Upper and Lower Bounds and Rounding for ESAT

Finding upper and lower bounds

When a number is rounded to a value, $x$, the Greatest Lower Bound (GLB) is defined as the smallest number that would round up to $x$. The Least Upper Bound (LUB) is the smallest number that would round up to a number larger than $x$. In practical contexts, we simply refer to these as the lower bound and the upper bound.

For example, if a number is 3.84 correct to 2 decimal places:

1. The lower bound is 3.835, which is the smallest number that rounds to 3.84.
2. The upper bound is 3.845, which is the smallest number that would round to a number larger than 3.84.

Worked Example: Finding bounds

Consider 236 grams, correct to the nearest gram.

The smallest number that rounds to 236 g when corrected to the nearest gram is 235.5 g, this is the lower bound. The number above 236 when rounded to the nearest gram is 237 g. The smallest number that rounds to 237 g is 236.5 g, this is the upper bound.

Alternatively, the bounds can be expressed as $236 \pm 0.5$ g, resulting in 235.5 g and 236.5 g.

Calculating with bounds: Multiplication and Addition

To find the upper bound of a calculation that involves only multiplication and addition, use the upper bounds of every quantity involved.

$\text{LUB}(A \times B) = \text{LUB}(A) \times \text{LUB}(B)$

$\text{LUB}(A + B) = \text{LUB}(A) + \text{LUB}(B)$

Similarly, to find the lower bound for addition and multiplication, use the lower bounds of all quantities.

$\text{GLB}(A \times B) = \text{GLB}(A) \times \text{GLB}(B)$

$\text{GLB}(A + B) = \text{GLB}(A) + \text{GLB}(B)$

Worked Example: Perimeter and Area

A rectangle has a length of 23.6 cm and a width of 14.7 cm, both correct to one decimal place.

The upper bound of the length is 23.65 cm and the upper bound of the width is 14.75 cm.

1. The upper bound of the perimeter is $2 \times 23.65 + 2 \times 14.75 = 76.8$ cm.
2. The upper bound of the area is $23.65 \times 14.75 = 348.8375$ cm$^2$.

Calculating with bounds: Division

To find the Least Upper Bound of a division, you must divide the upper bound of the dividend by the lower bound of the divisor.

$\text{LUB}\left(\frac{A}{B}\right) = \frac{\text{LUB}(A)}{\text{GLB}(B)}$

To find the Greatest Lower Bound of a division, divide the lower bound of the dividend by the upper bound of the divisor.

$\text{GLB}\left(\frac{A}{B}\right) = \frac{\text{GLB}(A)}{\text{LUB}(B)}$

Worked Example: Density

The volume of a piece of wood is 270 cm$^3$ (nearest 10 cm$^3$) and the mass is 540 g (nearest 10 g).

Bounds for volume: 265 and 275. Bounds for mass: 535 and 545.

Using $\text{density} = \frac{\text{mass}}{\text{volume}}$:

1. Upper bound of density: $\frac{275}{535} = 0.51$ g/cm$^3$.
2. Lower bound of density: $\frac{265}{545} = 0.49$ g/cm$^3$.

Calculating with bounds: Subtraction

To find the upper bound of a subtraction, subtract the lower bound of the second value from the upper bound of the first.

$\text{LUB}(A - B) = \text{LUB}(A) - \text{GLB}(B)$

To find the lower bound, subtract the upper bound of the second value from the lower bound of the first.

$\text{GLB}(A - B) = \text{GLB}(A) - \text{LUB}(B)$

Worked Example: $b^2 - 4ac$

Given $a = 6$, $b = 3$, and $c = 2$ (all to the nearest whole number), find the bounds of $b^2 - 4ac$.

Bounds for $a$: 5.5 to 6.5. Bounds for $b$: 2.5 to 3.5. Bounds for $c$: 1.5 to 2.5.

1. Upper bound calculation: $6.5^2 - 4 \times 2.5 \times 1.5 = 27.25$.
2. Lower bound calculation: $5.5^2 - 4 \times 3.5 \times 2.5 = -4.75$.

Rounding to Decimal Places and Significant Figures

Rounding to a specified level of accuracy involves checking the next digit in the sequence.

Rounding to Decimal Places (d.p.): To round to 3 d.p., if the 4th decimal digit is 5 or greater, add 1 to the 3rd decimal place. Otherwise, leave it unchanged.

Example: Rounding 31.56387 to 4 d.p.

$\begin{array}{r} 1\ 2\ 3\ 4 \\ 31.5638\ 7 \\ 31.5639 \end{array}$

Rounding to Significant Figures (sig. figs.):

1. Count from left to right, starting at the first non-zero digit.
2. Cut off digits after the required count.
3. If the first digit cut off is 5 or more, round up the last significant digit.
4. Use zeros to maintain the place value for numbers larger than 1.

Example: 365,892 to 2 sig. figs. becomes 370,000.

Rounding Measures and Truncation

Rounding Measures: Ensure units are consistent before rounding. To round metres to the nearest kilometre, divide by 1,000 and then round to the nearest whole number.

Example: 36,548 cm in metres, correct to the nearest metre. $36,548 \text{ cm} = 365.48 \text{ m}$. To the nearest metre, this is 365 m.

Truncation: This involves cutting off a decimal after a certain point without any rounding adjustment.

Example: Truncating 3.45699 after 3 d.p. results in 3.456.

Inequalities and Error Intervals

Rounding Intervals: If $x = 3.6$ (to 1 d.p.), the error interval is $3.55 \leq x < 3.65$. The lower bound is inclusive, but the upper bound is exclusive because 3.65 would round up to 3.7.

Truncation Intervals: If $x = 3.6$ (truncated to 1 d.p.), the error interval is $3.6 \leq x < 3.7$. The value must be at least 3.6 but must be strictly less than 3.7.