Question

Which zero is significant in the number \(50,540 ?\) What is the other zero called?

Short answer

The significant zero in the number 50,540 is the one between 5 and 4, which maintains the number's value and placement. It is called an "interior zero" or a "captured zero." The other zero, which is not significant, is the trailing zero at the end, and it only indicates the magnitude or order of the number.

Step-by-step solution

Identifying Significant Figures

In a given number, significant figures are: 1. All nonzero digits (1, 2, 3, 4, 5, 6, 7, 8, and 9). 2. Any zeroes between nonzero digits. 3. Leading zeroes (zeroes to the left of nonzero digits) are NOT significant. 4. Trailing zeroes (zeroes to the right of nonzero digits) are significant only if they appear to the right of a decimal point (in decimals).

Identifying significant zeroes in the number 50,540

Let's apply the rules mentioned above: 1. Nonzero digits are 5 and 4. They are significant. 2. The zero between 5 and 4 is significant as it is between two nonzero digits. 3. There are no leading zeroes. 4. The trailing zero (0) is NOT significant as it is to the right of a nonzero digit, but there is no decimal point.

Identify the significant zero and its name

The significant zero is the zero between 5 and 4, so the number can be represented as \(5 * 10^4\). This zero is called an "interior zero" or a "captured zero" as it lies between two nonzero digits and helps maintain the number's value and placement.

Identifying the other zero and its name

The other zero in the number 50,540 is the trailing zero. Since it is NOT significant (there is no decimal point), this zero is called a "trailing zero" or "end zero." It simply indicates the number's magnitude or order but does not add precision to the number.

Key concepts

Identifying Significant Figures

Understanding significant figures in measurements is vital in chemistry for accurately presenting data. A significant figure is any digit that contributes to the precision of a measurement. To identify significant figures, keep in mind that all non-zero digits (1 through 9) are always significant.

Zeroes present a more complex case — they can either be significant or non-significant depending on their position. Zeroes between non-zero digits are always significant because they are crucial for maintaining the correct value of the number. For example, in 207, the zero is significant because it signifies that the true value lies between 200 and 210, rather than 20 or 2000. It is important not to overlook these 'captured' or 'interior' zeroes as they directly impact the figure's precision.

Leading Zeroes in Numbers

When dealing with decimals, leading zeroes — the zeroes to the left of the first non-zero digit — are not considered significant. These leading zeroes simply serve as placeholders and do not affect the precision of the actual measurement.

For example, in the number 0.0052, the first two zeroes are not contributing to the precision; they are only indicating the decimal placement. The rule simplifies the identification process of significant figures in decimal numbers by eliminating any leading zeroes from our count. It's an important concept to grasp because it prevents the overestimation of the precision of a measurement, ensuring that only the meaningful digits are taken into account for calculations and data analysis.

Trailing Zeroes in Numbers

Trailing zeroes — the zeroes at the end of a number — can be significant or non-significant based on the context. In a decimal number, trailing zeroes that follow a non-zero digit and come after a decimal point are significant. They show the precision and extent of the measurement. For instance, in 2.500, the zeroes indicate that the measurement is precise to the thousandths place.

However, when a number has no decimal point, trailing zeroes become non-significant. They are only there to indicate the scale or magnitude and do not add precision. In the number 50,540 from our exercise, the last zero is non-significant. It tells us the value is in the tens place, but it doesn't contribute to the precision of the measurement. Understanding the distinction between when trailing zeroes count as significant figures or not is pivotal for accurate calculations and recording data in scientific fields.