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Rounding numbers is a fundamental mathematical skill that makes it easier to work with figures that may be long or cumbersome. The process involves adjusting the number to a specified degree of precision, making it more manageable for estimation, mental calculation, and readability. When we round a number, we follow certain rules to determine which way the number will be adjusted.

For instance, if we're rounding to the nearest tenth and our number is 2.86, we look at the digit in the hundredths place to decide whether to round up or down. As per standard rounding rules, if this digit is 5 or higher – in this case, 6 – we round the tenth place up, resulting in 2.9. Conversely, if the digit were lower than 5, we would round down, keeping the tenth place the same.

Decimal places refer to the positions to the right of the decimal point in a number. Each position represents a fraction of ten; the first decimal place is the tenths place, the second is the hundredths, the third is the thousandths, and so on.

When rounding to two decimal places, such as in the textbook exercise, the focus is on the hundredths place. Determining this place's value lets us know which digit, in terms of economic value, we are calculating down to, like calculating currency to the nearest cent. Rounding to more decimal places increases precision, which is particularly important in scientific and technical fields where measurements must be very precise.

Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all non-zero digits, any zeros between them, and any trailing zeros in the decimal portion. Using significant figures is important in science and engineering because they indicate how precise a measurement is. The more significant figures, the more precise the measurement.

In the textbook example, rounding to two decimal places affects the number of significant figures. The number 38.468 has five significant figures, but when we round to 38.47, we're left with four significant figures. This change reflects a decrease in precision but an increase in manageability of the number for various practical applications.