Question

Complete the following and explain each in your own words: leading zeros are (never/sometimes/ always) significant; captive zeros are (never/sometimes/always) significant; and trailing zeros are (never/sometimes/always) significant. For any statement with an answer of "sometimes," give examples of when the zero is significant and when it is not, and explain.

Short answer

Leading zeros are never significant, as they only indicate the position of the decimal point but do not contribute to the value of the number. Captive zeros, located between nonzero digits, are always significant because they contribute to the magnitude of the number. Trailing zeros are sometimes significant, depending on the presence of a decimal point and the context in which the number is used. If there is no decimal point, trailing zeros may be insignificant; if a decimal point is present, trailing zeros are significant as they provide more precise information about the number's value.

Step-by-step solution

Determining the significance of leading zeros

Leading zeros are the zeros that come before any non-zero digit in a number. They are used to indicate the position of the decimal point but do not contribute to the value of the number. Therefore, leading zeros are never significant. Example: In the number 0.0123, the two leading zeros (before the digit "1") are not significant.

Determining the significance of captive zeros

Captive zeros are the zeros that are located between nonzero digits. They contribute to the magnitude of a number and, therefore, are always significant. Example: In the number 2005, the two captive zeros (between the digits "2" and "5") are significant.

Determining the significance of trailing zeros

Trailing zeros are the zeros that come after the last non-zero digit in a number. They can be significant or not, depending on the context and the presence of a decimal point. In a whole number with no decimal point, trailing zeros may be insignificant since they only indicate the magnitude or place value without adding precision to the value. In this case, trailing zeros are sometimes significant. Example 1: In the number 3000, the three trailing zeros may be considered insignificant if their presence is only to indicate the magnitude of the number. However, when a number includes a decimal point, trailing zeros after the decimal point are considered significant because they provide more precise information about the value of the number. Example 2: In the number 45.00, the two trailing zeros after the decimal point are significant, as they indicate that the number is precisely equal to 45, not just an approximation. In summary: - Leading zeros are never significant. - Captive zeros are always significant. - Trailing zeros are sometimes significant, depending on the presence of a decimal point and the context in which the number is used.

Key concepts

Understanding Leading Zeros

Leading zeros are the zeros that appear before the first non-zero digit in a number. They are important because they help to show where the decimal point is located. However, these zeros do not affect the numerical value of a number. This is why they are never considered significant when counting significant figures.
For example, in the number \(0.0075\), the zeros before the \(7\) are leading zeros. They are just placeholders that show us the value is a small number, but they do not count as significant figures. To better grasp this:

• Leading zeros simply set the scale of the number.
• They play no role in the precision or accuracy of a measurement.

By removing leading zeros, the numerical value of a number does not change. It is crucial to remember that no example exists where leading zeros would ever become significant.

Exploring Captive Zeros

Captive zeros, often called "embedded zeros," are zeros located between two non-zero digits. These zeros are always significant because they represent an actual quantity and belong to the precise count of the number.
Take the number \(405.09\), for instance. Here, the zero between \(4\) and \(5\), as well as the zero between \(0\) and \(9\), are both captive zeros. They count as significant because:

• They exist within the bounds of non-zero digits.
• Their presence affects the measurement or value.

Counting them is straightforward: all zeros "trapped" between non-zero digits automatically contribute to the total count of significant figures.

Significance of Trailing Zeros

Trailing zeros, situated at the end of a number after any non-zero digits, can sometimes lead to confusion as they may be significant or not, depending on their placement and the presence of a decimal point.
A number like \(7500\) has two trailing zeros, and without a decimal point, they might be seen as just indicating the magnitude, not the precision. This is why they are often insignificant in whole numbers.
However, when the number is written with a decimal point, such as \(7500.0\), these zeros become significant. They imply that the measurement is accurate to a specific precision or value. Consider the examples:

• \(600\) without a decimal suggests two significant figures, while \(600.\) with a decimal implies three significant figures.
• \(45.00\) indicates the zeroes are part of the precise measurement, giving it four significant figures.

Knowing whether to count trailing zeros involves checking for decimal points or context in their usage.