Which of these numbers is an irrational number?
354570
D. The square root of 99 can't be expressed as a fraction. It isn't a perfect square, and it really can't be simplified...
1096569
The modular (and clock) arithmetic answers are correct, but I think they miss the point a bit, since in any modular arithmetic you still have 2+2 = 4 mod n. Let's suppose we are using the natural numbers with the usual addition.\n\nThe recurring decimal answers are just stupid, 1 = 0.9999... recurring (as shown in fact by the first answer here), so all of these answers are either irrelevant or wrong.\n\nThe '2' is just a symbol arguments possibly have some merit, but you should be careful. In Indianna in 1872 (or thereabouts) it was ruled by State legislature, that the value of pi is 4. Obviously this creates problems, if you go to Indianna you won't find many buildings that were designed in 1872. (Their mistake of course was that they didn't alter the representation of all the other numbers they where using accordingly).\n\nI'm not really a logician so I don't know the answer to this off the top of my head, I'm pretty sure that we don't have a satisfying answer here (sorry if we have but I missed it). \n\nPerhaps a good place to start looking for a serious discussion on this kind of topic would be to understand what a Peano system is (wiki it, I'm not explaining that here, 'tis an axiomatic description of the natural numbers). I've also heard that Bertrand Russell gave a proof that 1+1 =2 in Principa Mathematica, although I haven't read that and I don't know precisely where you can find it.
769748
The most famous example of an irrational number is pi (the symbol for which is the Greek letter ). You'll hear more about pi when looking at circles, for the time being it's enough to know that pi is approximately equal to 3.14159265358979……..going on and on and on. The numbers after the decimal place are never repeated, hence it's impossible to know exactly the value of pi. This hasn't stopped people trying to find out - pi has been calculated to more that 2,000 million decimal places using powerful computers and still no repeating pattern has been seen!
165715
Rational numbers are ones that can be expressed as the ration of two integers. i.e. as a fraction. 2/3, 113/114, etc. that is why they are called rational numbers. Irrational numbers cannot be expressed as the ratio of two integers, hence the name ir(not) rational.
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?
1161642
It's approximately equal to 1.61803398874989...,\nand is equal to [1 + sq.rt.(5)] / 2.\n\nThe ancient Greeks thought it was divine because the ratio of 1:1.618... provided what they termed the perfect rectangle in architecture.  The ratio 1:1.618... is equal to 0.618...:1\n\nMathematically, it's the only number that when squared is one more than itself.  \n1.61803398874989... * 1.61803398874989... = 2.61803398874989...,\nthough the numbers are irrational, continuing infininely with no repeating pattern.\n\nThe ratio appears in regular pentagons (starfish), Archimedean spirals (seashells), and in DaVinci's proportions in drawing the human figure.\n\n[Edited to comment]\nAll of jalonzmommy's crap doesn't answer the question asked.\nAlso, check out http://en.wikipedia.org/wiki/Golden_ratio
166362
The number that you have here is only a good approximation to the actual value of PI. Others exist that are even better. Their common feature is that unlike PI itself, they are all rational, which is why each has an easy pattern, like the one you noticed just now, for this particular approximation. No easy pattern can be found for PI however, because, as has been said in so many answers, PI is irrational That is not enough though, sqrt(2) is also irrational and yet it has a clearly noticeable pattern, through continued fractions. To add to PI's misery however, it is also transcedental(sp?) and that probably explains why it is so hard to deal with PI, or with E, for that matter. Both are irrational and transcendents and in that respect are different from many other irrational numbers.\nHope that helps.
