199. When the fractiona/bis not equal toc/d, but greater,ais said to have toba greater ratio thanchas tod; and whena/bis less thanc/d,ais said to have toba less ratio thanchas tod. We propose the following questions as exercises, since they follow very simply from this definition.
I. Ifabe greater thanb, andcless than or equal tod,awill have a greater ratio tobthanchas tod.
II. Ifabe less thanb, andcgreater than or equal tod,ahas a less ratio tobthanchas tod.
III. Ifabe tobascis tod, and ifahave a greater ratio tobthanchas tox,dis less thanx; and ifahave a less ratio tobthanctox,dis greater thanx.
IV.ahas toba greater ratio thanaxtobx+y, and a less ratio thanaxtobx-y.
200. Ifahave toba greater ratio thanchas tod,a+chas tob+da less ratio thanahas tob, but a greater ratio thanchas tod; or, in other words, ifa/bbe the greater of the two fractionsa/bandc/d,
will be greater thanc/d, but less thana/b. To shew this, observe that (mx+ny)/(m+n) must lie betweenxandy, ifxandybe unequal: for ifxbe the less of the two, it is certainly greater than
or thanx; and ifybe the greater of the two, it is certainly less than
or thany. It therefore lies betweenxandy. Now leta/bbex, and letc/dbey: thena=bx,c=dy. Now
is something betweenxandy, as was just proved; therefore
is something betweena/bandc/d. Again, sincea/bandc/dare respectively equal toap/bpandcq/dq, and since, as has just been proved,
lies between the two last, it also lies between the two first; that is, ifpandqbe any numbers or fractions whatsoever,
lies betweena/bandc/d.
201. By the last article we may often form some notion of the value of an expression too complicated to be easily calculated. Thus,
that is, between 1/xand 1/by. And it has been shewn that (a+b)/2 lies betweenaandb, the denominator being considered as 1 + 1.
202. It may also be proved that a fraction such as
always lies among
that is, is less than the greatest of them, and greater than the least.Let these fractions be arranged in order of magnitude; that is, leta/pbe greater thanb/q,b/qbe greater thanc/r, andc/rgreater thand/s. Then by (200)
whence the proposition is evident.
203. It is usual to signify “ais greater thanb” bya>band “ais less thanb” bya
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