Overview

Very small numbers are written and named using the same general ideas as very large numbers, but with negative exponents or special prefixes to indicate magnitude less than one. Two common approaches are plain decimal names (tenths, thousandths, millionths) and exponential forms such as scientific notation. The same conventions that help to compress and compare large values make it easy to represent quantities much less than one.

How scientific notation works

Scientific notation rewrites a number as a coefficient times a power of ten. For numbers smaller than one the exponent is negative. The coefficient is usually written so that it has one nonzero digit before the decimal point; such a form is called normalized. For example, 0.007 becomes 7 × 10−3 because the first nonzero digit (7) is three places to the right of the decimal point. Likewise, 0.0000452 is 4.52 × 10−5. When spoken aloud these are read as "seven times ten to the minus three" and "four point five two times ten to the minus five." For a concise introduction to scientific notation see scientific notation.

Decimal place names and basic fractions

Decimal place names describe small quantities without exponents: one tenth (0.1), one hundredth (0.01), one thousandth (0.001), one millionth (0.000001), and so on. These names come from fractions: a thousandth equals 1/1000. When reading numbers in words, leading zeros to the right of the decimal point are not pronounced; only the first nonzero digit and following digits are spoken as part of the fraction or the decimal sequence.

SI prefixes for small magnitudes

In science and engineering the International System of Units (SI) supplies standard prefixes that correspond to negative powers of ten. Using prefixes avoids writing many zeros and links the quantity directly to a unit:

  • 10−1 — deci (d)
  • 10−2 — centi (c)
  • 10−3 — milli (m)
  • 10−6 — micro (µ)
  • 10−9 — nano (n)
  • 10−12 — pico (p)
  • 10−15 — femto (f)
  • 10−18 — atto (a)
  • 10−21 — zepto (z)
  • 10−24 — yocto (y)

Thus 0.001 metre can be written as 1 millimetre (1 mm) and 0.000000001 second can be written as 1 nanosecond (1 ns). More on practical unit usage is available at SI prefixes.

Examples, reading tips and variants

Common conversions and reading tips: when converting a decimal to scientific notation, count how many places the decimal point must move to reach the first nonzero digit. That count becomes the magnitude of the exponent with a minus sign for small numbers. Example conversions:

  1. 0.007 = 7 × 10−3 (move decimal three places left → coefficient 7)
  2. 0.0000452 = 4.52 × 10−5 (move decimal five places left → coefficient 4.52)
  3. 0.42 = 4.2 × 10−1

An alternative is engineering notation, which keeps the exponent a multiple of three and aligns easily with SI prefixes (for example 4.52 × 10−5 = 45.2 × 10−6 = 45.2 µ×10−6 when attached to a unit). For further examples and practice problems see notation exercises.

Uses, significance and distinctions

Names for small numbers are essential in fields that measure microscopic or fleeting phenomena: chemistry (concentrations), physics (cross-sections, timescales), electronics (capacitance, currents), and metrology. Choosing an appropriate notation improves clarity: scientific notation emphasizes significant figures and order of magnitude; SI prefixes connect magnitude to units; plain decimal names are often simpler for everyday fractions. Note also that computer science sometimes uses binary prefixes for powers of two rather than powers of ten, which is a different convention and can lead to confusion between, for example, 10−3 and 2−10. For more on conventions and common pitfalls consult conventions and pitfalls.