Approximation, bounds and percentage error

Decimal places are counted to the right of the decimal point. To round to $d$ decimal places, keep $d$ digits after the decimal point, inspect the next digit, and round the final kept digit up when that next digit is $5$ or more.

The rounding unit for $d$ decimal places is $u=10^{-d}$. Thus 1 decimal place uses units of $0.1$, 2 decimal places uses $0.01$, and 3 decimal places uses $0.001$.

Trailing zeros can communicate precision. The values $7.3$ and $7.30$ are numerically equal, but $7.30$ signals reporting to two decimal places. In measurement contexts, that extra zero is meaningful.

For negative values, round the magnitude using the same rule and then restore the sign. For example, $-8.376\approx-8.38$ to two decimal places. The rounded value is farther from zero because the third decimal digit is $6$.

Significant figures are counted from the first non-zero digit. Leading zeros only locate the decimal point and are not significant. Zeros between non-zero digits are significant, and trailing zeros after a decimal point are significant.

For $0.004070$, the significant digits are $4,0,7,0$, so the number has four significant figures. In $58\,200$, trailing zeros can be ambiguous unless the accuracy is stated; scientific notation removes the ambiguity.

Scientific notation makes significant figures visible: $5.82\times10^4$ has three significant figures, while $5.8200\times10^4$ has five.

To round to a stated number of significant figures, find the first non-zero digit, count the required number of digits, inspect the next digit, and replace later place values with zeros when necessary.

Estimation replaces values with nearby, easy numbers to predict the scale of an answer. It is not a substitute for the final calculation when accuracy is required; it is a check that can reveal entry errors, incorrect units or an impossible model result.

For a quick estimate, round each factor to one significant figure or a convenient compatible number. For example, $19.8\times0.503\approx20\times0.5=10$. A calculator answer of $99.594$ would therefore be unreasonable.

Context supplies further checks: a length and a probability cannot be negative; a probability cannot exceed 1; a count of people should normally be a whole number; and a price may need the nearest cent.

The most appropriate accuracy depends on the purpose. Medicine and engineering may demand fine tolerances; a weather forecast or long-term economic projection may justify fewer digits because model uncertainty dominates.

A reported rounded value represents every original value that would round to it. If $r$ is rounded to the nearest unit $u$, the maximum displacement from $r$ is half a unit, $\frac{u}{2}$. Therefore the lower bound is $r-\frac{u}{2}$ and the upper bound is $r+\frac{u}{2}$.

The standard error interval is $r-\frac{u}{2}\le x<r+\frac{u}{2}$. The lower endpoint is included because it rounds up to $r$. Under the usual “5 rounds up” convention, the upper endpoint is excluded because it rounds to the next reported value.

The lower bound is the smallest possible value and the upper bound is the boundary the value cannot reach. In ordinary language one may say “the upper bound is $4.15$,” while the inequality correctly shows $x<4.15$.

A value reported as 4.1 to 1 decimal place represents the half-open interval $4.05 \le x < 4.15$.

For significant figures, first identify the place value of the final significant digit. That place value is the rounding unit. A value of 7300 to two significant figures is rounded to the nearest 100, so the half-unit is 50.

When a formula uses rounded measurements, the result also has a range. To find worst-case bounds, choose input endpoints that make the output as small or as large as possible. For positive quantities, sums and products grow when their inputs grow.

For positive $a$ and $b$, the product satisfies $a_{\mathrm L}b_{\mathrm L}\le ab<a_{\mathrm U}b_{\mathrm U}$. For a quotient with a positive denominator, the smallest quotient uses the smallest numerator and largest denominator; the largest uses the largest numerator and smallest denominator.

Subtraction requires crossed endpoints: $a-b$ is smallest at $a_{\mathrm L}-b_{\mathrm U}$ and largest at $a_{\mathrm U}-b_{\mathrm L}$. Do not simply combine both lower bounds.

These endpoint rules assume the relevant quantities remain positive and the formula is monotonic over the interval. If intervals cross zero or the formula changes direction, inspect the situation more carefully.

