ECON 1530 Lecture Notes - Lecture 1: Rational Number, Natural Number, Real Number

• School: York University
• Department: Economics
• Course: ECON 1530
• Professor: John Paul Rilstone

Document Summary

Other examples of rational numbers are: the rational numbers can also be represented on the number line. Imagine that we first mark 1/2 and all the multiples of 1/2. Then we mark 1/3 and all the multiples of 1/3, and so fo(cid:396)th. You (cid:272)a(cid:374) (cid:271)e e(cid:454)(cid:272)used fo(cid:396) thi(cid:374)ki(cid:374)g that (cid:862)fi(cid:374)all(cid:455)(cid:863) there will be no more places left for putting more points on the line. The ancient greeks already understood that (cid:862)holes(cid:863) (cid:449)ould remain in the number line even after all the rational numbers had been marked off. Fo(cid:396) i(cid:374)sta(cid:374)(cid:272)e, the(cid:396)e a(cid:396)e (cid:374)o i(cid:374)tege(cid:396)s p a(cid:374)d (cid:395) su(cid:272)h that (cid:1006) = p/(cid:395): he(cid:374)(cid:272)e, (cid:1006) is (cid:374)ot a (cid:396)atio(cid:374)al (cid:374)u(cid:373)(cid:271)e(cid:396). (cid:894)eu(cid:272)lid p(cid:396)o(cid:448)ed this fa(cid:272)t i(cid:374) a(cid:396)ou(cid:374)d the (cid:455)ea(cid:396) 300 bc. : the rational numbers are therefore insufficient for measuring all possible lengths, let alone areas and volumes. This deficiency can be remedied by extending the concept of numbers to allow for the so-called irrational numbers.