Authoritative Data Source and Methodology
Primary reference: NIST Special Publication 811 — Guide for the Use of the International System of Units (SI), 2008 Edition (updated 2019). National Institute of Standards and Technology. Official page · Direct PDF.
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This tool implements standard, internationally accepted definitions of scientific notation, significant figures, and engineering notation consistent with NIST SP 811. Arithmetic follows base-10 exponent rules with normalization and rounding to selected significant figures.
The Formula Explained
Scientific notation writes any nonzero number as:
$$N = m \times 10^{e}, \quad 1 \le |m| < 10,\; e \in \mathbb{Z}$$
For N ≠ 0, the exponent is: $$e = \left\lfloor \log_{10} \left(|N|\right) \right\rfloor,$$ and the mantissa is: $$m = \frac{N}{10^{e}}.$$
Engineering notation uses an exponent that is a multiple of 3: $$e_{\text{eng}} \equiv 0 \pmod{3}.$$
Arithmetic rules:
Add/Subtract: $$m_1 \times 10^{e_1} \pm m_2 \times 10^{e_2} = 10^{E}\Big(m_1 \times 10^{e_1 - E} \pm m_2 \times 10^{e_2 - E}\Big), \quad E=\max(e_1,e_2).$$
Multiply: $$(m_1 \times 10^{e_1}) (m_2 \times 10^{e_2}) = (m_1 m_2) \times 10^{e_1+e_2}.$$
Divide: $$\frac{m_1 \times 10^{e_1}}{m_2 \times 10^{e_2}} = \left(\frac{m_1}{m_2}\right) \times 10^{e_1-e_2}.$$
Rounding to significant figures s uses standard rounding on m to s digits, with renormalization if needed (e.g., 9.999→1.000×10^1).
Glossary of Variables
- N (Number): The original value in standard decimal form.
- m (Mantissa): The coefficient in scientific notation, constrained to 1 ≤ |m| < 10.
- e (Exponent): The integer power of 10 in scientific notation.
- Scientific notation: Representation N = m × 10^e with normalized m.
- Engineering notation: Scientific notation with e a multiple of 3.
- E notation: Computer-friendly form mE e (e.g., 4.57E-4).
- Significant figures (s): The number of meaningful digits retained in m.
- Operation (A ∘ B): Add, subtract, multiply, or divide two inputs A and B.
How It Works: A Step‑By‑Step Example
Example 1 — Convert
Convert N = 0.00045678 to scientific notation with s = 3 significant figures.
- Find e = ⌊log10|N|⌋ = ⌊log10(0.00045678)⌋ = ⌊−3.340…⌋ = −4.
- Compute m = N / 10^e = 0.00045678 / 10^-4 = 4.5678.
- Round m to s = 3 digits: 4.5678 → 4.57.
- Result: 4.57 × 10^-4 (E notation: 4.57E-4; engineering notation is the same here).
Example 2 — Arithmetic
Add A = 3.2 × 10^5 and B = 4.5 × 10^4, with s = 3.
- Align exponents: E = max(5, 4) = 5. A = 3.2 × 10^5; B = 0.45 × 10^5.
- Add mantissas: (3.2 + 0.45) × 10^5 = 3.65 × 10^5.
- Round to s = 3: 3.65 × 10^5 (already at 3 sig. figs.).
- Final: 3.65 × 10^5 (E notation: 3.65E5; decimal: 365000).
Frequently Asked Questions (FAQ)
What input formats are supported?
Type plain decimals (e.g., 12345.6), E notation (1.234E4), or the ×10^ form (1.234×10^4 or 1.234*10^4). Spaces and commas are ignored.
How do significant figures affect the result?
The mantissa m is rounded to your chosen number of significant figures. If rounding produces 10.0, the mantissa renormalizes to 1.0 and the exponent increases by 1.
What’s the difference between scientific and engineering notation?
Scientific notation requires 1 ≤ |m| < 10. Engineering notation additionally forces the exponent to be a multiple of 3, aligning with SI prefixes (kilo, mega, micro, etc.).
Can I perform arithmetic on mixed formats?
Yes. Each input can be in decimal, E notation, or ×10^ format. The calculator parses and normalizes them before computing.
What ranges are supported?
This tool uses IEEE 754 double-precision. Practical exponent range is roughly −308 to +308. Inputs resulting in overflow/underflow will trigger an error.
Is the output normalized?
Yes. Scientific results are normalized to 1 ≤ |m| < 10. Engineering results use exponent multiples of 3 and are also normalized within that constraint.
Why might the plain decimal appear rounded?
The plain decimal string is generated consistently with the selected significant figures and can be very long for large exponents. For extreme magnitudes, scientific or engineering notation is recommended.
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