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Scientific Notation Calculator

Convert numbers to scientific, E, and engineering notation. Free online tool — no sign-up. Mobile-friendly and privacy-first. Runs locally in your browser.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 5/5 declared partitions (convert, normalize, engineering, operate, invalid-domain) · Matrix
Numerical scope
Convert / normalize / engineering: ≤2 ULP on scientific coefficient, exact exponent. Operate: ≤2 ULP on reconstructed binary64 plus exact normalized exponent. Invalid domain: matching error codes (overflow / underflow / division by zero / missing). Not a whole-domain guarantee beyond recorded vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Conversions among decimal, scientific, E, and engineering notation; normalization; ±×÷ of two scientific values.
Scope
Finite IEEE-754 binary64 values. Inputs outside the supported floating-point range are rejected or reported as overflow/underflow. Very large/small in-range values use exponential display.
Verification
Engine tested · Source checked · v1.1.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.1 · CVP protocol 1.0.0-proposed · Evidence 2026-09-16.sources-no-wolfram
Verification revision
2026-09-16.sources-no-wolfram · 5/5 property · digest 4a53cbc5afe2
Legacy regression
6/6 tests · Production surface contract 3/3
Trust layers
Verification VERIFIED · Production CURRENT · overall VERIFIED
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
21/21 golden · 1/1 CVP boundary · 7/7 invalid · 5/5 property · 2/2 metamorphic · 2/2 round-trip · 4/4 cross-interface · 3/3 CVP contract · Manifest
Sources
Sources
Implementation conventions
  • Zero → 0 — Zero is written 0, not A × 10ⁿ. No coefficient satisfies 1 ≤ |A| < 10 while representing zero.
Methods
  • IEEE-754 binary64 parse of decimal / E-notation / A×10ⁿ
  • Normalize via Number#toExponential; engineering exponent ≡ 0 (mod 3)
  • Test suites are not interchangeable: 20 published examples on this page · 124 automated engine checks (IUT) · CVP verification suite on Evidence (Manifest is the verification source of truth)
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 21/21 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Scientificx = A × 10ⁿ, 1 ≤ |A| < 10, for x ≠ 0
Zero0 → 0
Multiply(A×10ᵐ)(B×10ⁿ) = (A·B)×10^(m+n)
Divide(A×10ᵐ)/(B×10ⁿ) = (A/B)×10^(m−n)
Add / subtractAlign exponents, then add coefficients
iZero is not written with 1 ≤ |A| < 10. Accepts plain decimals, e/E notation (3.2e5), Unicode A×10ⁿ, and forms like 3.2×10^5. Engineering exponents are …, −6, −3, 0, 3, 6, 9, …

How to use

1

Choose a mode

Convert a value, normalize A×10ⁿ, run ±×÷, or view engineering form.

2

Enter the number(s)

Paste 6020000, 3.2e5, or 3.2×10^5 — all work.

3

Read every format

Scientific, E-notation, engineering (with SI hint), real value, and steps.

Example calculations

Common configurations with formula and result.

ϟ

Small decimal

0.00042

decimal → scientific
4.2 × 10⁻⁴
ϟ

Large decimal

6,020,000

→
6.02 × 10⁶
ϟ

Normalize

32 × 10⁴

1 ≤ |A| < 10
3.2 × 10⁵
ϟ

Multiply

(3×10⁴)(2×10⁶)

3·2 × 10^(4+6)
6 × 10¹⁰
ϟ

Engineering

1.234 × 10⁸

n multiple of 3
123.4 × 10⁶ (mega)

Notation equivalents

Common values at a glance.

DecimalScientificE-notation
3570963.57096 × 10⁵3.57096e5
124.71.247 × 10²1.247e2
0.0056005.600 × 10⁻³5.600e-3
0.000989.8 × 10⁻⁴9.8e-4
−5×10⁹−5 × 10⁹−5e9
i Order of magnitude of A×10ⁿ is n (for the normalized scientific form).

