What Even Is a Number?
Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing).<p>In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too).<p>The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader: define appropriate arithmetical functions on the naturals.<p>Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its equivalence classes are called &quot;integers&quot;. The equivalence class containing (a,b) represents the integer b-a. For example, the integer -1 is the equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate arithmetical functions on the integers.<p>Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying (m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation. Its equivalence classes are called &quot;rationals&quot;. The equivalence class containing (m,n) represents the rational m&#x2F;n. For example, 1&#x2F;2 is the equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero integers k}. Exercise: define arithmetic operations on the rationals.<p>To get from the rationals to the reals, see <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dedekind_cut" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dedekind_cut</a>