Question

. Determine whether the zeros in each number are significant: (a) 2050 (b) \(9.00 \times 10^{2}\) (c) \(0.0530\) (d) \(0.075\) (e) 300 . (f) \(285.00\)

Short answer

2050: 1 zero significant; \(9.00 \times 10^{2}\): 2 zeros significant; \(0.0530\): 1 zero significant; 0.075: 0 zeros significant; 300: 0 zeros significant; 285.00: 2 zeros significant.

Step-by-step solution

Determine the significance of zeros in number 2050

Zeros between non-zero digits are always significant. Therefore, in 2050, the zero between 2 and 5 is significant. The trailing zero (the one after the 5) is not significant because there is no decimal point present. Thus, there is 1 significant zero in 2050.

Determine the significance of zeros in number \(9.00 \times 10^{2}\)

Zeros that are to the right of a decimal point and to the right of a non-zero number are significant. Therefore, in \(9.00 \times 10^{2}\), both zeros are significant. Thus, there are 2 significant zeros in \(9.00 \times 10^{2}\).

Determine the significance of zeros in number \(0.0530\)

Leading zeros (zeros before the first non-zero digit) are not significant. Zeros to the right of a decimal point and to the right of a non-zero digit are significant. Thus, the zero in \(0.0530\) is significant. Therefore, there is 1 significant zero in \(0.0530\).

Determine the significance of zeros in number 0.075

Leading zeros are not significant. Zeros that appear between non-zero digits in a decimal are significant. Therefore, there are no significant zeros in 0.075.

Determine the significance of zeros in number 300

Trailing zeros in a whole number without a decimal are not considered significant. Thus, there are no significant zeros in 300.

Determine the significance of zeros in number 285.00

Zeros to the right of a decimal point and at the end of the number are significant. Therefore, both zeros in 285.00 are significant. Thus, there are 2 significant zeros in 285.00.

Key concepts

Zeros in significant figures

Zeros can sometimes be tricky when determining significant figures. Significant figures are all the numbers that carry meaning in a value, contributing to its precision. Here are some key rules to determine whether zeros are significant:

• Zeros between non-zero digits are significant.
• Leading zeros (zeros before the first non-zero digit) are not significant.
• Trailing zeros (zeros after a non-zero digit) can be significant, but it depends on the presence of a decimal point.

Understanding these rules is crucial for accurate scientific calculations.

Decimal points

Decimal points play a crucial role in determining the significance of zeros. They directly affect whether trailing zeros are considered significant. Here are some details:

• If a number contains a decimal point, any trailing zeros to the right of a non-zero digit are significant. For example, in the number 40.0, the trailing zero is significant.
• If there is no decimal point, trailing zeros are not significant. In the number 400, the zeros are not significant due to the lack of a decimal point.

Trailing zeros

Trailing zeros, or zeros that appear at the end of a number, can be confusing. Their significance is influenced by whether or not the number has a decimal point. Here's how:
If trailing zeros are to the right of a decimal point, they are significant. For instance, in the number 9.00, both trailing zeros are significant as they contribute to the precision of the measurement.
Without a decimal point, trailing zeros do not count as significant figures. Take the number 300; the trailing zeros are not significant because there is no decimal.

Leading zeros

Leading zeros are found at the beginning of a number, before any non-zero digit. These zeros are not considered significant since they merely serve as placeholders. For example:

• In the number 0.0530, the leading zeros before the digit 5 are not significant. However, the zero after 3 is significant because it comes after a non-zero digit and the number has a decimal point.
• In the number 0.075, all leading zeros before the digit 7 are not significant.

Remembering the rule that leading zeros are never counted as significant can simplify your calculations.