A Handwaving Introduction to Infinitesimals

In the early 1960s, Abraham Robinson established nonstandard analysis rigorously. Not digging into the controversies around hyperreal numbers and infinitesimals, here I give a handwaving introduction to infinitesimals, following the construction in Lectures on the Hyperreals, an introduction to nonstandard analysis, GTM 188, by Robert Goldblatt, Springer 1998.

Assuming we know what natural numbers are, we create integers by doing addition and subtraction between two natural numbers. Then we define rational numbers as fractions, i.e., the ratios between an integer and a nonzero integer. Then we can do addition, subtraction, multiplication between two rational numbers, and can also do division by a nonzero rational number.

Then what are real numbers? The theory is very complicated but one way is to define real numbers by Dedekind cut, which essentially says that whenever you split the set of rational numbers into two disjoint parts by a single cut, then the “cut” is a real number. For example, you may split rational numbers into A={rational numbers x such that x³<3} and {rational numbers x such that x³>3}. Then the cut is the cube root of 3, which is not a rational number but a real number.

Another way is to define real numbers as equivalence classes of Cauchy sequences of rational numbers. If you don’t know what it means, that is OK, move on and read the following paragraphs.

Assuming that (we think) we know what real numbers are, consider all sequences of real numbers. We can do addition, subtraction, and multiplication between two real sequences in the following way: for {a_n}={a_{1 ,} a₂, a₃, a₄, a₅, …}, {b_n}={b_{1 ,} b₂, b₃, b₄, b₅, …}, we define

{a_n} + {b_n} ={a₁ + b₁, a₂ + b₂, …}, {a_n} – {b_n} ={a₁ – b₁, a₂ – b₂, …}, {a_n} x {b_n} ={a₁ x b₁, a₂ x b₂, …}.

But we can not define division in this way because some of the terms of {b_n} may be zero. Then we realize that when studying sequences, what we care are the “limiting behavior” of the sequence, not the exact values of the first several terms. So it is tempting to consider sequences whose terms are nonzero except for finitely many terms. For instance, we consider a sequence {b_n} such that b_n>0 for all n>10,000. Then we can do “division” between {a_n} and {b_n} by defining

{a_n} / {b_n} = {whatever you want for the first 10,000 terms, and a₁₀₀₀₁/b₁₀₀₀₁, a₁₀₀₀₂/b₁₀₀₀₂, a₁₀₀₀₃/b₁₀₀₀₃, a₁₀₀₀₄/b₁₀₀₀₄, …}.

This turns out to be a good definition for division between sequences, except that we cannot always require {b_n} to have at most finitely many zero terms. The layman’s words are that as long as almost all terms of {b_n} are nonzero, we can divide any sequence by {b_n}, by doing standard division on the corresponding nonzero terms of {b_n}, and defining other terms to be whatever you want, say just define them to be zero. Correspondingly, if almost all terms of a sequence {b_n} are zero, we will treat that sequence as a zero sequence.

(The rigorous theory for the concept of “almost all terms” is to use a nonprincipal ultra filter, and use equivalence classes. We will not discuss them here.)

Since we can do addition, subtraction, and multiplication between any two sequences, and division between any sequence and a nonzero sequence, the collection of all sequences can do the same arithmetic operations as “real numbers”, and we call them hyperreal numbers. Each real number r can be written as a sequence whose terms are all r, i.e., {r} = {r, r, r, r, r, …}. In this way real numbers are in fact hyperreal numbers.

In addition, hyperreal numbers are ordered in the sense that for any two sequences {a_n} and {b_n}, if almost all terms of {a_n} are smaller than the corresponding terms of {b_n}, then we say {a_n}<{b_n}, and if almost all terms of {a_n} are equal to the corresponding terms of {b_n}, then we say {a_n}={b_n}. By design, one and only one of the situations is true: {a_n}<{b_n}, {a_n}={b_n}, {a_n}>{b_n} (which means {b_n}<{a_n}).

Then what is an infinitesimal? An infinitesimal is a sequence {a_n} that is smaller than any positive real number and bigger than any negative real number, in the sense that for any real numbers r>0 and s<0, almost all terms of {a_n} satisfy s<a_n<r.

The only real infinitesimal is 0. There are many hyperreal infinitesimals. In fact, any real sequence that converges to 0 is a hyperreal infinitesimal under the above definition. Now the question is, what does it mean for a real sequence to converge to 0?

Reference: Lectures on the Hyperreals, an introduction to nonstandard analysis, GTM 188, by Robert Goldblatt, Springer 1998.