Question

For each of the mathematical expressions given: a. Tell the correct number of significant figures for the answer. b. Evaluate the mathematical expression using correct significant figures in the result.

Short answer

To find the correct number of significant figures and evaluate a given mathematical expression, follow these steps: 1. Determine significant figures for each number in the expression using these guidelines: - Non-zero digits are always significant. - Any zeros between significant digits are significant. - Leading zeros are not significant. - Trailing zeros are significant only if there is a decimal point. 2. Use the appropriate significant figure rules during calculations: - For addition/subtraction, round the result to the same number of decimal places as the measurement with the least number of decimal places. - For multiplication/division, round the result to the lowest number of significant figures among the measurements involved. 3. Evaluate the given mathematical expression using the correct significant figures by following the above rules. 4. Round the result according to the significant figure rules. For example, for the expression (3.20 * 10^3 kg) * (4.1 m/s), the result will be: 1.3 * 10^4 kg*m/s.

Step-by-step solution

Determine significant figures

Significant figures are the meaningful digits in a number which contribute to its precision. To identify significant figures follow these rules: - Non-zero digits are always significant. - Any zeros between significant digits are significant. - Leading zeros are not significant. - Trailing zeros are significant only if there is a decimal point. 2. Apply significant figure rules for addition/subtraction:

Rules for addition/subtraction

When performing calculations that involve addition or subtraction, you should round the result to the same number of decimal places (not significant figures) as the measurement with the least number of decimal places. 3. Apply significant figure rules for multiplication/division:

Rules for multiplication/division

When performing calculations that involve multiplication or division, the result should be rounded to the lowest number of significant figures among the measurements involved in the calculation. 4. Solve the given expression:

Evaluate given expression

Evaluate the given mathematical expression using the correct significant figures by following the above rules. Use the appropriate rules for significant figures during calculations, either addition/subtraction or multiplication/division. Finally, round the result according to the significant figure rules. Example: Suppose the given expression is: (3.20 * 10^3 kg) * (4.1 m/s) Step 1 - Determine significant figures for each number: - 3.20 * 10^3 kg has 3 significant figures (3, 2, and 0). - 4.1 m/s has 2 significant figures (4 and 1). Step 2 - Since the expression involves multiplication, the result should have the lowest number of significant figures among the given measurements (2 significant figures in this case). Step 3 - Evaluate the expression: (3.20 * 10^3 kg)*(4.1 m/s) = 13120 kg*m/s Step 4 - Round the result to 2 significant figures: 1.3 * 10^4 kg*m/s

Key concepts

Precision in Measurement

Precision in measurement refers to the level of detail and accuracy in the value of a measured quantity. When measurements are made, they are expressed with a certain number of significant figures that reflect both the precision of the measurement device being used and the care with which the reading is taken. Each significant figure in a measurement expresses a degree of certainty.

• Significant figures involve non-zero digits. For example, in the number 245, all digits are significant.
• Zeros located between significant figures are counted. The number 1003 has four significant figures because of the zero between the 1 and 3.
• Leading zeros are not counted as they merely position the decimal point. In 0.045, only the digits 4 and 5 are significant.
• Trailing zeros, only after a decimal point, are considered significant. For example, 2.300 has four significant figures.

Knowing these rules helps us determine the precision of any given measurement and how this reflects on our overall calculation.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact and easily understandable format. This format follows the pattern of a number between 1 and 10 multiplied by a power of ten, using the form: \[ a \times 10^n \]This method is especially helpful when dealing with measurements that involve extremely large numbers or tiny fractions.

• The number part \(a\) is the significant figures of the measurement.
• The exponent \(n\) represents the number of places the decimal point has "moved".

An example is the number 3,200 can be written as \(3.2 \times 10^3\). By focusing only on the significant figures, scientific notation clarifies the extent of precision in measured quantities. This prevents error in further operations with the number.

Addition Subtraction Rules

When performing addition or subtraction, precision is maintained by considering decimal places rather than the number of significant figures. You must adjust your result to align with the measurement that has the fewest decimal places. For example, when adding 123.45, 67.1, and 0.003:

• The numbers have two, one, and three decimal places, respectively.
• Therefore, the final sum should be reported with only one decimal place, following the number with the least decimal places: 123.45 + 67.1 + 0.003 equals approximately 190.6.

This rule ensures that final calculations don't overstate the precision of any component measurement. By aligning with the least precise measurement, we ensure that the precision of the result accurately reflects the inputs.

Multiplication Division Rules

In the case of multiplication or division, the precision is governed by the total number of significant figures in the values being calculated. Your final result should match the lowest number of significant figures present in any of the numbers used in the operation. To illustrate, imagine multiplying 4.56 and 0.0032:

• 4.56 has three significant figures, while 0.0032 has two.
• The result must therefore retain only two significant figures: \(4.56 \times 0.0032 = 0.014592\), which rounds to 0.015.

This rule helps maintain realistic precision, ensuring results don’t appear more accurate than the least precise measurement in the operation.