Scientific Notation: Complete Guide with Sig Figs, Conversion, and Arithmetic

Scientific notation writes a number as a mantissa between 1 and 10, multiplied by a power of 10 — for example, 45,000,000 becomes 4.5 × 10⁷. It exists to make very large and very small numbers manageable, but on a site about precision it earns its keep for a second reason: it’s the definitive fix for the one real ambiguity in significant figures. A number like 45000 can’t tell you on its own whether it has 2, 3, 4, or 5 significant figures; written as 4.5 × 10⁴, there’s nothing left to guess.

This page covers three things the title promises, in the order you’ll actually need them: how to convert numbers into and out of scientific notation, how significant figures work once you’re there (short version — cleanly, which is the whole point), and how to actually do arithmetic in this format without the two mistakes that trip up almost everyone: forgetting to match exponents before adding, and forgetting to renormalize after multiplying. Along the way it also covers e-notation (the “3.5e3” your calculator shows) and engineering notation (the “45 × 10⁶” your electrical engineering textbook prefers) — related formats that are easy to conflate with standard scientific notation but follow different rules.

<!– BLOCK: B02 – Inline Mini-Calculator –> <!– DEV NOTE: Embed the Scientific/E-Notation Converter tool here (Tier 1 tool #5 in the site blueprint). Shortcode: [sfc_notation_converter default_input=”45000000″ show_engineering=”true” show_sig_figs=”true”]. Should display standard form, scientific notation, E-notation, and engineering notation simultaneously, with the sig fig count of the mantissa shown alongside. –>

[Live Scientific/E-Notation Converter embeds here] — Enter any number to see it in standard form, scientific notation, E-notation, and engineering notation at once, with its significant figure count shown for each.

The Three Notation Formats

Standard scientific notation. Written as a × 10ⁿ, where the mantissa a satisfies 1 ≤ |a| < 10, and the exponent n is any integer — positive, negative, or zero. 3.5 × 10³ = 3,500. 3.5 × 10⁻³ = 0.0035.

E-notation. The exact same concept, written on a single line without superscripts: aEn or aen replaces “a × 10ⁿ.” 3.5E3 and 3.5e3 both mean 3.5 × 10³ = 3,500. This is a typographic convention, not a mathematical one — it exists because early computer displays and plain text couldn’t render a superscript exponent, and it stuck. The “e” here has nothing to do with Euler’s number (e ≈ 2.71828); it’s purely a separator meaning “times ten to the power of.” This is worth stating plainly because it’s a genuinely common point of confusion the first time someone sees a calculator or spreadsheet output “e” instead of “× 10.”

Engineering notation. The same underlying idea, with two added constraints: the exponent must be a multiple of 3 (…,−6, −3, 0, 3, 6, 9,…), and correspondingly the mantissa’s allowed range expands to 1 ≤ |a| < 1000. The payoff for the wider mantissa range is that the exponent now lines up exactly with a named SI prefix — 10³ is kilo, 10⁶ is mega, 10⁻³ is milli — which is why engineering and electronics work generally prefers it over standard scientific notation.

The exponent is always exact. This is worth stating explicitly because it connects directly back to our significant figures guide: the exponent is a defined integer describing magnitude, not a measurement. It never carries uncertainty, is never itself subject to sig fig rounding, and never counts toward — or against — a number’s significant figure total. Only the mantissa’s digits do that.

Orders of magnitude. Loosely, this just means the exponent itself. Two quantities that differ by one order of magnitude differ by roughly a factor of 10; three orders of magnitude apart means roughly a factor of 1,000. It’s a fast way to compare scale without needing exact values — a typical bacterium (on the order of 10⁻⁶ m) and a grain of sand (on the order of 10⁻⁴ m) differ by about two orders of magnitude, meaning the grain of sand is roughly 100 times larger, well before either measurement gets pinned down more precisely.

