Data Source and Methodology
Authoritative Data Source: NIST Special Publication 811 — “Guide for the Use of the International System of Units (SI),” 2008 Edition. National Institute of Standards and Technology, Gaithersburg, MD. Section 7: Rounding numbers. Direct link: https://physics.nist.gov/cuu/pdf/sp811.pdf. In addition, tie‑to‑even behavior aligns with IEEE Standard for Floating-Point Arithmetic, IEEE 754–2019.
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The Formula Explained
General quantization model (round to the nearest step q using operator R):
$$ y = \\frac{x}{q}, \\quad x_{\\text{rounded}} = q \\cdot R(y) $$
Where R is chosen as one of the following:
Nearest with ties to even (banker’s): $$ R(y)= \\begin{cases} \\lfloor y \\rfloor, & \\text{if } y-\\lfloor y \\rfloor < \\tfrac{1}{2} \\\\ \\lceil y \\rceil, & \\text{if } y-\\lfloor y \\rfloor > \\tfrac{1}{2} \\\\ \\text{even}(\\lfloor y \\rfloor,\\lceil y \\rceil), & \\text{if } y-\\lfloor y \\rfloor = \\tfrac{1}{2} \\end{cases} $$
Half away from zero: $$ R(y)= \\begin{cases} \\operatorname{sgn}(y)\\,\\lceil |y| \\rceil, & \\text{if } |y|-\\lfloor |y| \\rfloor = \\tfrac{1}{2} \\\\ \\operatorname{round\\_nearest}}(y), & \\text{otherwise} \\end{cases} $$
Always up (ceiling), down (floor), or toward zero (truncate): $$ R(y) = \\lceil y \\rceil,\\; R(y)=\\lfloor y \\rfloor,\\; R(y)=\\operatorname{trunc}(y) $$
Decimal places: choose q = 10^{-d} (d ≥ 0).
Significant figures: for x ≠ 0, let k = \\lfloor \\log_{10}(|x|) \\rfloor and set d = s - 1 - k; then round to d decimal places (q = 10^{-d}).
Glossary of Variables
- x: input number to be rounded.
- d: number of decimal places (0–15).
- s: number of significant figures (1–15).
- q: step size (quantization), e.g., 10^{-d} or a custom multiple like 0.05.
- R: rounding operator (nearest with a tie rule, ceil, floor, or truncate).
- x_rounded: resulting rounded value.
- Absolute difference: |x_rounded − x|.
- Relative difference: |x_rounded − x| / |x| × 100% (N/A if x = 0).
How It Works: A Step‑by‑Step Example
Goal: Round -123.4567 to 3 significant figures using Nearest, ties to even.
- Magnitude: k = ⌊log10(|x|)⌋ = ⌊log10(123.4567)⌋ = 2.
- Decimals needed: d = s − 1 − k = 3 − 1 − 2 = 0 → q = 10^{-0} = 1.
- Compute y = x/q = -123.4567. Fractional part is 0.4567 (not a tie), so R(y) = ⌊y⌋ = -124 or -123 depending on nearest: since 0.4567 < 0.5, nearest is -123.
- Result: x_rounded = q·R(y) = 1·(-123) = -123.
- Absolute difference: 0.4567; Relative difference: ≈ 0.37%.
Frequently Asked Questions (FAQ)
Does this tool support banker’s rounding?
Yes. Select “Nearest” and choose “Half to even (banker’s)” as the tie‑breaking rule. This minimizes bias in aggregated results.
How are significant figures determined?
We compute the order of magnitude, then convert the problem to rounding to a specific number of decimal places so that the total count of meaningful digits matches your request.
Can I always round up or down regardless of ties?
Yes. Choose the operation “Up (ceil)” to move to the next higher multiple, “Down (floor)” for the next lower multiple, or “Toward zero (truncate).”
What is “Half up” and how is it different from “Half away from zero”?
“Half up” sends exact halves toward +∞ (e.g., -1.5 → -1), while “Half away from zero” increases magnitude (e.g., -1.5 → -2). We expose both because conventions vary by domain.
Will the calculator preserve trailing zeros?
Enable “Pad trailing zeros” to format the output with zeros that convey precision, e.g., 5.20.
What about extremely large or small numbers?
Turn on “Scientific notation for extremes” to display results like 1.23×10^8 using compact scientific formatting.