Error Intervals

An error interval describes all original values that could produce a stated rounded or truncated result. The key idea is that rounding creates a symmetric interval around the reported value, while truncation creates an interval that starts at the reported value and extends upward by one unit of the retained place value. Error intervals are written with inclusive lower bounds and exclusive upper bounds because the next boundary value would be recorded differently.

• Error interval is the range of original values that could have produced a stated approximate value after rounding or truncation. It is usually written as a compound inequality such as $a \le x < b$, which shows every possible exact value the measurement or calculation might have had.
• Lower bound is the smallest possible original value, and upper bound is the first value that is too large to still give the stated approximation. This distinction matters because the upper endpoint is usually not included, since reaching that boundary would change the rounded or truncated result.
• Degree of accuracy tells you the size of the step used in the approximation, such as nearest unit, nearest $0.1$, or nearest $10$. Once that step size is known, the error interval can be built systematically by moving half a step for rounding or one full step upward for truncation.
• Inclusive and exclusive notation is essential in error intervals. The lower bound is written with $\le$ when that exact value still gives the stated approximation, while the upper bound is written with $<$ because that endpoint belongs to the next rounded or truncated value.

• Rounding creates a midpoint rule because values are assigned to the nearest available step. If the approximation unit is $u$, then all values within $\frac{u}{2}$ of the stated rounded value belong to the same rounded result, so the interval is centered on that value.
• For a rounded value $r$ to the nearest unit $u$, the general error interval is:

$r - \frac{u}{2} \le x < r + \frac{u}{2}$ This works because the lower midpoint still rounds up to $r$, but the upper midpoint would round to the next value instead.

• Truncation does not use nearest value logic; it simply cuts off digits after a chosen place. That means the stated truncated value is the smallest possible value in the interval, not the midpoint, because every larger value up to the next step begins with the same retained digits.
• For a truncated value $t$ with retained unit $u$, the general interval is:

$t \le x < t + u$ This works because truncation keeps the beginning of the number unchanged until the next retained-place step is reached.

• Step 1: identify the approximation type before doing any calculation. You must decide whether the value was rounded or truncated, because these lead to different interval structures even when the displayed number looks similar.
• Step 2: determine the unit of accuracy from the wording or notation. For example, nearest whole number means $u=1$, nearest tenth means $u=0.1$, nearest $10$ means $u=10$, and the retained place for truncation also determines the interval width.
• Step 3: apply the correct formula based on the approximation rule. Use $r - \frac{u}{2} \le x < r + \frac{u}{2}$ for rounding, and use $t \le x < t + u$ for truncation, where $r$ or $t$ is the stated value and $u$ is the place-value step.
• Step 4: write the answer in inequality form with the variable included. This is important because an error interval is not just two numbers; it is a statement about every possible exact value the quantity may take.
• Step 5: check whether the endpoints make sense by imagining what happens exactly at each boundary. If the upper boundary would be recorded as the next value, then the symbol must be $<$ rather than $\le$, which is one of the most common exam checks.

• Rounded values and truncated values are not interpreted in the same way. A rounded value represents the nearest permitted value, so the true value lies on both sides of it, while a truncated value only tells you the leading digits and therefore the true value lies at or above the stated number.
• Place value accuracy and significant figures accuracy require different first steps. With place value, the unit of accuracy is read directly from the position, but with significant figures you must first locate the first non-zero digit and then determine the place value of the last retained significant figure.
• Measurements and calculator outputs may look similar but are processed differently. A measurement given as correct to a stated accuracy should be treated using the rounding rule, whereas a displayed or shortened decimal described as truncated should be treated with the truncation rule.

• Comparison Table

• Always translate the wording into a unit before forming the interval. Phrases such as nearest whole number, nearest centimetre, nearest $0.01$, or correct to $3$ significant figures all encode the size of the final kept step, and getting that step wrong causes the entire interval to be wrong.
• Use a quick boundary check to catch symbol errors. Ask yourself whether the exact upper endpoint would still be written as the stated value; if not, then the interval must stop with $<$ rather than $\le$.
• Keep the reported precision visible in your thinking. If a value is written with decimal places or significant figures, that formatting tells you what place has been retained, and missing trailing zeros can make you misread the required accuracy.
• When significant figures are involved, locate the last kept significant digit before doing anything else. This prevents the common mistake of treating significant figures like decimal places, especially for numbers smaller than $1$ or numbers containing zeros.
• A sensible answer should match the approximation style. Rounded intervals should have the stated value exactly in the middle, while truncated intervals should begin exactly at the stated value, so this provides a fast self-check in exams.

• A common misconception is to include both endpoints in a rounded interval. This is incorrect because the upper endpoint is exactly where the value would round or truncate to the next result, so including it would make two intervals overlap.
• Students often confuse half a unit with a full unit when working with rounded numbers. For rounding, you move by $\frac{u}{2}$ from the stated value, not by $u$, because the changeover happens halfway between adjacent rounded values.
• Another frequent error is mixing up decimal places and significant figures. Decimal places count positions after the decimal point, whereas significant figures begin at the first non-zero digit, so the same written number can have very different accuracy depending on the wording.
• Truncation is sometimes mistaken for rounding down, but they are not identical concepts. Truncation simply removes later digits without considering whether they are large or small, so its interval starts at the stated value and extends upward to just below the next step.

• Error intervals connect approximation to measurement uncertainty. In practical contexts, a reported measurement is rarely exact, so the interval tells you the hidden range of possible true values and provides a more honest mathematical description of the quantity.
• They also support upper and lower bound calculations in later topics. Once an interval is known for each quantity, you can combine bounds to estimate the largest or smallest possible result of a formula or physical measurement.
• Error intervals reinforce place value understanding because they depend on knowing exactly what unit has been kept. This makes them a natural extension of rounding, decimal notation, and significant figures rather than a completely separate topic.
• The notation is part of mathematical communication as much as the calculation is. Writing $a \le x < b$ clearly expresses uncertainty while preserving precision about what is and is not possible.