IROQUOIS. 788 IRRAWADDY. tlidu^li due ill jiart to their geographic situation mid early aetiuireiiuiit of lireuriiis, was in great iiieaMire llie roMilt ot' their su|)erior system of irgaiiizutioii and of their individual foree of cliaraeler. The same traits have eiiahled them to hold their own iu-<he iiiidsl uf lui alien sur- rounding. Prominent features of their system were the couneil of matrons, the elaborate elan struoture, and the wholesale ail()])tion of cap- tives, who, if spared, were admitted to full trilial rights, instead of heiiig rciluced to seiui- slaverv, as among some other trilies. The best single source of information uiion the confed- eracy is probably Morgan, Lcuyue of the Ho- dOnosatinvr or Inxjiiois ( Hoehestcr, 1854). Dia- lectieally all the lro(|Uoian tribes closely resemble each other. The present number of the Iroi^iiois, inelnding those of llie t'atholie mis~ion eidonies which have cast oil' all allegiance to the aiuieiit league, is considerably above 17,000. distributed as fol- lows: Ontario. Canada. "Iroqunis and Algonquins of Gibson.' 12.5 (perhaps one-half being Iro- quois) ; OJohawks of the Bay of Quinte,' 12.'10; 'Oneidas of the 'niames,' 810: 'Six Nations on the Grand River,' ."JD.'iO. Quebec, Canada, 'Iro- quois of Caughnawaga.' lOHO: 'Iroquois of i^aint Regis,' l.'J2.'>: 'Iroquois and Algonquins of T,ake of Two Mountains,' 440 (jierhaps one-half being Iroquois). In the I'nited States — New York reservations, j.'UO: Wiseonsin (Oneida). 2030; liidi:ni i'cnitory (Seneca), .340. IROQUOIS, or 'Matii.da. A port of entry and manufacturiiig (own of llmidas County, Ontario, Call., 43 miles southeast of Ottawa, on the left bank of the Saint Lawrence (Map: Ontario, H 3). It has a station on the Grand Tnink Railway, and coiiimands the entrance to the Iro- qimis Canal. I'opiilation, in ISOl, 1047; in Umi, 1007. IRRATIONAL NUMBER. .iiy niimlxr that cannot be expressed as the quotient of two integers. A fraction or quotient of two integers may be expresM'd in the form of either a termi- nating or a non-terminating decimal, the latter always containing a repetend. (S«'e Dfcimal Sy.stem.) Thus. { eipials UG, a terminating decimal; % equals O.tiOfiO a non-terminating decimal. In either case the decimal may lie transformed again into the common fractional form by the fonnulas of series (q.v.). But when the process of evolution is ajiplied to in- tegers and the results are expressed decimally, there is often produced a decimal form that is non-terniinating, contains no repetend, and can- not be expressed as the quotient of two integers. For example, the 'surd' /2 equals a number (1.4142..) containing a non-teiiiiinating and non-repeating decimal, and cannot be expressed as the quotient of two integers. While, however, evolution thus often results in an irrational number, it is not every irrational number that can he expressed in the form of a surd. This may be plainly seen in the case of ir. the ratio of the circumference to the diameter of a circle. The value of this irrational number to five deci- mal places is 3. 141.50 See Circle. Certain operations with irrational numbers were perfoniicd by the ancients. The Pythago- reans proved the irrationality of the square roots of 3. 5. 7, . . 17. The arithmetic part of Euclid's Elements contains a geometric treatment of the subject, Archimedes apjiroximated the value of a great number of surds, stating, for example, that ia.Jl / 780 > i/a > 205 / lo3, but the method by which he arrived at his results is un- known. Ill the Middle Ages Kibonacci, and still hiter Stifel and Kudolir, devoted much attention to irrationals. But not until very recent times has a purely arithmetic theory of surds been produced, through the researches of Weierstrass, I)edekind, Cantor, and Heine, whose etTorts were inspired by a desire to fortify the basis of an- alytic nnithcmatics. No adequate explanation of these methods can bo given here. That of Weier- strass starts with a consideration of the forma- tion of dillerent kinds of number through arith- metical operations. Dedekind arranges positive and negative, integral and fractional numbers in order of magnitude, and observes that any rational number, as a, divides the system into two classes, d and C„ so that every number in C, is less than every number in C., and a is either the greatest number in C, or the least ill C;. These rational numbers are then repre- sented by points on a straight line. But there are still an infinite mimher of points on the line for which there are no corresponding rational num- bers, lie then shows that to every one of tlie.se points corresponds a unique irrational number. Cantor and Heine introduce irrational number through the concept of a fundamental .series. I'ollowing is an example of the series method. may be expressed thus: 4 2- 4 — < i/2 10 ^ 100 ^ 1000 10000 ^10^ 100 ^ 1000 ^ 10000 or more generally Nt,N;_,N, Np , ]0^1()^10»^"' 10"^^ "^ Ni^N, N, Np+1 10 ""io^ ^10^ "^ lOp _ Now, as ]i becomes indefinitely great, ^/2 evi- dently becomes the common limit of the two series, and m.ay therefore be defined by them. Expressing the sum of the series on the left bv P/Q, and that on the right by (P+ 1)/Q, the square root of 2 may be expressed by the relation P/'Q< i/2<(P + l)/Q. Similarly, any irra- tional number I may be expressed bv the relation P / Q< I<(P -j- 1)/Q, where P and Q are de- rived from the corresponding series. Consult: Dedekind, Essni/s on iY«m?)cr, trans, by Beman (Chicago. 1901); Dirichlet, V'orle- siniiien iibrr Znhlonthrnrie (Brunswick, 1870); Stolz, Vnrlrf!>i)iqrn iihrr nllqomeine Arithmctik (2 voN., Leipzig. lSS5-8()). IRRAWADDY, Ir'A-wii'di, or lEAWADI. The principal river of Burma. It rises /m the extreme northeast border, near Mount Dapha- bum, a peak of the Nam-kiu Mountains, and flows southward with a tortuous course over 1.500 miles, till it enters the Bay of Bengal through a large delta between the cities of Ran- goon and Bassein (Map: Burma. B 3). The scenery along its banks is extremely varied: the lower valley, especially the delta, is occupied by wide and level rice-fields; farther up there arc undulating, fertile, and thickly populated
