What all should I say about irrational numbers for a Calculus presentation?
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)
942166
a transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. the most well known example of a transcendental number, of course, being pi.\ni hope this helps?
702697
it is irrational, the numbers will never repeat in a similar fashion
787164
is there a specific area that you need help with in rational numbers?  I guess you can use a site that i know of.  Here: \nwww.algebra-online.com\n^ after that put a slash and write: rational-numbers-1.htm\nthat might help you (:
550047
rational numbers are numbers that have decimals that end or have decimals that repeat.\n\ne.g. 1.33333333..., 1/3, 2, sq root 4, 3.1212121212..., 3.3233343536\n\nirrational numbers have decimals that cannot be terminated, meaning they can continue on forever.\n\ne.g. pi (mathematical term, not pie for eating!), 3.3233343536... (... is very important because it indicates that the decimals go on, if it doesn't have that then it means decimal ends) sq root 3\n\nhope you understand now :)
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.
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?
471450
weezy is on the right track, that glow1989 is on the right track.  For everyone who thinks that writing 1+1=2 is so obvious that it constitutes a self-contained proof of itself(!)... if you undertake a serious study of mathematical proof, you might find that 357 pages is a good length for a quick introduction to the ideas that are involved in considering questions in this area.  Of course, I am talking about actually UNDERSTANDING something - very different to simply passing an exam, regardless of the 'level' of maths you are enrolled in.\n\nYes, as you mentioned, this is a hard question.  'Proving' that 1+1=2 is not as simple as it sounds, because of the inherent question of what we mean by 'prove'.  You might have noticed that some proofs you have used depend on other proofs, which in turn depend on other proofs, ... which ultimately rest on a handful of axioms.  It is the nature of operations that are allowed by any axiomatic system, that determine the characteristics of mathematical problems that can be said to be SOLVABLE.  It is no small task to understand the axioms that underlie our systems of mathematics.  For example, maths students learn: that division by zero is undefined; that any object raised to a null power is equal to unity; some differences between rational and irrational numbers; what we mean when we consider objects to exist in 'sets'; and so on.  Those who consider themselves to be excellent students are usually able to list these rules in rapid-fire style.  But if we are asked to 'prove' why those rules are the way they are, the questions start to seem fuzzy, and you typically run out of convenient formulas to rely on.  You could easily write 357 pages just to explain what you actually MEAN by the numbers 1 or 2, without worrying about the definition of the addition operation!  As usual, people might do better to stop and consider what the question is asking before booting in a smug answer.\n\nLook into Hilbert's Problem (David Hilbert - German mathematician), which was a more workable and compact form of a proposal by Russell and Whitehead to formalise mathematics on a general basis, according to well-established axioms and procedures.\n\nOther notable mathematicians to work in this area were Alan Turing, Alonzo Church (look for Church's Lambda Calculus), Emil Post, and the brilliant Kurt Gödel.  In more recent decades, look for Gandy (1988) and of course many others.\n\nLol, molawby is spot-on.  I wish he'd posted b4 me... would've saved me a bit of typing.  If all you need to do is learn to sing the right song, to pass your exams and get the job that stipulates that you need to have completed such-and-such a course, then you don't have to worry about any of this.  Of course, that doesn't mean you have to dismiss it either.  But if you are chasing a deeper understanding, it probably does take the 20 yrs, and you won't necessarily get paid for it, heheheh.\n\nFor bedside reading, I suggest Roger Penrose etc...
699709
Rather than tackle all these topics where no one else can see, I will briefly summarize some thoughts here for you.  Of course, you'll get much more from checking out a good Beginning Calculus study aid (like "Schaum's Outline" series) at your local bookstore.\n\nINTERVAL NOTATION\n\nWe can use brackets [ ] and parentheses ( ) to denote portions of the x-axis or any other continuous set of numbers.\n\nExamples are probably best to learn from.  I'll say it in English, then write the interval using notation.\n\n"All the numbers from 0 to 5, but not including 5 itself."  [0,5)\n\nSee how the parenthesis ) means "not including this number"?  And the [ means "include the zero" on the left-hand side of the interval.\n\n"Numbers greater than 6 but less than 11."\n(6,11)\nHere, neither of the interval's endpoints is included in the set, but everything in between is.