Worked Examples

Example 1 — Converting to scientific notation

Large number: 45,000,000 → move the decimal point left until one non-zero digit remains before it (7 places) → 4.5 × 10⁷

Small number: 0.0000032 → move the decimal point right until one non-zero digit remains before it (6 places), and the exponent is negative because the original number is less than 1 → 3.2 × 10⁻⁶

Example 2 — Converting back to standard form

From a positive exponent: 3.2 × 10⁵ → move the decimal point 5 places right → 320,000

From a negative exponent: 5.6 × 10⁻⁴ → move the decimal point 4 places left → 0.00056

Example 3 — Significant figures carry over exactly

Number: 5.60 × 10⁴

Only the mantissa’s digits count: 5, 6, 0. The trailing zero counts because it’s written after the decimal point in normalized form — the same Case 3 rule from our zeros guide.

Answer: 3 significant figures. This is also the fix for the ambiguous 45000 case: writing it as 4.50 × 10⁴ instead states, unambiguously, that 3 sig figs were intended — no guessing required.

Example 4 — Multiplication

Calculation: (2.0 × 10³) × (3.0 × 10²)

Multiply the mantissas, add the exponents: (2.0 × 3.0) × 10^(3+2) = 6.0 × 10⁵.

Answer: 6.0 × 10⁵ (600,000). No renormalization needed here since 6.0 already falls within 1–10.

Example 5 — Multiplication that needs renormalizing

Calculation: (5.0 × 10⁴) × (4.0 × 10³)

Multiply and add exponents as before: (5.0 × 4.0) × 10^(4+3) = 20 × 10⁷. But 20 is outside the 1–10 mantissa range, so this isn’t finished — shift the decimal point and adjust the exponent to compensate: 20 × 10⁷ = 2.0 × 10¹ × 10⁷ = 2.0 × 10⁸.

Forgetting this last step is one of the most common errors in scientific-notation arithmetic — the math is right, but the answer isn’t in valid scientific notation until it’s renormalized.

Example 6 — Division

Calculation: (8.0 × 10⁷) ÷ (2.0 × 10³)

Divide the mantissas, subtract the exponents: (8.0 ÷ 2.0) × 10^(7−3) = 4.0 × 10⁴.

Example 7 — Addition: the exponent-matching trap

Calculation: 5.2 × 10⁶ + 3.1 × 10⁵

These have different exponents, so the mantissas cannot be added directly — 5.2 + 3.1 = 7.3 is not a meaningful intermediate step here, and neither 7.3 × 10⁶ nor 7.3 × 10⁵ is correct. Convert to matching exponents first. Rewrite 3.1 × 10⁵ with an exponent of 6: 3.1 × 10⁵ = 0.31 × 10⁶. Now add: 5.2 × 10⁶ + 0.31 × 10⁶ = 5.51 × 10⁶.

Check against standard form: 5,200,000 + 310,000 = 5,510,000 = 5.51 × 10⁶. Confirmed. Unlike multiplication and division, addition and subtraction in scientific notation always require this exponent-matching step first — there’s no shortcut around it.

Example 8 — Raising a scientific-notation number to a power

Calculation: (3.0 × 10²)³

Raise the mantissa to the power, multiply the exponent by that power: 3.0³ × 10^(2×3) = 27 × 10⁶. The mantissa (27) is outside the 1–10 range, so renormalize: 27 × 10⁶ = 2.7 × 10¹ × 10⁶ = 2.7 × 10⁷.

Check against standard form: 3.0 × 10² = 300, and 300³ = 27,000,000 = 2.7 × 10⁷. Confirmed. The same renormalizing habit from Example 5 applies here — raising a mantissa to a power is exactly the kind of step that pushes it outside the valid range.

Example 9 — Converting to engineering notation

Number: 45,000,000 (the same number from Example 1, where it became 4.5 × 10⁷ in standard scientific notation)

Engineering notation needs the exponent to be a multiple of 3. The nearest one at or below 7 is 6, so shift the decimal point to compensate: 45,000,000 = 45 × 10⁶. The mantissa (45) is now outside the 1–10 range that standard scientific notation requires, but well within engineering notation’s 1–1000 range — and 10⁶ maps directly onto the “mega” prefix, so this is also correctly read as 45 megaunits of whatever’s being measured.

