What rational number is NOT the product of 2 irrational number?
405630
I'll take a guess and say zero, since if a*b = 0, then either a is zero or b is zero.
1144806
The set {1, 2, 3, ...,} has been closed under multiplication, because any product belongs to that set.\n24 and 25 belong to {1, 2, 3, ...,}. 24 x 25 = 600 as well.\n\nThe identity of the multiplication is 1, because n x 1 = 1 x n = n.\n\nAll numbers n have the inverse 1/n except n=0.\n\nAll the precious answers have no sense.
468001
x+2 must not be 0\nso x must not be -2\ndomain is \nall real numbers - (-2)\nall real no except -2
435410
2i or -2i, where i = sqrt(-1)\n\nIf you haven't learned imaginary numbers, then your answer would be that the sqrt(-4) is not a real number.
633963
Let (a,b) be any element of S.\n\nIf a is irrational, then there is a path from (a,b) to (pi, pi) which goes from (a,b) to (a, pi) keeping the first coordinate constant and thus staying in S; then going from (a, pi) to (pi, pi) keeping the second coordinate constant thus also staying in S.\n\nOtherwise a is rational and so b must be irrational.  Then there is a path from (a,b) to (pi,pi) in S which goes from (a,b) to (pi,b) keeping the second coordinate constant and thus staying in S, then from (pi,b) to (pi,pi) keeping the first coordinate constant and thus staying in S.\n\nNow if (a,b) and (c,d) are any points in S, there is a path in S from (a,b) through (pi,pi) and on to (c,d).\n\nQuestion to you: is S arc connected?
795285
first of all, you need to know that there are basically two types of numbers. Real and imaginary. \nreal numbers are those which can be represented in one form or another, but they exist in real world. for example, if you say that there are 4 trees, that means that you can count the number of trees to be a distinct value, ie., 4.\nor consider this. the probability of an event happening is 0.58. this means that the event will be successful 58 times if it is performed 100 times. \nreal numbers consist of all the rational, irrational numbers. that means all these can be realised to be present.\n\nnow coming to imaginary numbers, these are not actually present. \nconsider a number like sqrt(-1). we all know that a square root of a negative number doesnt exist. so, we just say that sqrt(-1) doesnt exist. such numbers can be conveniently represented as Imaginary Numbers. these numbers donot mean anything in the real world, but help in our mathematical calculations.\nnow, hope this answers your question....
188047
you cannot factor this using integers or rational numbers.
395570
You can't!  \n\nDefine a recursive sequence as follows:\n\nx_{n+1} = 1/2(x_n - 1)\n\nThen for any x_0 you plug in for x, you can determine the values of all x_i (i nonnegative integer).  The general formula for this sequence is as follows:\n\nx_n = (1/2)^n * (x_0 - n)\n\nNote then that if you plug in a rational number, all numbers in this sequence will be rational while if you plug in an irrational number, all numbers in this sequence will be irrational.\n\nTherefore, all such sequences that share a common element MUST both have a rational seed, x_0 so that they only contain rational numbers, or both have an irrational seed x_0, so that they only contain irrational numbers.\n\nWithout the condition of continuity then, the function g defined by g(x) = 1 if x is rational and g(x) = 0 if x is irrational is a nonconstant function that will satisfy the given condition.
902410
Irrational numbers are those which cant be expressed as terminating decimals\n\nSQRT(3) and SQRT(2) both can't be expressed as terminating decimals.\n=> they are irrational.\n\n2 * SQRT(2) = Integer * Irrational Number =  Irrational Number\n\nSQRT(2) + 1 = Irrational Number + Integer =  Irrational Number
253144
square root of 2 is irrational here is a proof.  But first the definition.\nRational number:  Any number that can be written in the form a/b for a,b integers such that a,b have no common factors.\nIrrational number:  Any number that is not rational.\n\nProof that sqrt(2) is irrational:\n  Suppose sqrt(2) is rational\nthat means that sqrt(2)=a/b for some a,b integers\nsquare both sides leads to 2=a^2/b^2\nby multiplying by b^2 throughout we get:\n    2*b^2=a^2.\n since the left hand side is divisible by 2 so must be the right hand side.  so,\n       a=2*c for some integer c\nthen 2*b^2=a^2 becomes:\n             2*b^2=4*c^2 or b^2=2*c^2 which shows that b is a multiple of 2 which means that a/b have a common factor which is a contradiction.  So, sqrt(2) must then be irrational.
682294
transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. The most prominent examples of transcendental numbers are π and e.\n\nTranscendental numbers are never rational. However, not all irrational numbers are transcendental: the square root of 2 is irrational, but is a solution of the polynomial x2 − 2 = 0.Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients. This means that pi could not exactly satisfy equations of the type: pi2 = 10, or 9pi4 - 240pi2 + 1492 = 0. These are equations involving simple integers with powers of pi. The numbers pi and e can be expressed as an endless continued fraction or as the limit of an infinite series. The remarkable fraction 355/113 expresses pi accurately to six decimal places.
981666
A number is rational if and only if it can written as a ratio of integers, otherwise it is irrational.\n\nHere are two rational numbers between 5 and 7: \n6 and 6.5.\n(6 can be written as 6/1 and 6.5 as 13/2.)\n\nThat's easy.  I'll give you one irrational.  You'll need to think of one more.\nsquare root of 47 = 6.855654...
386206
Real numbers includes any number positive, negative, or zero, rationl or irrational.  Basically, any possible number you can think of on a number line.  A non-real number would then be imaginary, a complex number whose square is a negative real number.  Good example would be the square root of -1.
912338
Neither, it is undefined, i.e. it has none of the properties of all rational or whole numbers.\n\nI might be wrong, but I think this is one of those trick questions that are so common in math courses.
385685
Consider the numbers\nA1,\nA1+A2,\nA1+A2+A3\n..\nA1+A2+A3+...+An.\n\nDo the computations modulo n. If none of these is divisible by n, then we have n numbers and n-1 non-zero residue classes (mod n), so there are two of these numbers which are equivalent (mod n) by the pigeon hole principle. But then subtracting gives a sum of the required form that is 0 (mod n), in other words, so that n divides that sum.\n\nLooking at the proof, there is no need for the sequence to be monotonic.
198310
It depends on what you mean by "flanked". When you are talking about the rationals and the irrationals, there is no thing as "next to". \n\nWhat you can show is the following:\n\nGiven a rational number q, and ε > 0, you can find an infinite number of irrationals numbers x such that |x-q| < ε. That is now matter how small an interval around q you pick, it will contain irrationals.\n\nSimilarly, given a rational number x, and ε > 0, you can find an infinite number of rationals numbers q such that |x-q| < ε. That is now matter how small an interval around x you pick, it will contain rationals.\n\nBetween any two rational numbers, there is an irrational number.\n\nBetween any two irrational numbers, there is a rational.\n\n\nBy the way, .99999... is not next to 1, it is 1. If you want to know about that, go back through older answers. It's been answered dozens of times.
61490
The square root of 2, , also known as Pythagoras' constant, is the positive real number which, when multiplied by itself, gives the product 2. Its numerical value approximated to 65 decimal places (sequence A002193 in OEIS) is:\n\n1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799. \n was the first known irrational number. Geometrically,  is the length of a diagonal across a square with sides of one unit of length; this follows from Pythagoras' theorem.\n\nA number called the silver ratio has the same expansion only starting with 2.4 rather than 1.4.