Question

For each number, underline the zeroes that are significant and draw an \(\mathbf{x}\) through the zeroes that are not. MISSED THIS? Read Section 1.7; Watch \(\mathrm{KCV} 1.6,\) IWE 1.5 a. \(1,050,501 \mathrm{~km}\) b. \(0.0020 \mathrm{~m}\) c. \(0.000000000000002 \mathrm{~s}\) d. \(0.001090 \mathrm{~cm}\)

Short answer

a. 1,\textbf{05}0,\textbf{5}0\textbf{1} km; b. 0.00\textbf{20} m; c. 0.000000000000002 s; d. 0.00\textbf{10}\textbf{90} cm.

Step-by-step solution

- Understanding Significant Figures

Significant zeroes are those that are between other non-zero digits, at the end of a decimal number, or at the beginning of a decimal number only if there's a decimal point explicitly stated. Zeroes at the beginning of a decimal number without an explicit decimal point are not considered significant.

- Analyzing Number a

For the number 1,050,501 km, the two zeroes after the 1 are significant as they are between non-zero digits, and the last zero is also significant since it comes after a non-zero digit and there's a decimal number assumed at the end (i.e., 1,050,501.0).

- Analyzing Number b

For the number 0.0020 m, the first two zeroes are not significant as they come before any non-zero digits and are in the decimal portion of the number. The last zero is significant as it comes after a non-zero digit.

- Analyzing Number c

For the number 0.000000000000002 s, none of the zeroes are significant as there are no non-zero digits after them, they are only serving as placeholders.

- Analyzing Number d

For the number 0.001090 cm, the first three zeroes are not significant because they come before any non-zero digits and are part of the decimal portion. The last two zeroes are significant because they come after non-zero digits.

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, engineers, and mathematicians to handle vast or minuscule numbers efficiently. In scientific notation, numbers are written as the product of two factors: a coefficient that is greater than or equal to 1 and less than 10, and a power of 10. This format is convenient because it simplifies arithmetic operations and makes the comparison of magnitudes straightforward.

For example, the number 0.000000000000002 s from the exercise could be written in scientific notation as \(2 \times 10^{-15}\) s. This representation clearly shows the precision of the measurement and the scale of the value compared to regular decimal form.

Measurement Precision

Measurement precision reflects the accuracy and specificity with which a value is expressed. In other words, it's the degree to which the result of a measurement conforms to the correct value within the limits of the measurement's uncertainty. When measurements are recorded, the number of significant figures conveys the precision of the measurement. This is critical in scientific and technical fields, where the exactness of data is paramount.

Precision is not just about the instrument's capability, but also about how the measurement is reported. For instance, the number 1,050,501 km suggests a precision to the nearest kilometer, whereas 0.0020 m indicates precision to the thousandth of a meter. Understanding the concept of significant figures is essential to properly communicating this precision.

Zeroes in Significant Figures

Zeroes play a special role in determining the significant figures of a number. They can either represent actual measured values or serve as placeholders to indicate the scale of the number. Differentiating between the two is crucial for understanding the precision of a measurement. Here are some general rules:

• Zeroes between non-zero digits are always significant (e.g., 1,050,501 km).
• Zeroes at the end of a number and to the right of a decimal point are significant (e.g., 0.0020 m).
• Zeroes at the beginning of a decimal number are not significant; they merely indicate the position of the decimal point (e.g., 0.001090 cm).

To improve understanding, visual cues are helpful. For example, underlining significant zeroes and marking non-significant zeroes with an 'X' can clarify their relevance within a given number.