\n\nIf I could write the symbol for "infinity" I could come up with open-ended intervals:\n"All the numbers smaller than -3" would be the interval (-inf,-3)\n\nIMPORTANT:  When infinity is one of the 'endpoints', a parenthesis is ALWAYS used.  Because you can never 'include' infinity.\n\nLIMITS\n\nThis is normally a lengthy topic, and limits can seem complicated because there are so many different occasions where limits apply.  Even the notation is funny-looking (like an arrow --> that means 'approaches') and takes a while to get used to.\n\nBut the simplest way to think of a limit is as "a possible value that is being approached somehow."  We could speak of x values becoming closer and closer to, say, 15 -- and 15 might turn out to be "the limit" in the problem, in which case x might eventually EQUAL 15 and stop there (or x might just get arbitrarily CLOSE to 15 and something stops it from BEING 15).\n\nIf we choose simple functions like y=3x and then ask "What will be the limit of this function as x approaches 7?" ... the answer is very straightforward.  21!  Because there's nothing to stop x from being 7, and it doesn't frankly matter whether it 'got there' after being 7.1 or 6.9.  The function has no holes (discontinuities) or breaks (asymptotes) or restrictions (such as x<5) ... so life was really easy with y=3x!\n\nBut for functions that are more "interesting" or have funky graphs right around a particular x-value, it pays to be aware how the limit (the y-value) may be a challenge to discover.  The limit may even CHANGE as we let x approach from the 'left' (below) or the 'right' (above).\n\nHere's the last example:\n\nf(x)=(x-1)/(x+5)\nBeing a rational function, f is defined only for values of x that are permitted ... and, like any fraction, the denominator may NEVER be 0.  That eliminates x=-5.  In interval notation, we'd say that the permissible values of x come from the set (-inf,-5) U (-5,inf).  As you can see, sometimes 'interval notation' is not so brief!\nIn those cases, we often simply write x / -5 (where the 'equal sign' has a slash drawn through it).\nNevertheless, f(x) has values we can calculate for every other possible value of x besides -5!  If you graph f(x), you will see that the function behaves 'strangely' right around x=-5.  Those are asymptotes ... a break in the graph that cannot be "fixed" simply by filling in a missing hole!  You could say the function f(x) "increases without limit" or "decreases without limit" on either side of x=-5 ... some teachers prefer "f has infinite limits."\n\nMAXIMA and MINIMA\nThese are merely the greatest and least values that a function might attain over a given interval.  Sometimes a function's least or greatest value occurs at the interval's endpoint(s).  But just as often, the 'lowest' (valley) or 'highest' (peak) might occur somewher
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.
190744
A rational number is a real number that can be written as a ratio of two integers. 0.1 and 1/3 are rational numbers. \nNumbers that cannot be written as ratios of two integers are called irrational. The square root of 2 is not rational, and there is a cute little proof of this fact. 1/0 is not a rational number because it is not a real number. Pi (the ratio of a circle's diameter to its circumference) is also irrational.
171301
The square root of 3 is an irrational number, however, we can approximate it using a bit of calculus.\n\nWe know the sqrt of 4 = 2.  4 is pretty close to 3, so we can estimate the sqrt of 3 to be,\nsqrt(4) - (1/(2*sqrt(4)))*(1) = 1.75\n\nBy plugging sqrt(3) into a calculator, we get about 1.732060808, which is pretty close to our estimate.  The close the numebr you choose to 3, the closer your estimate will be to the "true" value.
280301
Nb2718 and ixat01 have understood the concept well.\n\nYour question has kindled me to refresh my knowledge about numbers.\n\nFirst we shall deal with rational number.\n\nA rational number is in the form P/Q. (P and Q are integers and Q is not equal to zero) \n\nAny rational number is an integer when Q = 1. \n\nDecimals are simply a different way of expressing a rational number.\n\nSquare root of two is an irrational number and hence it cannot be expressed as an irrational number \n\nWhat we can do best is to find a rational number which is more or less equal to that irrational number. \n \nThe number we find can either be written in P/Q form or in decimal form because both represent one and the same.\n\nFor example, the ratio of the circumference of a circle to its radius is an irrational number. \nIf we write 22/7 then it is a rational number.\n If we write 3.1415927, then it is 31415927/10000000. Again it is a rational number. \n\nThen how to represent an irrational number in a simple way? The great mathematicians have guided us.\n\nThe simple way of writing this is Pi. \n\nSimilarly how to write the square root of two?  Write 2^ (1/2).\n\n In fact, the figures 1,2 , 3 etc are all symbols only.\n\nThe multiplication factor 1000000 is written as 10^6 or simply 1M.\n\nIn algebra we use x or y to denote any quantity. Thus there exists many numbers of ways to express any quantity in a simple way.
