Question

For each measured quantity, underline the zeros that are significant and draw an \(X\) through the zeros that are not. (a) \(0.00010320 \mathrm{~s}\) (b) \(1,322,600,324 \mathrm{~kg}\) (c) \(0.0001240 \mathrm{in}\). (d) \(0.02061 \mathrm{~m}\)

Short answer

Significant zeros: (a) 0.0001X0320 s, (b) 1,322,600,324 kg (all zeros are significant), (c) 0.0001X240 in, (d) 0.0X2061 m.

Step-by-step solution

- Understand the rules of significant figures for zeros

Firstly, recognize the types of zeros that are significant: (1) Zeros between non-zero digits are always significant. (2) Zeros at the beginning of a number are never significant - they are merely placeholders. (3) Zeros at the end of a number and to the right of a decimal point are significant. (4) Zeros at the end of a number but to the left of a decimal point may or may not be significant, and they are not significant if there's no decimal point mentioned.

- Analyzing zeros in (a) 0.00010320 s

For the number 0.00010320 s, the first four zeros are not significant because they are at the beginning of the number. The zeros between '1' and '3' and after '2' are significant because they are between non-zero digits and at the end of the number after a decimal.

- Analyzing zeros in (b) 1,322,600,324 kg

For the number 1,322,600,324 kg, all zeros are between non-zero digits, therefore, they are all significant. There is no need to consider ending zeros separately because this number does not have a decimal point mentioned.

- Analyzing zeros in (c) 0.0001240 in

For the number 0.0001240 in, the first four zeros are not significant as they are at the beginning of the number. The zero after '4' is significant because it is at the end of the number and after a decimal point.

- Analyzing zeros in (d) 0.02061 m

In the number 0.02061 m, the first two zeros are not significant because they are at the beginning of the number. Thereafter, all digits, including the zero between '2' and '6', are significant because the zero is between non-zero digits.

Key concepts

Measuring Quantities and Significant Figures

Accurate measurement is a fundamental aspect of science and engineering. When we measure quantities, we must also consider the precision of those measurements, which is indicated by the number of significant figures. Significant figures, or 'sig figs', reflect the confidence in a measurement by showing which digits are trustworthy. Understanding the rules for identifying significant figures helps in consistently reporting the precision of measured quantities.

The key rules for determining significant figures include:

• Non-zero digits are always significant.
• Leading zeros (zeros before non-zero digits) are not significant; they merely indicate the position of the decimal point.
• Captive zeros (zeros between non-zero digits) are always significant.
• Trailing zeros (zeros at the end of a number after a non-zero digit) are significant if the number has a decimal point.

By following these rules, you can underline or mark the significant zeros in measurements, just as in the provided exercise examples. It is critical to apply these rules accurately to avoid misrepresenting the precision of your measurements.

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. This notation is frequently used in science and mathematics because it simplifies numerical calculations and clearly indicates the number of significant figures.

Here's how scientific notation works:

• A number is written as the product of two factors.
• The first is a number greater than or equal to 1 and less than 10; it includes all of the significant figures.
• The second is a power of 10, which places the decimal point correctly.

For example, the number 0.00010320 s would be written in scientific notation as \(1.0320 \times 10^{-4}\) s, indicating that there are five significant figures in the measurement. Writing numbers in scientific notation can also help in quickly identifying significant zeros since all of the digits in the first factor are significant by default.

Decimal Place Value

Place value in decimals helps us understand the position's value each digit holds relative to the decimal point. With each step to the right of the decimal point, the value of the digit becomes ten times smaller, moving from tenths to hundredths to thousandths, and so on.

Understanding decimal place value is crucial when considering significant figures:

• The placement of a zero can change its significance, as the place value determines whether it's a leading, captive, or trailing zero.
• A trailing zero communicates that measurements were taken to a certain level of precision (e.g., 0.020 kg indicates precision to the hundredths place).

The value that each place holds directly impacts the interpretation of precision in a measurement. When reporting significant figures, one must pay close attention to these decimal place values to correctly convey the level of precision.