Keep scale and precision separate
How many digits should a small or large number display?
Entering midway? A positive power of ten moves scale upward; a negative power represents division by ten repeatedly.
Start with a concrete question
Represent an original synthetic positive quantity as mantissa times a power of ten, then round to a chosen number of significant digits.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict whether fewer displayed digits automatically means a less accurate instrument.
Change one thing. Inspect what happens.
- Write 3.14×10⁻³ as a decimal, then round to two significant digits.
- Keep mantissa fixed and change exponent; compare relative versus absolute rounding size.
- Try 9.99 rounded to two digits; explain why 10×10ᵉ can be renormalized to 1.0×10ᵉ⁺¹.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Scientific notation: 3.14 × 10^-3 = 0.00314.
- Keep 2 significant digits: round mantissa using scale 10; 3.1 × 10^-3 = 0.0031.
- Absolute error = |0.0031 − 0.00314| = 0.00004.
- Half a rounding unit = 0.5 × 10^(-3 − 2 + 1) = 0.00005.
- The bars compare coefficients 3.14 and 3.1 with the same multiplier 10^-3; multiply either bar value by that shared factor to recover its represented magnitude.
Significant digits describe a rounding convention, not proof of measurement accuracy. Carry digits through a calculation and round at the end. Trace numbers are displayed to ten significant digits; calculations retain their underlying precision.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Keep scale and precision separate
Recap for entering here
A positive power of ten moves scale upward; a negative power represents division by ten repeatedly.
The concrete question
Represent an original synthetic positive quantity as mantissa times a power of ten, then round to a chosen number of significant digits.
Predict whether fewer displayed digits automatically means a less accurate instrument.
Work through the mechanism
Scientific notation separates the significant digits from magnitude. 3.14×10⁻³ is 0.00314, not 0.314.
Two significant digits produce 3.1×10⁻³ = 0.0031. The absolute rounding error is 0.00004.
The nearest-rounding half-unit bound is 0.00005 here. It bounds rounding of this represented value, not total measurement error or systematic bias.
Formula and symbols
Value = m×10ᵉ. The mantissa m is at least 1 and strictly less than 10; e is the integer power of ten. With d significant digits, the rounding unit is 10^(e−d+1).
The comparison bars show the original and rounded coefficients with the same power-of-ten multiplier. For example, coefficients 3.14 and 3.1 both use the multiplier 10⁻³. This preserves their relative sizes without making a small represented quantity disappear against a unit-sized chart axis. The numerical summaries still show the full represented values and rounding error.
Test the idea
Write 3.14×10⁻³ as a decimal, then round to two significant digits.
Keep mantissa fixed and change exponent; compare relative versus absolute rounding size.
Try 9.99 rounded to two digits; explain why 10×10ᵉ can be renormalized to 1.0×10ᵉ⁺¹.
Assumptions and boundaries
Only positive normalized inputs are modeled; zero and signed values need their own notation handling.
Decimal rounding uses bounded binary floating-point arithmetic; exact decimal ties may require a specified decimal-arithmetic rule.
Extend your investigation
High school: Compare distances or file sizes spanning several powers of ten.
College: Analyze absolute versus relative error and propagation through subtraction.
Working professionals: Specify display precision separately from instrument calibration and tolerance.
Try a fresh case
Write 4.2×10⁻⁴ as an ordinary decimal.
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Compare distances or file sizes spanning several powers of ten.
College investigation
Analyze absolute versus relative error and propagation through subtraction.
Workplace investigation
Specify display precision separately from instrument calibration and tolerance.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.