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But other kinds of numbers can be made from that: - Differences of counting numbers (integers) - Ratios of numbers (rationals) - Limits to sequences of numbe

But other kinds of numbers can be made from that: - Differences of counting numbers (integers) - Ratios of numbers (rationals) - Limits to sequences of numbers (reals) - Solutions to polynomials made from numbers (complex) You can think of each of these as adding a layer of behavior to the basic “each number is nothing or one more than another number”.

bitwize 8y ago
One way to explain the real numbers to a child might be: There are whole numbers (integers, terminology not accurate but it'll do for now), and then there are numbers called fractions (rationals) which are either whole numbers themselves, or they fall between two whole numbers, and we get them by dividing one whole number by another. The real numbers are either fractions, or they are numbers that fall between two fractions. If it's one inch from the center of a circle to its edge, the distance around the circle is not a fraction. In inches it's about 3.14; it falls between 3 and 3 1/7, and no fraction you can possibly choose will ever get it exactly, it will be at best a little under or a little over. You can get other non-fraction real numbers by finding square roots, but there are infinitely many of these numbers between any two fractions, not all of which are square roots. And if you try to take the square root of a negative number, you won't get a real number at all, and we call these numbers "imaginary". Hearing stuff like this as a kid blew my mind, a bit like learning about black holes. It set me up to enjoy math throughout my life.
kccqzy 8y ago
Nitpick but sequences of numbers can diverge (converge to infinity) or be stuck in a cycle; it's more accurate to say limits of Cauchy sequences.
btilly 8y ago
I agree with your sequence until the last two. The second to last, that should be Cauchy sequences. And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through complex analysis. For those who don't know the proof, the idea is this. Liouville's theorem says that if a function is differentiable everywhere, and it is bounded, then it must be constant. Now suppose that p(z) is a polynomial. Consider the function 1/p(z). You can show that as z approaches infinity, it approaches zero. It is not constant. Therefore it must not be differentiable or not bounded. It doesn't take too much work from there to prove that it blows up somewhere, and the spot that it blows up is a point where p(z) is 0. Apply unique factorization for polynomials (see http://sites.millersville.edu/bikenaga/abstract-algebra-2/polyufd/polyufd.html http://sites.millersville.edu/bikenaga/abstract-algebra-2/po... for that proof) and you quickly get the fact that the complex numbers are algebraically closed.
- 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.

bitwize 8y ago

One way to explain the real numbers to a child might be: There are whole numbers (integers, terminology not accurate but it'll do for now), and then there are numbers called fractions (rationals) which are either whole numbers themselves, or they fall between two whole numbers, and we get them by dividing one whole number by another. The real numbers are either fractions, or they are numbers that fall between two fractions. If it's one inch from the center of a circle to its edge, the distance around the circle is not a fraction. In inches it's about 3.14; it falls between 3 and 3 1/7, and no fraction you can possibly choose will ever get it exactly, it will be at best a little under or a little over. You can get other non-fraction real numbers by finding square roots, but there are infinitely many of these numbers between any two fractions, not all of which are square roots. And if you try to take the square root of a negative number, you won't get a real number at all, and we call these numbers "imaginary". Hearing stuff like this as a kid blew my mind, a bit like learning about black holes. It set me up to enjoy math throughout my life.

kccqzy 8y ago

Nitpick but sequences of numbers can diverge (converge to infinity) or be stuck in a cycle; it's more accurate to say limits of Cauchy sequences.

btilly 8y ago

I agree with your sequence until the last two. The second to last, that should be Cauchy sequences. And for the last, demonstrating that the algebraic closure of the reals is the complex numbers from first principles is much harder than describing complex numbers as pairs of reals with a multiplication rule that (a,b) * (c,d) = (ac - bd, ad + bc) and then much later proving that it is algebraically closed through complex analysis. For those who don't know the proof, the idea is this. Liouville's theorem says that if a function is differentiable everywhere, and it is bounded, then it must be constant. Now suppose that p(z) is a polynomial. Consider the function 1/p(z). You can show that as z approaches infinity, it approaches zero. It is not constant. Therefore it must not be differentiable or not bounded. It doesn't take too much work from there to prove that it blows up somewhere, and the spot that it blows up is a point where p(z) is 0. Apply unique factorization for polynomials (see http://sites.millersville.edu/bikenaga/abstract-algebra-2/polyufd/polyufd.html http://sites.millersville.edu/bikenaga/abstract-algebra-2/po... for that proof) and you quickly get the fact that the complex numbers are algebraically closed.

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.