3 ms·

A much easier to understand definition for the set of reals is probably to use infinite decimal expansions that don't end in 9999 etc. Equivalence classes of ca

by est31 8y ago

A much easier to understand definition for the set of reals is probably to use infinite decimal expansions that don't end in 9999 etc. Equivalence classes of cauchy sequences is tougher to explain I'd say.

btilly 8y ago
In that case, good luck coming up with a good definition around multiplication where 3 * 0.33333... works out right. And then proving arithmetic properties like the associative law. And then proving that when you do the reals in decimal, you get the same system as the reals in binary. It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representation naturally gives rise to a Cauchy sequence. So, for example, 3.1415926535... gives you (3, 31/10, 314/100, 3141/1000, ...). And as Cauchy sequences, of course, (1, 1, 1, 1,...) is easily proved to be the same as (9/10, 99/100, 999/1000, ...).
- est31 8y ago
  Oh right, 3 * 0.33333 is an issue, yeah. Still, you need to define that equivalence relation on the Cauchy sequences and then explain what a set modulo a relation means.
  - btilly 8y ago
    This is all standard mathematics. The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to infinity of x_n - y_n = 0. Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of all Cauchy sequences which are equivalent to that one. Since "equivalent to" is transitive, any Cauchy sequence in that set will define the same set. Addition and multiplication are defined elementwise. Proving that they are well-defined is relatively straightforward. Their algebraic properties follow for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q, ...) which leads to a unique real number that we somewhat sloppily call q again. I've left some details out, but this construction is well-understood, and is how we define the completion of a metric space.

by est31 8y ago

btilly 8y ago

In that case, good luck coming up with a good definition around multiplication where 3 * 0.33333... works out right. And then proving arithmetic properties like the associative law. And then proving that when you do the reals in decimal, you get the same system as the reals in binary. It sounds harder, but is actually easier to go through the Cauchy sequence definition and then point out that the decimal representation naturally gives rise to a Cauchy sequence. So, for example, 3.1415926535... gives you (3, 31/10, 314/100, 3141/1000, ...). And as Cauchy sequences, of course, (1, 1, 1, 1,...) is easily proved to be the same as (9/10, 99/100, 999/1000, ...).

est31 8y ago
Oh right, 3 * 0.33333 is an issue, yeah. Still, you need to define that equivalence relation on the Cauchy sequences and then explain what a set modulo a relation means.
- btilly 8y ago
  This is all standard mathematics. The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to infinity of x_n - y_n = 0. Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of all Cauchy sequences which are equivalent to that one. Since "equivalent to" is transitive, any Cauchy sequence in that set will define the same set. Addition and multiplication are defined elementwise. Proving that they are well-defined is relatively straightforward. Their algebraic properties follow for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q, ...) which leads to a unique real number that we somewhat sloppily call q again. I've left some details out, but this construction is well-understood, and is how we define the completion of a metric space.