Question

Determine whether the zeros in each number are significant: (a) \(0.005\) (b) 1500 (c) 250 . (d) \(10.000\) (e) \(6.070 \times 10^{4}\) (f) \(0.2300\)

Short answer

Zeros in (a) and leading in (f) are not significant; zeros in (b) are not; trailing zeros in (c), (d), (e) and the middle in (e), (f) are significant.

Step-by-step solution

- Understand Significant Figures

Significant figures are the digits in a number that are reliable and necessary. This includes all non-zero numbers, any zeros between significant digits, and trailing zeros in a decimal number.

- Analyze Each Number

Let's go through each number one by one to determine if and which zeros are significant.

- Determine Zeros for (a) 0.005

For 0.005, the leading zeros are not significant. Only the digit 5 is significant, so the zeros in this case are not significant.

- Determine Zeros for (b) 1500

For 1500, there are two trailing zeros. These zeros are not significant unless there is a decimal point specified, which there is not.

- Determine Zeros for (c) 250 .

For 250., the trailing zero after the decimal point is significant. So, the zero in this case is significant.

- Determine Zeros for (d) 10.000

For 10.000, all zeros after the decimal point are significant. Therefore, all four zeros here are significant.

- Determine Zeros for (e) 6.070 x 10^4

For 6.070 x 10^4, the zero between the digits 7 and 6 is significant because it is between significant digits. Also, the trailing zero after the 7 in the decimal part is significant.

- Determine Zeros for (f) 0.2300

For 0.2300, the leading zero is not significant, but the trailing zeros and the zero between 2 and 3 are significant.

Key concepts

leading zeros

When we talk about leading zeros, we refer to zeros that appear at the beginning of a number. For example, in the number 0.005, the zeros before the 5 are leading zeros.
Leading zeros are not considered significant figures. They only serve to place the decimal point.
For 0.005, we can identify that the leading zeros are not significant. Therefore, only the digit 5 is a significant figure here.

trailing zeros

Trailing zeros are zeros that appear at the end of a number. Whether or not they are significant depends on the presence of a decimal point.
For instance, in the number 1500, the zeros are trailing zeros. Without a decimal point, these zeros are not considered significant.
Meanwhile, in the number 250., the zero after the decimal point is a trailing zero and is significant. Similarly, in the number 10.000, all trailing zeros are significant because there is a decimal point.

decimal point significance

The decimal point plays a crucial role in determining the significance of zeros.
When a number contains a decimal point, any trailing zeros after a non-zero digit become significant. For example, in 0.2300, the zeros after the digit 3 are significant because of the presence of the decimal point.
On the other hand, leading zeros like in 0.005 do not gain significance regardless of the decimal point.

scientific notation

Scientific notation is a way of expressing very large or very small numbers conveniently and it can clarify the significance of zeros.
For example, the number 6.070 x 10^4 is in scientific notation. Here, we notice that the zero between 6 and 7 is significant because it is between significant digits. Similarly, the trailing zero in the decimal part also becomes significant.
Scientific notation helps in maintaining the number of significant figures clearly.

significant digits analysis

Analyzing significant digits involves understanding which digits in a number are necessary for precision. Here are some steps to analyze:

• Non-zero digits are always significant.
• Any zeros between significant digits are significant (e.g., 6.070 x 10^4).
• Trailing zeros in a decimal number are significant (e.g., 0.2300).
• Leading zeros are not significant (e.g., 0.005).

This analysis helps ensure calculations and measurements are precise and accurate, maintaining the integrity of significant figures throughout computations.