Where This Still Trips People Up

• “e” is not Euler’s number. 3.5e3 means 3.5 × 10³ = 3500, full stop. It has no relationship to e ≈ 2.71828, despite using the same letter. This isn’t just a calculator quirk, either — it’s valid floating-point literal syntax in essentially every modern programming language (Python, JavaScript, and C all accept 1e10 directly in source code), which is part of why the convention is worth understanding even outside a calculator display.
• Addition and subtraction always need matching exponents first; multiplication and division never do. These are opposite requirements for the two pairs of operations, and mixing them up (matching exponents before multiplying, or skipping the match before adding) is the single most common scientific-notation arithmetic error. See Examples 4–7.
• A multiplication or division result can land outside the valid mantissa range and need renormalizing before it’s actually finished — see Example 5. An un-renormalized answer like “20 × 10⁷” isn’t wrong arithmetic, it’s just not valid scientific notation yet.
• Engineering notation’s mantissa range is 1 to 1000, not 1 to 10. It’s easy to apply the standard scientific-notation range by habit and “fix” a perfectly correct engineering-notation number like 45 × 10⁶ into something that breaks the multiple-of-3 exponent rule.
• A negative sign on the mantissa and a negative sign on the exponent mean very different things. −4.5 × 10³ is a negative number (−4,500). 4.5 × 10⁻³ is a small positive number (0.0045). Confusing which part carries the minus sign is an easy transcription error with a large effect on the result.
• Calculators don’t all display this the same way — raised superscript exponents, E-notation, and a small “×10” indicator are all common, and some calculators only switch to scientific notation automatically past a certain magnitude threshold. See our software and calculator guide for model-specific behavior.

Comparing the Three Formats

All three formats represent identical values — the difference is purely presentational, driven by what’s easiest to type, read, or map onto a unit prefix in the context you’re working in.

Standards Note

NIST’s own SI style guide devotes dedicated sections to the mechanics covered here: prefix usage and the rules for combining a prefix symbol with a unit symbol are set out in Chapter 6, while the general rules for expressing a quantity’s numerical value — including how many digits are meaningful — are in Chapter 7, the same chapter cited in our zeros guide for the ambiguous-trailing-zero problem this page’s Example 3 resolves.

Common Mistakes

1. Adding or subtracting in scientific notation without matching exponents first — see Example 7.
2. Forgetting to renormalize a multiplication or division result that lands outside the 1–10 mantissa range — see Example 5.
3. Treating the “e” in e-notation as Euler’s number.
4. Applying the wrong mantissa range — using 1–10 where 1–1000 (engineering notation) was called for, or vice versa.
5. Confusing a negative mantissa with a negative exponent — one makes the number negative, the other makes it small.
6. Losing significant figures during conversion — writing 4,500,000 as 4 × 10⁶ when the original data justified 4.5 × 10⁶ (2 sig figs), silently discarding a real digit in the process of reformatting.

Practice Problems

Concept: Converting to scientific notation

Q1. Convert 6,700,000 to scientific notation. A) 6.7 × 10⁵ B) 6.7 × 10⁶ C) 67 × 10⁵ D) 0.67 × 10⁷ Answer: B) 6.7 × 10⁶.

Q2. Convert 0.000045 to scientific notation. A) 4.5 × 10⁻⁵ B) 4.5 × 10⁻⁴ C) 4.5 × 10⁴ D) 45 × 10⁻⁶ Answer: A) 4.5 × 10⁻⁵.

Concept: Significant figures in scientific notation

Q3. How many significant figures does 5.60 × 10⁴ have? A) 2 B) 3 C) 4 D) 5 Answer: B) 3 — the trailing zero in the mantissa counts.

Q4. Which of these unambiguously states 4 significant figures for a value of 32,000? A) 3.2 × 10⁴ B) 3.20 × 10⁴ C) 3.200 × 10⁴ D) 32 × 10³ Answer: C) 3.200 × 10⁴ — four mantissa digits: 3, 2, 0, 0.

