Question

Indicate if the zeros are significant or not in each of the following measurements: a. \(20.05^{\circ} \mathrm{C}\) b. \(5.00 \mathrm{~m}\) c. \(0.00002 \mathrm{~g}\) d. 120000 years e. \(8.05 \times 10^{2} \mathrm{~L}\)

Short answer

a. Significant, b. Significant, c. Not Significant, d. Not Significant, e. Significant.

Step-by-step solution

Identify Significant Zeros

Determine whether zeros are significant or not in each of the given measurements.

Analyze 20.05°C

The number 20.05 has a zero between two non-zero digits. Such zeros are always significant. Both zeros in 20.05°C are significant.

Analyze 5.00 m

The number 5.00 has zeros after a decimal point and a non-zero digit. Such zeros are significant as they indicate precision. Both zeros in 5.00 m are significant.

Analyze 0.00002 g

The number 0.00002 has leading zeros before the first non-zero digit. Leading zeros are not significant. None of the zeros in 0.00002 g are significant.

Analyze 120000 years

The number 120000 has trailing zeros that can be ambiguous without additional context (e.g., decimal points or scientific notation). Usually, such trailing zeros are considered not significant unless specified otherwise. The zeros in 120000 years are not significant.

Analyze 8.05 × 10² L

The number 8.05 has a zero between two non-zero digits. Such zeros are significant. The zero in 8.05 × 10² L is significant.

Key concepts

Significant Zeros

In the world of measurements, zeros play a crucial role in indicating precision and accuracy. Significant zeros are zeros that hold meaning in their positions within a number. Let's break this down further:

• **Between non-zero digits:** Zeros that appear between two non-zero digits are always significant. For instance, in the number 20.05°C, the zeros are significant.
• **After a decimal point:** Zeros that come after a decimal point and after non-zero digits are also significant. In 5.00 meters, the zeros indicate that the measurement was made with precision.

Understanding significant zeros helps ensure that we communicate measurements accurately and consistently.

Precision in Measurements

Precision in measurements refers to how detailed or exact a measurement is. More significant figures mean more precision. Knowing which zeros are significant is essential for determining this.

• **Decimal Points:** When a zero comes after a decimal point and after a non-zero number, it shows the precision of the measurement, like in 5.00 meters.
• **Scientific Notation:** This helps to clearly show precision. For example, writing 8.05 × 10² L indicates all digits, including zeros, are significant.

The more precise a measurement, the more confident we can be in its accuracy.

Leading Zeros

Often, numbers have zeros at the beginning. These are called leading zeros, and they are not significant. Their primary role is simply to position the decimal point.
For example:

• In the number 0.00002 grams, the zeros before the 2 are leading zeros. They don't add any significant value to the measurement.

Leading zeros serve as placeholders and don't count towards the precision or significance of the number.

Trailing Zeros

Trailing zeros are zeros at the end of a number. Their significance depends on context:

**Without a Decimal Point:** Trailing zeros in a whole number without a decimal point are usually not significant, such as in 120000 years.

**With a Decimal Point:** When there is a decimal point, trailing zeros are significant. They indicate measurement precision. For instance, in 5.00 meters, the zeros are important.

**Scientific Notation:** Clearly shows when trailing zeros are significant. For example, 8.05 × 10² L makes it evident that the zero is significant.

Understanding where trailing zeros are significant helps in correctly interpreting the precision of measurements.