Scientific Notation calculator specification

Version 1.1.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
Scientific notation writes a number as A × 10ⁿ where 1 ≤ |A| < 10 (except 0) and n is an integer (order of magnitude). E-notation replaces ×10 with e (e.g. 3.45e5). Engineering notation uses the same idea but forces n to a multiple of 3 so values align with SI prefixes.
What it calculates
Conversions among decimal, scientific, E, and engineering notation; normalization; ±×÷ of two scientific values.
Inputs
  • A number in decimal, e-notation, or A×10^n form
  • Optional second operand and operation for arithmetic mode
Outputs
  • Normalized scientific notation
  • E-notation
  • Engineering notation with SI prefix hint
  • Real / standard decimal when practical
  • Order of magnitude
Formula
x = A × 10ⁿ, 1 ≤ |A| < 10, for x ≠ 0; 0 → 0; eng: n ≡ 0 (mod 3)
Assumptions
  • Finite IEEE-754 binary64 values. Inputs outside the supported floating-point range are rejected or reported as overflow/underflow. Very large/small in-range values use exponential display.
  • UI, share URLs, REST, and validation cases all call this same engine.
Units
  • Dimensionless (or consistent physical units carried separately)
Boundary conditions
  • Division by zero rejected (DIVISION_BY_ZERO)
  • Empty or unparsable input rejected (MISSING_REQUIRED_INPUT / INVALID_NUMBER)
  • Inputs outside the IEEE-754 binary64 range rejected (RESULT_OVERFLOW / RESULT_UNDERFLOW)
  • Zero is returned as 0, not A × 10ⁿ with 1 ≤ |A| < 10
Numerical precision
  • IEEE-754 binary64 parse of decimal, E-notation, and A×10ⁿ (including Unicode × and superscript exponents).
  • Normalize uses binary64's own decimal conversion (Number#toExponential) so subnormals and values just below 10 stay in 1 ≤ |A| < 10. Engineering shifts the exponent to a multiple of 3 (SI prefixes).
  • Shareable URLs (?numInput=4.2e-4&mode=convert) restore inputs and are server-rendered with the same deterministic engine. Query `v` is ignored — it is not a calculation_version pin. Pin a version with REST/MCP `CalculatorX-Spec-Version`.
  • Primary result uses true superscripts (4.2 × 10⁻⁴), not ASCII 10^-4.
Example
0.00042 → 4.2×10⁻⁴; 0 → 0; (3×10⁴)(2×10⁶)=6×10¹⁰
Validation cases

20 published on this page · 6/6 tests · Production surface contract 3/3 · View evidence

  • 0.00042 → 4.2 × 10⁻⁴
  • 4.2e-4 → 4.2 × 10⁻⁴
  • 6020000 → 6.02 × 10⁶
  • 32 × 10^4 → 3.2 × 10⁵
  • (3e4)*(2e6) → 6 × 10¹⁰
  • 1.234e8 engineering → 123.4 × 10⁶
  • 0 → 0
  • 1 → 1 × 10⁰
  • -1 → −1 × 10⁰
  • -0.00042 → −4.2 × 10⁻⁴
  • 9.999 → 9.999 × 10⁰
  • 10 → 1 × 10¹
  • 1e0 → 1 × 10⁰
  • 1E+20 → 1 × 10²⁰
  • 3.2×10⁵ (Unicode) → 3.2 × 10⁵
  • 1e-3 engineering → 1 × 10⁻³
  • 999 engineering → 999 × 10⁰
  • normalize 10 × 10^0 → 1 × 10¹
  • 1e-30 → 1 × 10⁻³⁰
  • (6e4)/(0) → error DIVISION_BY_ZERO
Methods
  • IEEE-754 binary64 parse of decimal / E-notation / A×10ⁿ
  • Normalize via Number#toExponential; engineering exponent ≡ 0 (mod 3)
  • Test suites are not interchangeable: 20 published examples on this page · 124 automated engine checks (IUT) · CVP verification suite on Evidence (Manifest is the verification source of truth)
Sources
  • OpenStax Elementary Algebra 2e — Scientific Notation — 5.2 An application of exponents: scientific notation · accessed 2026-08-30
    Supports: A number in scientific notation is written A × 10ⁿ with 1 ≤ |A| < 10 and integer n; 0 is the exception
  • BIPM SI Brochure (9th edition) — Table 8 — SI prefixes (powers of 10³) · accessed 2026-08-30
    Supports: Engineering notation uses exponents that are multiples of 3 so values line up with SI prefixes
  • NIST SI prefixes — SI prefixes from 10⁻³⁰ to 10³⁰ · accessed 2026-08-30
    Supports: Prefix names (milli, micro, kilo, mega, …) attached to engineering exponents
Implementation conventions
  • Zero → 0 — Zero is written 0, not A × 10ⁿ. No coefficient satisfies 1 ≤ |A| < 10 while representing zero.
Calculation version
1.1.1

