IRRATIONAL NUMBER. :\ny number that cannot he eXIIIVssell as the quotient of two integers. A fraction or quotient of two integers may be expre•,ed in the form of either a termi nating or a non-terminating Ileeinial, the latter always containing a repet end. (See DECI)1.11, SYsTEN1.1 T1111a, 1,(1. a terminating decimal; equals 0.1106i1 a non•terminating deeinials In either ease the decimal may lie transformed again into the common fractional form by the formulas of series (q.v.). But when the process of evolution is applied to in tegers and the results are expressed deeimally, there is often produced a decimal form that is non-terminating, contains no repetend. and can not be expressed as the quotient of two integers. For example. the 'surd' 1/2 equals a number (1.4142.. ) containing a non-telminating and non-repeating. decimal. and cannot he expressed as the quotient of two integers. \\Mile. however, evolution thus often results in an irrational number, it is not every irrational number that can be expressed in the form of a surd. This may be plainly seen in the ease of 7, the ratio of the circumference to the diameter of a circle. The value of this irrational number to five deci mal plqees is 3.14159..... See elm-LE.
Certain operations with irrational numbers were performed by the ancients. The Pythago reans proved the irrationality of the square roots of 3. 5. 7. .. 17. The arithmetic part of Euclid's Elonents contains a geometric treatment of the subject. Archimedes approximated the value ut a great number of surds, stating, for example, that 1:1:51 / ititt -> 20;5 / 1 53, but, the method by which he argued at his results is un known. In the Aliddle Ages Fibonacci, and still later Stifel and Rudolf'', devoted inuch attention to irrationals. Btit not until very recent times has a purely arithmetic theory of surds been produced, through the researches of \Veierstrass, Dedekind. Cantor, and Heine. whose efforts were by a desire to fortify the basis of an tie mathematics. No adequate explanation of
these methods can be given here. That of ‘Veier strass starts with n 11111Siderat inn of the forma tion of different kinds of number through arith metical operations. Dedel:ind 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, C, and C„, so that every 111111111er in is less than eV( ry number in C,, and a is either the greatest number in C, or the least in C,. These rational numbers are then repre sented try points on a straight line. But there are still an infinite number of points on the line for which there are 110 corresponding rational num bers. Ile then shows that to every one of these points corresponds a unique irrational number. Cantor and Heine introduce irrational number through the concept of a fundamental series. Following is an example of the series method. The surd lying 1.41-12 and 1.4143 may be expressed thus; 14 1 4 3 – • - –10 100 10000 1 4 + • 100 1000 11)000or more generally • N. N N• V 10 1 0' 10 10P < , N., Np+ 1 10P Now, as p becomes indefinitely great, evi dently becomes the common limit of the two series, and may therefore be defined by them. Expressing the slim of the series on the left by P/Q, and that on the right lay (P + 1)/Q, the square root of 2 may be expressed by the relation P /Q < <(1' +1)/Q. Similarly, any irra tional number I 111:1• be expressed by the relation 1' / Q<1<(1' + 11/Q. where 1' and Q arc de rived from the corresponding series.
Consult: Dedekind. Essays on Number, trans. by Beinan (Chicago, 1901) ; Dirichlet, l'orTe Nang•n idler Zablf.ntheorie (Brunswiek, 1S791: Stolz. l'orlesuny,n iiber allgemeine .irithmetik (2 vols., Leipzig, 1SS5.S11).