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Here are 10,000 digits... I could do many millions with ease, but Yahoo probably does not like huge posts.\n\nBy the way, the sqrt(Pi) has an infinite number of digits.  This can be proved using a "diagonalization proof" that proves that the number is irrational.\n\n1.77245385090551602729816748334114518279754945612238712821380778\n9852911284591032181374950656738544665416226823624282570666\n2361528657244226025250937096027870684620376986531051228499\n2517302895082622893209537926796280017463901535147972051670\n0190185234018585446974494912640313921775525906216405419332\n5009063984076137334774751534336679897893658518364087954511\n6516173876005906739343179133280985484624818490205465485219\n5613251561647467515042738761056107996127107210060372044483\n6723652966137080943234988316684242138457096091204204277857\n7806869476657000521830568512541339663694465418151071669388\n3321942929357062268865224420542149948049920756486398874838\n5059306402182140292858112330649789452036211490789622873894\n0324597819851313487126651250629326004465638210967502681249\n6930595420461560761952217391525070207792758099054332900662\n2230676144696612481887430699788352050614644438541853079735\n7425717918563595974995995226384924220388910396640644729397\n2841345043002140564233433039261756134176336320017037654163\n4763206692765418128357624903269045084853201341924359897308\n7119379948293873011126256165881888478597787596376136321863\n4246546641333954355703201522654193952186030497310513829498\n4396591656142459554212266151024785360980955103956007894021\n8809961338285402501680074580272911936642519282051000193635\n0073914643295493433951928853735459200563766502880540575532\n1231890091263228191509149808366956244831008522219239736463\n2484286326114576693242537157737789441409054457359535122562\n6391080239236909732127905807617134603914574791879794124850\n2184451458113418888804132209553321846467097274910285652627\n0784545326222784880098238583630075495095476406237708338835\n7225436621567481327668384244972420874516161833205077991480\n1846668142366936519028454638576148278570377743883762974799\n8273770543158368241099868322850380552635536972229313380526\n4428410372312043967004307612454138311792278275363715598398\n3768845370278429857070905112238405367790133854145853162080\n7304313806973998743669316601381707927205604195488285806309\n3111636297047867814026963272962701226135985897754505289483\n1130166840015320748519824024633375558517135680193412289759\n8071956874025057150214178379254364303036592821125092588061\n8903117074543127903953553660682611001188965742048727593919\n9769955383521150866962555964413705038292449535903106362345\n3056471711621685872545868744002961175792172319055405719868\n1727588419089649657906696515601728351482903856551169807210\n7953309161308435985243894654406821655003275379960238665037\n9888648152118657999585718656377511331597475359604341377664\n5119143460134292508116324806409073773212629335747472967679\n3412716029665120809898090577996603051021816255579789074870\n7621107623802626785429701502710915350498535149239083248428\n0828987595575678848892608420885521269510357370208661259115\n5413204403735608643388371239620642939023786663116326167888\n4192279819499524039424578422044303042043042071096927339294\n6085104969289739161855607837870336428902342932718872968029\n7215816594261297283663959050411303747457435097490580163269\n1653757690981097485625377850342879942192237718584431832793\n7312980065870368619230102174545052030845533038619506602364\n3341153205974122426426535421655815003647269194638254412455\n7259473900719227187097960549897246028226024291050774020812\n3756748076276377437063450463147737694803316835707620587495\n7817335099676287338739670296881906614261445737132282280393\n5444608377601058646738943855776887203971767059273166300903\n1126081448843086241142922479654540261637551586587351984372\n1755055272950641504357687583814935940105565912150377044197\n3880370568097143463163605434136507669997576211981184258742\n252081404364062117230919743945524596778174219146433350671
360581
One way is to construct the rational numbers so that each is written as a fraction that does not need to be simplified.  \n\nBegin with all fractions whose numerator and denomenator sum to 2.  There is only one such fraction (1/1).  Map this fraction to the integer 1\n\n1/1 --> 1\n\nThen consider all fractions whose numberator and denomenator sum to 3.  There are two such fractions:  2/1 and 1/2.  Map these two fractions to the next available integers:\n\n2/1 --> 2\n1/2 --> 3\n\nThen map the three fractions whose numerators and denominators sum to 4 to the next available integers:\n\n3/1 --> 4\n2/2 --> 5\n1/3 --> 6\n\nAnd so forth.\n\nThis algorithm will map all of the rational numbers onto the set of natural numbers.  However, it is a many-to-1, not a 1-to-1 mapping (it maps 1/1 to 1 and also maps 2/2 to 5, but 1/1 and 2/2 have the same value as rational numbers).  You could get around this by skipping over rational numbers that are not simplified fractions.
770041
For "log base 2 of 3", I'm just going to write "log 3".  We'll let the 2 be assumed.  I'll give you a quick sketch of the proof, but you're going to have to fill in the details yourself.\n\nYou're asked to prove that that number is irrational by contradiction.  To prove something by contradiction, you begin by assuming the opposite of what you want to prove and show that that assumption leads to an impossibility.  In that case, the other case must be true, and you've proved it.  This also assumes that there are only two possibilities.\n\nThere are only two possibilities - either a number is rational or it is irrational.  We want to prove that the number is irrational, so we begin by assuming that log 3 is rational.\n\nIf it's rational, that means it can be expressed as a ratio of two integers.  Call those integers a and b.  Then we have that\n\nlog 3 = a/b\n\nRewrite this in exponential form:\n\n2^(a/b) = 3\n\nThis can be rewritten (although not on these forums - at least, not easily) as\n\nThe b'th root of 2^a = 3\n\nA little algebra then says that\n\n2^a = 3^b\n\nwhere, again, a and b are integers.\n\nThere's an obvious contradiction here - figure out what it is and you're done.\n\nHope that helps.
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