Short Answer
Rule Statement
An exact number is a value known with no uncertainty: a counted quantity, a defined constant, or an exact conversion factor. Exact numbers have infinite significant figures (infinite sig figs). They do not limit the precision of a calculation. In scientific notation, write an exact number as a × 10n, where 1 ≤ |a| < 10, and include every digit that is exact. Do not round the coefficient to match measured values. If you need to signal exactness visually, use an overline over the exact digit(s), e.g. 2.54 × 100 cm, but because overline is also used for repeating decimals, the safest practice is to write ‘exact’ or cite the defining standard. Exact numbers are exempt from the usual significant figures rules: they never reduce sig figs in multiplication or division, and they never limit decimal places in addition or subtraction.
Core conventions
- Exact count: 12 eggs = 1.2 × 101 eggs (exact).
- Defined constant: c = 299 792 458 m/s = 2.99792458 × 108 m/s (exact).
- Exact conversion: 1 in = 2.54 cm = 2.54 × 100 cm (exact).
- Measured value: 2.54 cm could also be a measurement with 3 sig figs; context and notation matter.
Worked Examples
Example 1: Exact conversion factor in multiplication
Convert 3.0 inches to centimeters. The conversion factor 1 in = 2.54 cm is exact. Write it as 2.54 × 100 cm/in. The measured length 3.0 in has 2 sig figs. Calculation: 3.0 × 2.54 = 7.62 cm. Because 2.54 is exact, it does not limit the result. The limiting value is 3.0 (2 sig figs), so round to 7.6 cm. Do not keep 7.62 cm unless the original measurement justifies it.
Example 2: Defined SI constant
Since the 2019 SI redefinition, the Avogadro constant is exact: NA = 6.022 140 76 × 1023 mol−1. In scientific notation: 6.02214076 × 1023 mol−1. Every digit is exact. If using overline notation, write 6.02214076 × 1023 mol−1, but ‘exact’ is clearer.
Example 3: Exact count in addition
Add 12 (exact count) and 3.45 g (measured). The exact 12 has no uncertainty and does not limit decimal places. Result: 12 + 3.45 = 15.45 g. If 12 were a measured value with 2 sig figs, the addition rule would limit the result to the ones place: 15 g. The distinction is critical.
Example 4: Overline for significant trailing zeros
A measurement reported as 1500 with 4 sig figs can be written 1.500 × 103. Some authors write 1500 to bar the significant zeros. In scientific notation, 1.500 × 103 is unambiguous. Do not confuse this with an exact count of 1500, which has infinite sig figs and can be written 1.5 × 103 (exact) or 1500 (exact).
Counter-Examples
- Wrong: ‘2.54 has 3 sig figs, so 3.0 in × 2.54 cm/in = 7.62 cm should be 7.62 cm.’ Correct: 2.54 is exact; it does not limit. The result is 7.6 cm because 3.0 has 2 sig figs.
- Wrong: Writing an exact count as 1.50 × 103 when the count is exactly 1500. Correct: 1.50 × 103 implies 3 sig figs and a measured value. Use 1.5 × 103 (exact) or 1500 (exact).
- Wrong: Treating π as an exact decimal. π is exact as a symbol, but 3.14159 is an approximation. In calculations, use enough digits so π does not limit the result, but do not call the decimal exact.
- Wrong: Assuming every overline means exact. In mathematics, 0.3 means 0.333…, a repeating decimal. In sig-fig notation, an overline may mark significant trailing zeros. Context and a stated convention are required.
- Wrong: Rounding a defined constant to 3.00 × 108 m/s and calling it exact. The exact speed of light is 299 792 458 m/s; 3.00 × 108 m/s is a rounded approximation.
Convention Comparison Table
| Notation | Meaning | Sig Figs | Typical Use |
|---|---|---|---|
| 1.5 × 103 | Measured value | 2 | Rounded measurement |
| 1.50 × 103 | Measured value | 3 | Measurement with significant trailing zero |
| 1.500 × 103 | Measured value | 4 | High-precision measurement |
| 1.5 × 103 (exact) | Exact count or definition | Infinite | Counted items, definitions |
| 1.5 × 103 | Exact coefficient (author-dependent) | Infinite | Optional exactness marker; avoid if ambiguous |
| 1.50 × 103 | Trailing zero is significant | 3 | Alternative to 1.50 × 103 |
Best practice: scientific notation already removes trailing-zero ambiguity. Use 1.50 × 103 for 3 sig figs, and write ‘exact’ for exact values. Reserve overline notation for contexts where its meaning is defined.
