How do we determine significant figures in calculations involving both addition/subtraction and multiplication/ division?

Understanding the concept of significant figures is crucial in measurements and calculations to correctly indicate precision. To identify significant figures, first look at the whole number portion from the left side, counting from the first non-zero digit. For example, in the number 0.0035, the significant figures are '35', as the leading zeros are not counted. Similarly, in 4500, if there was a decimal point (4500.), all digits would be significant. However, without a decimal point, zeroes at the end of a number are ambiguous and their significance can depend on the context or given information.

When dealing with numbers less than one, discard any leading zeros and begin your count with the first non-zero digit. Decimal places play a role as well, as they indicate precision — in 45.600, all digits are significant because the zeroes after the decimal show accuracy in measurement. Recognizing these patterns allows a student to navigate through calculations with precision and present their results with the correct level of detail.

After identifying significant figures, the next step is to apply rules that govern their use in various calculations. There are general guidelines to keep in mind: Non-zero digits are always significant; any zeros between non-zero digits are significant; leading zeros (those before the first non-zero digit) are never significant; trailing zeros in a number containing a decimal point are significant.

Understanding these basic rules is foundational for performing accurate calculations. Additionally, exact numbers like defined constants or quantities counted out (such as 20 apples) have an infinite number of significant figures as they are considered precise without ambiguity. Grasping these rules ensures that results maintain integrity and reflect the limitations imposed by the initial measurements or constants in an equation.

When adding or subtracting, we must pay attention to the decimal places to maintain precision. The rule is straightforward: the result should not have more decimal places than the least precise number in the operation. For instance, when adding 22.3 (with one decimal) and 7.689 (with three decimals), the result should be rounded to one decimal place, thus 29.9.

This preserves the integrity of the least precise measurement and acknowledges its impact on the overall calculation´s precision. When performing these types of calculations, ensure each step adheres to this rule — especially before proceeding to further operations like multiplication or division — so that the final result is as accurate as possible given the data provided.

In contrast to addition and subtraction, for multiplication and division, the focus shifts from decimal places to the number of significant figures. The golden rule here is that your answer must have the same number of significant figures as the factor with the least significant figures. If you multiply 3.987 (four significant figures) by 2.0 (two significant figures), your result should have two significant figures, rendering it as 8.0 after rounding.

This method acknowledges the inherent imprecision in the least accurate measurement and carries that level of precision throughout the calculation. During combined operations that involve both types of calculations, ensure you perform and round off each operation separately according to its significant figures rules to prevent compounding errors in your final answer.