Question

For each number, underline the zeroes that are significant and draw an \(\mathbf{x}\) through the zeroes that are not. a. \(180,701 \mathrm{mi}\) b. \(0.001040 \mathrm{~m}\) c. \(0.005710 \mathrm{~km}\) d. \(90,201 \mathrm{~m}\)

Short answer

a. 1(80),701 mi b. 0.00(1)040 m c. 0.00(57)10 km d. 9(0)201 m. In parentheses are the significant zeroes.

Step-by-step solution

- Understand zeros in significant figures

In significant figures, any zeroes between non-zero digits are significant. Leading zeroes are not significant, and trailing zeroes are significant only if the number contains a decimal point.

- Analyze a. 180,701 mi

The zero between 8 and 7 is significant as it is between non-zero digits. There are no leading or trailing zeroes to consider in this number.

- Analyze b. 0.001040 m

The initial three zeroes are leading zeroes and not significant. The zero between 4 and the terminal zero is significant as it is between non-zero digits. The terminal zero is also significant since it comes after the decimal point and after a non-zero digit.

- Analyze c. 0.005710 km

The initial three zeroes are leading zeroes and not significant. There are no zeroes between non-zero digits. The final zero is significant as it is a trailing zero in a decimal number.

- Analyze d. 90,201 m

The zero between 9 and 2 is significant because it is between non-zero digits. There are no leading or trailing zeroes to consider in this number.

Key concepts

Scientific Notation

When working with extremely large or small numbers, scientific notation is an efficient way to express them.

It involves writing numbers in the form of \( a \times 10^n \) where \( a \) is a number, usually between 1 and 10, that includes all the significant digits of the original number, and \( n \) is an integer that represents the number of places the decimal point has been moved.

For instance, in the number \( 0.001040 \text{~m} \) from the exercise, if we were to express it in scientific notation, it would be \( 1.040 \times 10^{-3} \text{~m} \) once all leading zeroes are omitted as they are not significant in this context.

Measurement Accuracy

Measurement accuracy refers to how close a measurement is to the true or accepted value. In expressing accuracy, significant figures play a crucial role because they tell us which digits in a measurement are reliable and therefore meaningful.

The more significant figures a measurement has, the more precise it is thought to be. For example, \( 0.005710 \text{~km} \) suggests a greater degree of precision than \( 0.006 \text{~km} \) because it has more significant figures, implying that the measurement was taken with a finer instrument or technique.

Decimal Places

Decimal places are the number of digits to the right of the decimal point in a number.

They are important in determining which zeroes are significant in a decimal measurement. Trailing zeroes, those at the end of a number after a decimal point, are considered significant because they affect the precision of the measurement.

Understanding Decimal Places in Measurements

For example, in the given exercise, \( 90,201 \) has no decimal places, and the zero is significant because it's between non-zero digits. In contrast, \( 0.005710 \) has four decimal places, and the final zero is significant because it denotes a certain level of precision in the measurement.