Question

To three significant figures a number is \(3.60\). State the maximum and minimum possible values of the number.

Short answer

Answer: The original number must be within the range of 3.595 to 3.605.

Step-by-step solution

Identify the significant figures

In this case, the number is given to have three significant figures: \(3.60\). The significant figures are the non-zero digits and any zeros between them (the trailing zero after the decimal point is considered significant here).

Determine the rounding boundaries

Now we must determine what numbers would round to \(3.60\) when considering three significant figures. The boundaries for rounding are set at half the distance to the next significant figure, so we need to find the halfway point between \(3.59\) and \(3.61\).

Calculate the maximum and minimum possible values

To find the halfway points, we add and subtract \(0.005\) (half of \(0.01\), the difference between the significant figures) from the given number. This gives us: Minimum possible value: \(3.60 - 0.005 = 3.595\) Maximum possible value: \(3.60 + 0.005 = 3.605\)

Present the final answer

The minimum possible value for the number is \(3.595\) and the maximum possible value is \(3.605\). So the number originally must be within this range before being rounded to three significant figures.

Key concepts

Rounding Numbers

Understanding how to round numbers is crucial for various mathematical and real-world applications. The process simplifies a number to make it easier to work with while maintaining its approximate value. When instructed to round a number to a certain number of significant figures, it essentially dictates the clarity or precision with which you wish to express a number.

Rounding involves looking at the digits to the right of the desired significant figure. If the next digit is 5 or higher, we increase the significant figure by one; if it is lower than 5, we leave it unchanged. For example, to round 3.604 to three significant figures, we look at the fourth figure (4 in this case) and since it's less than 5, the rounded number is 3.60. It's also key to note that zeros can play a role in rounding, especially when they follow a non-zero digit after the decimal point.

Mathematical Precision

Mathematical precision refers to the level of detail and exactness in the representation of numbers. It is especially important in fields like science and engineering where precise measurements are paramount. Significant figures are a way to indicate precision. If a measurement is given with more significant figures, it suggests a higher degree of precision.

For the number 3.60, having two significant digits after the decimal point implies that the measurement is precise to within one hundredth of a unit. However, it is vital to differentiate between precision and accuracy; a value can be precise with many significant figures but may not be an accurate representation of the true value due to systematic errors or other factors.

Minimum and Maximum Values

When discussing minimum and maximum values in the context of rounding and significant figures, we refer to the range within which the actual value could lie before being rounded. This range provides a scope of uncertainty that acknowledges the inherent imprecision in the rounded number. After rounding a number like 3.60 to a specific number of significant figures, in this case three, it's useful to know the extent of possible values that could lead to this rounded figure.

To compute these ranges, we find the halfway point between the number expressed to one fewer significant figure (minimum range) and one more (maximum range). As seen in the solution step, adding or subtracting 0.005 (which represents half of 0.01—the step between two successive values with three significant figures) determines the boundaries. Thus, the original number is between 3.595 and 3.605 before rounding, with 3.595 being the minimum and 3.605 the maximum possible pre-rounded values.