479145
1) Natural Numbers are numbers that you count.  They are 1,2,3,4,5,6,7 and so on and so forth starting with one.\n2) Whole numbers are numbers that have no decimals.  They are like -1,0,1,2,3,4,5 and so on towards positive and negative infinty.\n3) Integers are the same as whole numbers.\n4) Rational numbers are numbers that can be expressed in a fraction.  Like 21/7ths.\n5) Irrational numbers are numbers that cannot be expressed in a fraction.  Like the square root of 6.5.\n6) Decimals are pieces of a number like .12 dogs or .75 cups of water.\n7) Repeating decimals are decimals the repeat forever.  Such as 1/3 is .3333333 repeating.\n8)  Terminating decimals do not repeat forever.\n9) Solution sets are sets that solve a problem.\n10) Finite sets are sets that have a final answer.\n11) Infinite sets go on forever.\n12) Empty or null set is a problem with no answer.  It looks like a circle with a line through it.\n13) Domain is what x values the function passes through.\n14) An open sentence is a sentence the doesn't end.\n15) An equation is two things connected by a equals sign.  Such as 4x+y=13-7a\n16) Inequalites are two things connected by a greater then or less then sign.  15>2.\n17) Addition is adding numbers 4+5=9.\n18) Multiplication is timesing numbers like 4x5=20.\n19) Polynomials are numbers with more then one term.  Like 4x+5+6y
187972
when a number can be expressed as a terminating (like 0.76) or repeating (like 2.343434....) it is classifyed rational. An irrational can be expresssed as a non-terminating, non-repeating decimal (like pi--3.141592..... or the square root of a non-perfect square.)\n\n2/3 can be expressed as 0.6666........ which is a repeating decimal, so it is rational.
268558
square root of 3 is an irrational number\nirratioal number can't be described as fraction
852187
Saw your edit, see below...\n\nSo if I understand correctly, you start with a decimal number that has a finite number of decimals, like 0.41 or 0.87293, and you want to know if you can build an irrational number that begins with those same decimals, expressing it as a function of integers only.\n\nIn your example, this can be done with 0.41, because sqrt(2) = 1.41421... (the left side of the decimal doesn't matter to you).\n\nSure then.  There are infinitely many ways to do this, but here's just one:\n\nGiven the number "0,a" having n decimals.  So for 0.41, a=41 and n=2.  Then the given number can be expressed as a/10^n.\n\nIf you take sqrt(2)=1.41421... and divide it by 10 often enough, you'll get the number 0.000...0141421... which is irrational.  All you need to do then is add this to the original number.  Just make sure the first 1 appears on the (n+1)th decimal so as not to perturb the first n decimals.  This means dividing by 10^(n+1).\n\nSo the number a/10^n + sqrt(2)/10^(n+1) satisfies the conditions.  Everything is there is integers.\n\nIn the 0.41 case, this would yield the solution 41/100 + sqrt(2)/1000 = 0.41141421356...\n\nEdit: So you don't like it because it uses a.  OK, how about this:\n\nDefine "0,x" as the square of "0,a".  By adding an extra decimal to "0,x" and taking its square root, you will get "0,a" plus an irrational component.  This can be expressed as \n\nsqrt(x/10^n+1/10^(n+1))\n\nwhere n is the number of decimals for "0,x".  In the 0.41 example, this yields the number sqrt(1681/10000 + 1/100000) = sqrt(0.16811) = 0.410012194...
860947
sqrt ( -1 * -1 ) doesn't equal sqrt(-1) * sqrt(-1).\n\nanother example:\n\nsqrt ( 5 * 5 ) = sqrt ( 25) = 5\n\nbut it doesn't equal sqrt(5) * sqrt(5) because this is an irrational answer..\n\nhope this helps
203784
In an odd-number base the number x is odd if Q(x) is odd, and even if Q(x) is even.\n\nIn an even base the number is odd if the last digit is odd, and even if the last digit is even.
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.
57974
Not neccessarily. For instance, the square root of two (which is irrational) multiplied by the square root of two is two, which is very rational.\n\nGood Luck!\n\nP.S. Great Screenname. I love the phantom.
267044
The square root of pi is about 1.772. So the negative square root would be -1.772. I don't believe infinity has a square root, though.
23358
A=Pi*R^2, where A=area of the circle, R=radius of the circle. * stands for multiplication and ^ for exponent. so the formula for Pi would be Pi=A/R^2.\n\nAnother formula is L=2*Pi*R, where L=length of the circle, R=radius of the circle. so the formula would be Pi=L/2*R.\n\nAnd Pi is an irrational number, meaning that it has an infinity of numbers after the comma. It cannot be exactly calculated, only aproximized. Not even using the ultimate computers, because of a single fact: its memory is not infinite, unlike the number, so you cannot phisically represent it.