with no continuous way to complete it. The limit is not indeterminate for this function; it does not exist.\nThe expression 0/0 is said to be indeterminate as a limiting form because one can have many functions that approach the form 0/0 that have convergent limits of different values. For example, the limit as x approaches 0 of [sin(x)]/x yields the indeterminate form 0/0, but L'Hopital's rule reveals the limit to be 1. The expressions listed on this page: http://mathworld.wolfram.com/Indeterminate.html are called indeterminate because they do not have a fixed behavior when studied as limits of expressions.
472830
agree, but in case of number its not true, it is infinite
973435
A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field. \n\nover real number x^2+1 is irreducible but not in complex number domain. This can be exteded to modulus. Please refer to
835911
The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of 2; a real number can be given by an infinite decimal representation, such as 2.4871773339…, where the digits continue in some way; the real numbers may be thought of as points on an infinitely long number line.http://en.wikipedia.org/wiki/Real_number\nall real numbers is just ALL of the real numbers\nsomething that is not a real number would be like the square root of -1 http://en.wikipedia.org/wiki/Imaginary_number\nbut don't go too deep into that, because it might confuse you\n\nidentity is when both sides of the equation are EXACTLY the same, so you can't solve it like\n2(4x-6)+3=8x-9\n8x-12+3=8x-9\n8x-9=8x-9 if you try to get rid of the 9 you get 8x\n8x-9+9=8x-9+9\n8x=8x then if you try to get rid of 8x, you get 0\n8x-8x=8x-8x\n0=0\nso, to skip all the unnecessary steps, when you get the EXACTLY\nthe same values on each side of the equation, put identity for the answer
1088429
Any demonstration of this false fact will be flawed, and often uses a hidden division by zero\n\nfor instance:\na and b are real numbers different from zero:\na = b\n\na^2 = ab\na^2 - b^2 = ab - b^2\n(a - b)(a + b) = b(a - b)\n\nhence: \na + b = b,\n2b = b, and:\n2 = 1\n\nHere, i divided by "(a-b)", which is illegal since a -b =0
457166
This is a Frequently Asked Question. Following is my previous answer:\n\nThis question may be asked so frequently (along with the infamous 0.999...) because no concrete axioms or definitions are ever given for the real number system until one studies higher algebra or real analysis; it is assumed that everyone knows what a real number is until then. The proliferation of these questions show that this assumption is no good.\nFor the sake of conciseness, I will not define what a real number is here, as only the algebraic structure of the real numbers is necessary to answer this question. The algebraic structure of the real numbers is what we call a field. A field is an algebraic structure that follows a finite set of axioms. These axioms define what statements are true and false in this system based on the laws of logic. You can find the field axioms here: http://mathworld.wolfram.com/Field.html .\nEach line is preceded by "For all real numbers a, b and c, the following is true...". Note that the proper form of the additive identity and multiplicative identity is that it prescribes the existence of some number 1 and some number 0 such that the equations hold. These same numbers are then used to prescribe the existence of additive and multiplicative inverses.\nDivision a/b is just shorthand for a*(b^(-1)). So in your question, you are looking for 1*0^(-1). It is easy to show that there is no such number as 0^(-1) in any field. Suppose 0^(-1) exists.\nThen 0*0^(-1) = 1. But multiplying both sides by 0^(-1) gives 0 = 0^(-1), which means 0*0 = 0 = 1. This is obviously false.\nSuppose 0 = 1. 0*a = a for all numbers a. But 0*a = (1 + (-1))*a = a + (-a) = 0. Thus 0 is not equivalent to 1 and our sole assumption that 0^(-1) exists must be false. Thus the expression 1*0^(-1) is undefined in a field.\nProof by contradiction is very powerful for many types of statements.\nAs to the limit argument, note that you are evaluating a limit of the function f(x) = 1/x. As we approach 0 from the negative x-axis, this limit decreases without bound, but as we approach 0 along the positive x-axis, the limit increases without bound. Thus, the limit of f(x) at 0 does not exist, so f(0) remains undefined with no continuous way to complete it. The limit is not indeterminate for this function; it does not exist.\nWhile 1/0 is not defined in any number system, |1/0| is defined as positive infinity in extended number systems (which are not fields by the above argument) for various purposes. See http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html .
885409
A number that can be written as a fraction.
75345
Yes. They all have Z_1 many points, which is the number of all real numbers. It has been proved that R(set of real numbers) has same size as R^2(set of pairs of real numbers, such as complex numbers), R^3, ... , R^n.
103035
yes, numbers are infinate
756332
Hi there!\n\nI've heard the name "hyper real" before, but I don't know much about them.  I had a look on the net though, and came up with this link, which looks quite in depth and reliable: http://mathforum.org/dr.math/faq/analysis_hyperreals.html#construction\n\nAlso try having a look on wikipedia, the link is below.\n\nFrom what I can make out from having a quick read of these webpages is that hyperreal numbers include infinite numbers and infinitesimal numbers (i.e. numbers that are smaller than all positive real numbers and greater than all negative real numbers, and it is possible to have non-zero infinitesimal numbers).  \n\nIt helps to understand a little of real analysis, where you have what's known as an epsilon delta definition of the limit of a function, i.e. the real number R is the limit of f (a function from the real numbers to the real numbers) at a if given epsilon greater than zero there exists a delta greater than zero such that if \n0<|x-a|<delta, then |f(x)-R|<epsilon. (where |x-a| means the modulus of x-a, i.e. the [positive] distance between x and a on the real number line).  Basically, this rigorously defines the idea of "getting as close as you like" to the limit, and it was important in the development of calculus, because it put the limit concept on a rigorous setting, and not just an intuitive one.\n\nHowever, according to Wikipedia, Abraham Robinson gave a rigorous definition of infinitely large and infinitely small numbers, in what's known as non-standard analysis, and this is where the hyperreal numbers come in.\n\nHope this is helpful for you!
120877
well i dont fully understand what you mean by structure but the real numbers are pretty much everything that is not imaginary so \n(-infinity,infinity) this includes rationals, irrationals, integers, and natural numbers.  Numbers that are not included are imaginary number (things that have an i attached to it), these numbers occur when you take the square root of -1 (sqrt(-1))\n\nhope that helps
3940
1. It's one multiplied by ten to the twenty-third power.\n  -Otherwise ten to the twenty-third power.\n\n2. It is also an example of multiplicative identity.\n\nSorry if that is not what you are looking for.
50930
I know it.\nIts one more than the Infinity.\nIf you know the value of infinity then substitute it and you'll get the answer.