Which set is bigger, the set of rational or irrational numbers?
610549
Both have infinite numbers so both are big and equal
361225
I am assuming you are an early undergraduate or late highschooler.\n\nI would recommend you read the first 30 pages of Maxwell Rosenlicht's Introduction to Analysis (but see below for alternatives).  The book is written with advanced undergraduates in mind, but is a lot more excessible then Analysis books by Rudin, Apostle, or Royden.  Best of all, it is only $15 dollars.  \n\nThe book assumes the existence of the real number line rather than going through the tedious construction and delves straight into the axioms (pp. 15-30).  These will teach you a lot of interesting information about the properties of irrational numbers.  In particular, you want to talk about the least upper bound property and the theorems thereof.  You can build your presentation to the following lemma: An irrational (or real) number may be approximated as closely as we wish by a rational number.\n\nThe book is a great keep if you are interested in an intro to more advance material but if you don't have the time (or don't want to spend the money) I would suggest you look at the topics below in Wikipedia, Math World or another internet source:\n\nRational Number\nIrrational Number\nReal Number \nSupremum (or Least Upper Bound)\nInfemum (or Greatest Lower Bound)
232073
Pi is what is know as an "irrational" number because it cannot be written as a ratio of two integers.  However, it can be easily defined as the ratio of the circumference of a circle (the distance around the circle) to the diameter of the circle (the circle's width).  This makes it an extremely useful number to "approximate" (since it cannot be expressed fully with a 100% accurate number) when dealing with anything that is cyclical (planetary orbits or repeating patterns for example).  I've included a link with more fun facts.
328117
Yes. This is a Frequently Asked Question. Following is my previous answer:\n\nThe real numbers can be defined using Cauchy sequences. Basically, the real numbers make a number line continuous by filling in all possible "holes". If you have a set of real numbers A, and that set is bounded above by some real number (there is a real number that is greater than every real number in A), then we define the set to be a real number that is the smallest real number greater than or equal to every real number in A. This is how one constructs the real number system, by defining every point on a line segment to be a real distance from the start of the line segment (thus all square roots, corresponding to hypotenuses of right triangles, have been incorporated into our number system). \nIn other words, every bounded (from above) subset of rational numbers corresponds to a real number. Now, rational numbers are defined as the set of all ratios of integers. Just like two ratios of integers can correspond to the same rational number (for example, 2/4 and 500/1000 are the same rational number), so can two bounded subsets of rational numbers correspond to the same real number. \nA common way of representing these sets is to use decimal notation. Decimal notation is an infinite series in powers of ten. Each partial sum of the series is taken to be an element of the set in study. A series is said to converge if the set of partial sums is bounded as described above.\nFor example, the number 0.333... is the decimal way of writing the series 0 + 3/10 + 3/(10^2) + 3/(10^3) + ... which corresponds to the set of partial sums {0, 3/10, 33/100, 333/1000, ...}. It is trivial to show that this set is bounded by 1, so this set corresponds to a real number. It can be shown by algebraic manipulation that this set actually represents the rational number 1/3.\nLet us look at the set corresponding to 0.999..., or {0, 9/10, 99/100, 999/1000, ...}. This set corresponds to a real number because each element is of the type (1 - (9/10)^n)/(1/10) = 10(1 - (9/10)^n) (it is a geometric sum; the expression can be derived after a moment's thought). Obviously, no power of any fraction less than 1 is greater than 1 since all such powers are less than the original number, so this is bounded by 1.\nWe hypothesize that 1 is the smallest number that bounds this sequence. Consider the difference 1 - L where L is the smallest number that bounds this sequence. By the field axioms (common algebra), if 1 - L = 0, then 1 = L. Now L is greater than every element of the set in question by definition. Thus, consider 1 - 10(1 - (9/10)^n) = 9^n/10^(n - 1) - 9 = 9((9/10)^(n - 1) - 1). Suppose 1 - L = d for some positive number d. Then (9/10)^N - 1 > D, or (9/10)^N > E ( = d/9 + 1 ) for some constant E (E is not a function of N, only of the fixed difference d between 1 and L) for every possible choice of N. This is absurd, as it implies that after some N, (9/10)^(N + 1) either increases or remains the same as (9/10)^N which is nonsense. Thus, our assumption that there exists a positive difference (and thus any difference) between 1 and L must have been false, so the difference must be 0. \nBy the construction of real numbers, that makes 1 = 0.999... . This is a bit more tedious than proving two rational numbers equivalent (and thus why there exist many hand-waving arguments or misunderstandings of the issue), but is conceptually the same.
