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Infinitesimal

Infinitesimal

A hypothetical number that is larger than zero but smaller than any positive real number. Although the existence of such numbers makes no sense in the real number system, many worthwhile results can be obtained by overlooking this obstacle.

Note: Sometimes numbers that aren't really infinitesimals are called infinitesimals anyway. The word infinitesimal is occasionally used for tiny positive real numbers that are nearly equal to zero.

 

See also

Differential, infinity, infinite

Key Formula

0<ε<1nfor every positive integer n0 < \varepsilon < \frac{1}{n} \quad \text{for every positive integer } n
Where:
  • ε\varepsilon = An infinitesimal quantity
  • nn = Any positive integer (1, 2, 3, …)

Example

Problem: Use infinitesimals informally to find the derivative of f(x) = x² at a general point x, the way Leibniz originally reasoned.
Step 1: Let dx be an infinitesimal increment in x. Compute the change in f:
f(x+dx)−f(x)=(x+dx)2−x2=2x dx+(dx)2f(x + dx) - f(x) = (x + dx)^2 - x^2 = 2x\,dx + (dx)^2
Step 2: Form the ratio of the change in f to the change in x:
f(x+dx)−f(x)dx=2x dx+(dx)2dx=2x+dx\frac{f(x + dx) - f(x)}{dx} = \frac{2x\,dx + (dx)^2}{dx} = 2x + dx
Step 3: Because dx is infinitesimal, it is negligibly close to zero. Drop the infinitesimal term:
dydx=2x\frac{dy}{dx} = 2x
Answer: The derivative of x² is 2x. The infinitesimal dx let us set up and simplify the difference quotient without needing a formal limit.

Another Example

Problem: Explain why 1/n for large n is sometimes loosely called an infinitesimal, and why it technically is not one.
Step 1: Pick a very large number, say n = 1,000,000. Then 1/n is extremely small:
11,000,000=0.000001\frac{1}{1{,}000{,}000} = 0.000001
Step 2: However, you can always find a smaller positive real number, for instance:
110,000,000=0.0000001<0.000001\frac{1}{10{,}000{,}000} = 0.0000001 < 0.000001
Step 3: A true infinitesimal must be smaller than every such fraction 1/n, no matter how large n is. No real number satisfies this condition (except zero, which is not positive).
Answer: No positive real number qualifies as an infinitesimal. Values like 1/1,000,000 are merely very small; a true infinitesimal is smaller than all of them.

Frequently Asked Questions

Do infinitesimals actually exist as numbers?
Within the standard real number system, no — there is no real number that is positive yet smaller than every fraction 1/n. However, mathematicians have constructed extended number systems (notably the hyperreal numbers, formalized by Abraham Robinson in the 1960s) in which infinitesimals do rigorously exist. So their status depends on which number system you are working in.
What is the difference between an infinitesimal and a limit?
A limit describes the value a quantity approaches as some variable changes; it stays entirely within the real numbers. An infinitesimal is a new kind of number that is already at that 'infinitely small' size. Limits replaced infinitesimals in standard 19th-century calculus because they avoid the logical difficulties of non-real quantities, but both approaches yield the same results.

Infinitesimal vs. Zero

An infinitesimal is positive — strictly greater than zero — yet smaller than any ordinary positive number you can name. Zero itself is not an infinitesimal because it is not positive. In practice, after you finish a calculation with infinitesimals you often 'discard' them, treating the result as if those tiny quantities were zero, but during the calculation they behave as distinct from zero (for example, you can divide by an infinitesimal but not by zero).

Why It Matters

Infinitesimals were the original language of calculus. When Newton and Leibniz invented calculus in the 1600s, they spoke of infinitely small changes dx and dy to derive derivatives and integrals. Although modern courses usually teach limits instead, the infinitesimal viewpoint often gives faster intuition — and it has been made fully rigorous through non-standard analysis. Understanding infinitesimals helps you see why notation like dy/dx looks like a fraction: historically, it was one.

Common Mistakes

Mistake: Treating an infinitesimal as exactly equal to zero during a calculation.
Correction: An infinitesimal is not zero. You can divide by it, multiply by it, and manipulate it algebraically. Only at the end of a derivation do you 'take the standard part' (discard the infinitesimal remainder). Treating it as zero mid-calculation would make expressions like dy/dx meaningless.
Mistake: Believing that very small real numbers like 0.0001 are infinitesimals.
Correction: Any positive real number, no matter how tiny, is still larger than a true infinitesimal. A real number like 0.0001 is just small; an infinitesimal is smaller than every positive real number.

Related Terms

  • Differential — Notation dx, dy built on infinitesimal idea
  • Infinity — Reciprocal of an infinitesimal is infinite
  • Zero — Infinitesimals are near but not equal to zero
  • Real Numbers — Standard system with no infinitesimals
  • Limit — Modern replacement for infinitesimal reasoning
  • Derivative — Originally defined using infinitesimals
  • Positive Number — An infinitesimal is positive yet smaller than all