Concept: Multiplication and division

Q5. (2.0 × 10³) × (3.0 × 10²) = ? A) 6.0 × 10⁴ B) 6.0 × 10⁵ C) 5.0 × 10⁵ D) 6.0 × 10⁶ Answer: B) 6.0 × 10⁵.

Q6. (8.0 × 10⁷) ÷ (2.0 × 10³) = ? A) 4.0 × 10⁴ B) 4.0 × 10²¹ C) 6.0 × 10⁴ D) 4.0 × 10¹⁰ Answer: A) 4.0 × 10⁴.

Concept: Addition and subtraction

Q7. What has to happen before you can add 5.2 × 10⁶ and 3.1 × 10⁵? A) Nothing — add the mantissas directly B) Convert both to the same exponent first C) Add the exponents together D) Multiply one of them by 10 first, then add exponents Answer: B.

Q8. 5.2 × 10⁶ + 3.1 × 10⁵ = ? A) 8.3 × 10⁶ B) 5.51 × 10⁶ C) 8.3 × 10¹¹ D) 5.2 × 10⁶ Answer: B) 5.51 × 10⁶.

Concept: Engineering notation

Q9. Which of these is correctly written in engineering notation for 67,000? A) 6.7 × 10⁴ B) 67 × 10³ C) 0.67 × 10⁵ D) 670 × 10² Answer: B) 67 × 10³ — exponent 3 is a multiple of 3, and 67 falls within the 1–1000 mantissa range.

Q10. Convert 250,000,000 to engineering notation. A) 2.5 × 10⁸ B) 25 × 10⁷ C) 250 × 10⁶ D) 2500 × 10⁵ Answer: C) 250 × 10⁶.

One Number, Four Formats

Number: 45,000,000

• Standard form: 45,000,000
• Scientific notation: 4.5 × 10⁷
• E-notation: 4.5E7
• Engineering notation: 45 × 10⁶ (= 45 megaunits)

Number: 0.0000032

• Standard form: 0.0000032
• Scientific notation: 3.2 × 10⁻⁶
• E-notation: 3.2E-6
• Engineering notation: 3.2 × 10⁻⁶ (already aligned — exponent −6 is a multiple of 3)

Quick Reference

Format rules:

Arithmetic rules:

Common SI prefixes (engineering notation exponents):

Continue Learning

Related fundamentals:

• Significant Figures: The Complete Guide
• Which Zeros Are Significant? Every Case, Settled

Go deeper on one topic at a time:

• How to Convert Any Number to Scientific Notation
• E-Notation vs Scientific Notation: What 3.5e3 Actually Means
• Engineering Notation vs Scientific Notation
• Adding and Subtracting in Scientific Notation
• SI Prefixes Table with Significant Figure Notes
• Orders of Magnitude Explained
• Scientific Notation on the TI-84 and Casio fx-991

Tools:

• Scientific / E-Notation Converter
• Engineering Notation Converter

Sources and Further Reading

• NIST Guide to the SI, Chapter 6, Rules and Style Conventions for Printing and Using Units — covers SI prefix usage and symbol-combination rules referenced in the Quick Reference table above. (nist.gov)
• NIST Guide to the SI, Chapter 7, Rules and Style Conventions for Expressing Values of Quantities — the same chapter cited in our zeros guide, relevant here for how many digits of a quantity’s numerical value are meaningful. (nist.gov)
• Engineering LibreTexts, 1.3: Scientific and Engineering Notation — an open engineering-education reference confirming the engineering-notation mantissa range (1–1000) and multiple-of-three exponent rule used throughout this page. (eng.libretexts.org)

Review and Methodology

Reviewed by: [Pending — reviewer assignment required before publication] Last reviewed: [Pending] Methodology: Format definitions and worked examples are cross-checked against the NIST and open-education sources listed above. Calculator results referenced on this page use an arbitrary-precision decimal engine, not native floating-point math, validated against the site’s versioned regression fixture set.

Changelog

v1.0 — Initial draft completed, 2026-08-10.