662185
Irrational numbers don't have patterns, either.\n\nYou could take pi or e or sqrt(2) out to a hundered digits and stat test it.\n\nI just looked at sqrt(2), sqrt(3), and e.  I couldn't see any pattern to occurances.\n\nAlso, I can't think of any reason there would be more of some number than another.  For years, sqrt(prime) has been used as a source of random numbers.
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...
602511
radicals are root of a number and rational exponents are powers of a number.
1063198
√11 = 3.3166... It's an irrational number.
873608
What makes you think it is?\n\nBecause it is not.\n\nNow, if you want to ask why is it irrational.  That is an unanswerable question.  It is, because it is.
300988
There are different degrees of infinity.\nFor instance,\nThere are more real numbers between 0 and 1 than there are natural numbers (1,2,3,4,...)\n\nTry searching for Georg Cantor.\n\nAlso, a countable infinite set is smaller than an uncountable infinite set.  This is what my example is based on.  The set of numbers between 0 and 1 is uncountable.  The natural numbers are countable.
234765
There is no problem with having an element of a set that is itself a set. Element does not imply anything about being indivisible; it simply means it satisfies the property defining the set. The set of all natural numbers is a set, and the natural numbers can themselves be defined as equivalence classes of sets having certain cardinalities. See the definition of set in the source below.\nI think you should go over your definitions. It is trivial that your set has countable cardinality, and it is not an example of any paradox.
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The type of size you are referring to is the cardinality of sets. Two sets have the same cardinality if each element of one set is related to exactly one element of the second set and all elements of the first and second set are covered by this relation. Such a function is called a bijection. In this way, when talking about countable sets, it is just like pointing to each object and saying 1, 2, etc., relating each object to exactly one natural number. It is a bijection because every object is assigned a natural number and you can find the number of a given object.\nIf a set has the same cardinality as the set or some subset of the natural numbers, then the set is said to be countable. All other sets are uncountable.\nThe set of even numbers is obviously countable, as is the set of natural numbers where each number is increased by 1. The very definition of the set is the bijection necessary. The set of rational numbers is also countable. A given rational number p/q can be written as an ordered pair (p, q). If we make a lattice (natural numbers as axes), we see that we can easily snake a series of straight lines through all rational numbers. See http://www.homeschoolmath.net/teaching/rational-numbers-countable.php .\nThe set of real numbers, the rationals together with all convergent series of rationals, is uncountable by an elegant argument first presented by Georg Cantor. It is the diagonal argument presented at http://en.wikipedia.org/wiki/Cantor's_diagonal_argument . It amounts to showing that given an infinite countable (and thus totally ordered) list of real numbers, one can construct a real number that is not on that infinite list by simply making it differ in the nth term of its decimal sum from the nth real number on the list for each real number on the list. This is the "diagonal number" in the diagonal argument.\nBy the above discussion, we see that all countable sets can be totally ordered (one item is first, a given item is the nth item for some n). However, the converse statement is not true. Not all totally ordered collections are countable. The proper class of ordinals is a totally ordered class that is uncountable (I do not call it a set because the class of ordinals is "too large" to be a set. This is a technicality that you may study in logic and foundations). Ordinals are described here: http://en.wikipedia.org/wiki/Ordinal_numbers .
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It does not, however like Pi there are reasons why it shows up very often.  Phi is the "most irrational" of the irrational numbers.  What I mean by that is that it is the most difficult irrational number to approximate with the ratio of two integers.  All other irrational numbers have better approximations with ratios of fixed magnitude integers. \n\nhttp://goldennumber.net/
425287
No. A rational number is one that can be written as p/q where p and q are integers. Zero can be written as 0/1. In fact,every integer is rational.
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
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.
72448
The set of all rational numbers are countable in the following pattern..\nhttp://www.homeschoolmath.net/teaching/rational-numbers-countable.php\nNow from this you can easily verify if you multiply all the numbers by 2/3 then you will get the set you want..So this set is also countable..