Rounding and Error Intervals for the TMUA

Rounding to Decimal Places

Rounding to a specified number of decimal places involves looking at the digit immediately following the required decimal place. If this digit is $5$ or more, we increase the digit in the final decimal place by $1$. Otherwise, the digit remains unchanged.

For example, to round the number $31.56387$ to $4$ decimal places, we count four places to the right of the decimal point. The fifth digit is $7$. Since $7$ is greater than or equal to $5$, we round the fourth digit up from $8$ to $9$. This indicates that $31.56387$ is closer to $31.5639$ than to $31.5638$.

Conversely, to round $31.56387$ to $2$ decimal places, we look at the third decimal place, which is $3$. Since $3$ is less than $5$, the second decimal digit remains unchanged. The result is $31.56$, as the original number is closer to $31.56$ than to $31.57$.

Rounding to Significant Figures

Rounding to significant figures follows a similar logic to decimal places but begins at the first non-zero digit when reading from left to right. This digit is the first significant figure. Once the required number of significant figures are identified, all digits to the right are cut off. If these digits are to the left of the decimal point, they are replaced with zeros to maintain the place value of the number. If the digit following the final significant figure is $5$ or more, the last significant figure is increased by $1$.

Examples with Decimals and Large Numbers

Consider rounding $0.0023824$ to $4$ significant figures. The first significant figure is $2$. Counting four figures to the right ($2, 3, 8, 2$), we see the next digit is $4$. Since $4$ is less than $5$, we make no correction, resulting in $0.002382$.

To round the same number, $0.0023824$, to $2$ significant figures, we count from the first non-zero digit ($2, 3$). The next digit is $8$. Since $8$ is $5$ or more, we increase the $3$ to a $4$, resulting in $0.024$.

For large numbers, such as $365,892$ rounded to $2$ significant figures, we count the first two figures ($3, 6$). The next digit is $5$, which requires us to round the $6$ up to a $7$. We must then fill the remaining places to the left of the decimal point with zeros to preserve the magnitude, giving $370,000$.

Rounding Measures

When rounding measures to a specified accuracy, it is vital to first ensure the units are correct. For example, to write $36,548$ cm in metres correct to the nearest metre, we first divide by $100$ to convert to metres, giving $365.48$ m. Correcting this to the nearest whole metre involves cutting off digits after the decimal point. Since the first digit removed is $4$, no correction is needed, and the result is $365$ m.

Truncation of Decimals

Truncation is the process of cutting off a decimal at a specific point without any rounding or correction. For example, truncating $3.45699$ to $3$ decimal places involves simply removing everything after the third decimal digit. The result is $3.456$. No attempt is made to round up, even though the next digit is $9$.

Error Intervals and Inequalities

Rounding or truncating a number creates an error interval, which is the range of possible values the original number could have taken. We use inequality notation to define these intervals.

Intervals for Rounding

If a number $x$, rounded to $1$ decimal place, is $3.6$, the range of possible values for $x$ is $3.55 \leq x < 3.65$. The lower bound ($3.55$) is the smallest value that rounds up to $3.6$. The upper bound ($3.65$) is the threshold where a value would round to $3.7$. Note the use of the 'less than' symbol ($<$) for the upper bound: $3.65$ itself would round to $3.7$, so it is not included in the interval for $3.6$.

Intervals for Truncating

If a number $x$, truncated to $1$ decimal place, is $3.6$, the logic differs. Because truncation simply removes digits, the smallest possible value for $x$ is $3.6$ itself. The largest possible value is anything just below $3.7$. Thus, the inequality is $3.6 \leq x < 3.7$. Any value in this range, such as $3.6999$, would result in $3.6$ when truncated to $1$ decimal place.