Question

Underline any trailing zeros in these measurements: (a) \(12.202 \mathrm{~km}\) (b) \(0.01 \mathrm{~mL}\) (c) \(205^{\circ} \mathrm{C}\) (d) \(0.010 \mathrm{~g}\)

Short answer

Trailing zeros are the zeros that come after a non-zero digit in a decimal number. In the given measurements: (a) There are no trailing zeros in \(12.202 \mathrm{~km}\). (b) The trailing zero in \(0.01 \mathrm{~mL}\) is the last zero: \(0.0\underline{1}\mathrm{~mL}\). (c) There are no trailing zeros in \(205^{\circ} \mathrm{C}\). (d) There are two trailing zeros in \(0.010\mathrm{~g}\): \(0.0\underline{1}\underline{0}\mathrm{~g}\).

Step-by-step solution

1. Identify Trailing Zeros in (a)

In the number \(12.202 \mathrm{~km}\), the decimal part consists of the digits \(202\). There are no zeros after the last non-zero digit, which means there are no trailing zeros in this number.

2. Identify Trailing Zeros in (b)

In the number \(0.01 \mathrm{~mL}\), the decimal part consists of the digits \(01\). There is one zero after the last non-zero digit (1), so the trailing zero in this number is the last zero: \(0.0\underline{1}\mathrm{~mL}\).

3. Identify Trailing Zeros in (c)

In the number \(205^{\circ} \mathrm{C}\), there are no decimal points, so there are no trailing zeros. However, if we wanted to express the number with a decimal point, it would be \(205.0^{\circ} \mathrm{C}\), but still, there is no zero after the last non-zero digit, so there are no trailing zeros.

4. Identify Trailing Zeros in (d)

In the number \(0.010\mathrm{~g}\), the decimal part consists of the digits \(010\). There are two zeros after the last non-zero digit (1): \(0.0\underline{1}\underline{0}\mathrm{~g}\).

Key concepts

Trailing Zeros

Trailing zeros are zeros at the end of a number, but only if they appear after a decimal point. They can be tricky because they are significant in some cases and not in others. When measuring with significant figures, trailing zeros are only counted if they come after a decimal point. These zeros can indicate the precision of a measurement.

For example, in the measurement \(12.202\ \mathrm{km}\), there are no trailing zeros. However, in \(0.010\ \mathrm{g}\), the zeros after the '1' in the decimal part (010) are considered trailing zeros. In this case, they show that the measurement precision goes to three decimal places.

Remember:

• If a number has a decimal point and zeros follow a non-zero digit, those zeros are trailing.
• Trailing zeros without a decimal point aren't usually considered significant.

Decimal Points

Decimal points are crucial in determining the significance of zeros in a number. They serve as a boundary that distinguishes between whole numbers and fractional parts. Knowing how to appropriately handle decimal points is essential for precision in scientific measurements.

In the number \(0.01\ \mathrm{mL}\), the decimal point informs you to only consider numbers after it as significant, such as the trailing zero here. Without a clear decimal point, like in \(205\degree \mathrm{C}\), it's often difficult to assume if a zero is significant. In practice, a number like \(205\degree \mathrm{C}\) could be expressed as \(205.0\degree \mathrm{C}\) to make it clear that the measurement precision is up to one decimal place.

This clarity is why decimal points matter so much.

• Decimal points separate whole numbers from decimal fractions.
• Their presence or absence affects whether trailing zeros are considered significant.

Measurement Units

Measurement units indicate what dimension is being measured, like length, volume, temperature, or mass. They are imperative in ensuring that numbers are meaningful and contextually accurate. Just stating \(205\) provides little context without its unit, which could be degrees Celsius \((\degree \mathrm{C})\).

Units also play a role in significance and conversion accuracy. Consider \(12.202\ \mathrm{km}\) and \(0.01\ \mathrm{mL}\); each entails specific units that tell you whether you're measuring kilometers or milliliters.

Units must always accompany measurements:

• They provide context to raw numbers, allowing for accurate interpretation.
• Without them, data is often uninterpretable or misleading.

Scientific Notation

Scientific notation is a way to express extremely large or small numbers in a more convenient form. It makes calculations easier and clarifies the significance of zeros. It is written as \(m \times 10^n\), where \(m\) is the significant figure and \(n\) shows the power of ten.

For example, \(0.010\ \mathrm{g}\) can be expressed in scientific notation as \(1.0 \times 10^{-2}\ \mathrm{g}.\) It highlights that the precision of the figure is to one significant decimal place.

Benefits of using scientific notation include:

• Easy communication of very large or small numbers.
• Clearly communicates which digits are significant.
• Avoids misunderstanding about zeros, distinguishing leading from trailing zeros.

Using scientific notation not only simplifies large calculations but also emphasizes which figures carry meaning in precise measurements.