Background

Interpretation and common distinctions.

What is scientific notation?

Scientific notation expresses a number as

A × 10ⁿ

with 1 ≤ |A| < 10 (except 0) and integer n. The integer n is the order of magnitude of the normalized form.

Form Example
Scientific 3.456 × 10¹¹
E-notation 3.456e11
Engineering 345.6 × 10⁹ (giga)
Real / decimal 345600000000

How to convert a decimal

  1. Move the decimal until one nonzero digit lies to the left → this is A.
  2. Count the moves → this is n (left ⇒ positive; right ⇒ negative).
  3. Write A × 10ⁿ.

Examples: 357096 = 3.57096 × 10⁵; 0.005600 = 5.600 × 10⁻³.

Arithmetic

  • Multiply: multiply coefficients; add exponents.
  • Divide: divide coefficients; subtract exponents.
  • Add / subtract: match exponents first, then combine coefficients.
  • Always normalize afterward.

Engineering notation

Same idea, but n is a multiple of 3 (…,−6,−3,0,3,6,9,…) so values line up with SI prefixes (kilo, mega, milli, …).

Other calculators in this family: Division Calculator, Fractions Calculator, Multiplication Calculator, Subtraction Calculator, Sum Calculator . Explore all Basic Arithmetic.

→ Division Calculator Divide numbers with quotient, remainder, and decimal results for quick arithmetic checks. Runs locally in your browser. Free online tool — no sign-up. › → Fractions Calculator Add, subtract, multiply, or divide fractions and mixed numbers with step-by-step LCD work. Free online tool — no sign-up. Runs locally in your browser. › → Multiplication Calculator Multiply two numbers instantly, or see long multiplication with partial products, carrying, decimals, and sign rules. Runs locally in your browser. › → Subtraction Calculator Subtract minuend − subtrahend to get the difference. Free online tool — no sign-up. Clear formulas and worked examples. Interactive calculations run locally in your browser. › → Sum Calculator Add a list of numbers (sum), see the mean, or work long addition with place-value steps. Paste values separated by commas, spaces, or new lines — decimals and negatives. ›
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Frequently asked questions

Key distinctions behind the calculation.

What is scientific notation?

A compact form A × 10ⁿ with 1 ≤ |A| < 10 (except zero). Example: 4.2 × 10⁻⁴ for 0.00042.

What does the exponent mean?

Positive n moves the decimal point n places to the right; negative n moves it left. n is the order of magnitude of the normalized form.

What is E-notation?

The same idea with e or E instead of ×10, common in calculators and programming: 4.2e-4.

What is engineering notation?

Like scientific notation, but the exponent must be a multiple of 3 so the value matches SI prefixes (kilo, mega, milli, …).

How do you multiply or divide?

Multiply/divide coefficients; add exponents for × and subtract for ÷. Then normalize so 1 ≤ |A| < 10.

How do you add or subtract?

Rewrite both numbers to the same power of 10, add/subtract coefficients, then normalize.

What does normalize mean?

Adjust A and n until 1 ≤ |A| < 10. Example: 32 × 10⁴ → 3.2 × 10⁵.