Standards Citation
International and national standards consistently separate exact values from measured values. ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, Section 6, treats exact numbers as not limiting significant digits. ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 6.4, distinguishes exact and approximate numerical values and gives rounding guidance. NIST SP 811, Guide for the Use of the International System of Units (SI), Section 7.7, advises against rounding exact conversion factors and defined constants. JCGM 100:2008 (GUM), Clause 3.3, defines uncertainty for measured values; exact numbers have zero uncertainty and therefore infinite significant figures. JCGM 200:2012 (VIM) defines exact value and measurement uncertainty in its vocabulary. These references support the rule: exact numbers do not reduce precision, and overline notation must be used with an explicit convention.
Common Mistakes
- Counting sig figs in exact factors. Exact conversion factors such as 2.54 cm/in, 60 s/min, and 1000 mL/L have infinite sig figs.
- Forgetting that exact counts are exact. 5 students, 12 eggs, and 3 trials are exact; they do not limit a calculated average.
- Using overline without a legend. A reader may interpret 1.5 as repeating or as exact. State the convention.
- Confusing exact constants with rounded values. c = 299 792 458 m/s is exact; 3.00 × 108 m/s is a rounded approximation.
- Applying addition decimal-place rules to exact numbers. Exact numbers do not limit decimal places; measured values do.
- Writing exact numbers with unnecessary trailing zeros. 1.500 × 103 (exact) is redundant and can imply a measured precision. Use 1.5 × 103 (exact) or ‘1500 exact’.
Quick Reference Table
| Situation | Correct Notation | Sig Figs |
|---|---|---|
| Counted items: 1500 atoms | 1.5 × 103 atoms (exact) | Infinite |
| Defined constant: speed of light | 2.99792458 × 108 m/s (exact) | Infinite |
| Exact conversion: 1 in = 2.54 cm | 2.54 × 100 cm/in (exact) | Infinite |
| Measured length: 2.54 cm | 2.54 × 100 cm | 3 |
| Measured value with 4 sig figs: 1500 | 1.500 × 103 | 4 |
| Optional exactness marker | 2.54 × 100 (exact) | Infinite |
Sources & Further Reading
- ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, ASTM International.
- ISO 80000-1:2009, Quantities and units — Part 1: General, International Organization for Standardization.
- NIST SP 811, Guide for the Use of the International System of Units (SI), National Institute of Standards and Technology.
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
- JCGM 200:2012, International vocabulary of metrology — Basic and general concepts and associated terms (VIM).
For more precision rules, see Rounding Rules, Sig Figs in Scientific Notation, and Measurement Uncertainty.
FAQ
Do exact numbers have infinite significant figures?
Yes. Exact numbers—counts, defined constants, and exact conversion factors—have no uncertainty, so they have infinite significant figures. They do not limit the sig figs of a result.
How do I write an exact number in scientific notation?
Write it as a coefficient between 1 and 10 times a power of 10, including all exact digits. For example, 299 792 458 m/s becomes 2.99792458 × 10^8 m/s. Add 'exact' or, if your convention defines it, an overline over the exact digits.
Does overline notation always mean exact?
No. In mathematics, an overline over digits often means a repeating decimal, e.g. 0.3 with an overline = 0.333... In significant-figure notation, an overline may mark significant trailing zeros or, in some texts, exact digits. Always check the author's convention.
Can an exact number limit significant figures?
No. Exact numbers never limit significant figures in multiplication or division, and they never limit decimal places in addition or subtraction. Only measured or approximate values limit precision.
Is π an exact number?
π is exact as a mathematical constant, but any finite decimal such as 3.14159 is an approximation. In calculations, treat π as having enough digits not to limit the result, but do not call the decimal exact.
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