When an exact or accepted reference value is known, the absolute error is the magnitude of the difference: $E_{\mathrm a}=|x_{\mathrm{approx}}-x_{\mathrm{exact}}|$. It uses the same units as the measured quantity.

The relative error compares the absolute error with the size of the exact value: $E_{\mathrm r}=\frac{E_{\mathrm a}}{|x_{\mathrm{exact}}|}$, provided the exact value is not zero. Relative error has no units.

The percentage error is the relative error expressed as a percentage: $E_{\%}=100E_{\mathrm r}\%$. It makes errors in differently sized quantities easier to compare.

An error is not automatically a mistake. Measurement error can arise from instrument precision, calibration, environmental conditions, sampling, model assumptions and rounding. State the reference value and source of uncertainty when interpreting the result.

If the exact value is unknown but a measurement was rounded to unit $u$, its maximum possible absolute rounding error is $\frac{u}{2}$. This is a bound on the uncertainty, not the actual error.

A maximum percentage error for the measured quantity can be estimated by comparing the half-unit with an appropriate reference. Some problems instead ask for the maximum percentage error in a calculated quantity such as area; then evaluate the formula at its input bounds and compare the output with the value calculated from the reported measurement.

For nonlinear formulas, equal input changes need not cause equal output changes. Squaring a radius makes the upper-area deviation slightly different from the lower-area deviation, so calculate both or reason which is larger.

Every time an intermediate result is rounded, information is discarded. Later multiplication, division or repeated iteration can magnify the resulting difference. Keep the full calculator value or several guard digits and round once, at the end.

For example, $\frac{1}{7}$ is recurring. Using $\frac{1}{7}\approx0.14$ in $350(\frac{1}{7})$ gives $49$, while using the stored value gives exactly $50$. The early rounding creates a final error of $1$.

Adding many individually rounded values may accumulate error. A total made from original readings is normally more reliable than a total made from separately rounded display values.

Technology reduces arithmetic effort but does not decide appropriate accuracy. Calculator displays are finite approximations, and copying too few digits between steps can still cause avoidable error.

A measurement is affected by instrument resolution, technique and conditions. A ruler marked in millimetres, a digital scale displaying 0.01 g and a weather sensor each support different precision. Repeated digits on a screen do not guarantee that the instrument is accurate.

In construction, manufacturing and medicine, a tolerance specifies an acceptable range rather than a single target. Bounds help decide whether every possible value meets a safety or quality requirement.

Currency normally uses the smallest available unit for final transactions, but exchange rates and tax calculations should retain guard digits before the final monetary rounding. Repeated rounding across thousands of transactions may have a material effect.

Statistical and scientific claims should state data quality and uncertainty. Selective rounding can mislead: reporting only favourable precision, hiding a wide interval, or presenting a model output with excessive digits can exaggerate confidence.

A scientific calculator or GDC can round, store values, evaluate endpoint expressions and compare results, but the written method must identify the relevant bounds or error formula. Technology output without a setup is not a complete mathematical explanation.

Use full display precision during working. Parenthesize numerators and denominators in percentage-error calculations, and confirm that the calculator is not set to a display mode that silently rounds every value before you copy it.

For a bound problem, store lower and upper endpoints as variables and evaluate the model at the required combinations. A table or graph can confirm whether a function increases throughout the interval, but monotonic reasoning should be stated when it determines the endpoints.

Spreadsheet formulas are useful for many measurements: compute absolute error, divide by the accepted value, and format the result as a percentage. Keep original data separate from rounded display columns.

These questions move from foundational rounding to multi-step application and evaluation. Attempt each problem before opening the worked method, and keep full precision until the final reporting instruction.

For rounding, locate the required place, inspect the next digit and preserve meaningful trailing zeros. For bounds, convert the accuracy into a rounding unit $u$, then use the half-unit interval $r-\frac{u}{2}\le x<r+\frac{u}{2}$.

For error, use $\text{absolute}=|\text{approximate}-\text{reference}|$, divide by the reference magnitude for relative error, and multiply by $100\%$ for percentage error. If no true value is known, state a maximum possible error from the rounding information.

For applications, keep guard digits, propagate input bounds through the formula, attach units, choose meaningful final accuracy and check the result against an estimate and contextual constraints.