Question

Underline the significant zeros in the following numbers: (d) \(4.0100 \times 10^{4}\) (a) \(0.41 ;\) (b) \(0.041 ;\) (c) 0.0410

Short answer

In (d) 4.0100 × 10^{4}, underline 0100; in (c) 0.0410, underline the last zero.

Step-by-step solution

- Identify all zeros in each number

Observe where the zeros are located in each number. Determine which zeros are between non-zero digits or after a decimal point.

- Analyze the role of each zero

Significant zeros include any zero between non-zero digits, and any trailing zero to the right of the decimal point after a non-zero digit.

- Underline significant zeros

Apply the rules from Step 2 to underline significant zeros: - For (d) \(4.0100 \times 10^{4}\), significant zeros are underlined as \(4.0100 \times 10^{4}\). The last two zeros are significant because they are trailing zeros after the decimal point. - For (a) \(0.41\), there are no significant zeros. - For (b) \(0.041\), there are no significant zeros. - For (c) \(0.0410\), significant zero is \(0.041\textbf{0} \). The last zero is significant because it is trailing after the decimal point following non-zero digits.

Key concepts

Identifying Significant Zeros

Understanding which zeros in a number are considered 'significant' is crucial in many areas of science and mathematics. In any given number, not all zeros will contribute meaningfully to the precision of that number.

Here are some key ideas to bear in mind when identifying significant zeros:

• Any zero between non-zero digits counts as significant. For example, in the number 105, the zero is significant because it is between the digits 1 and 5.
• Zeros that appear after a non-zero digit and to the right of a decimal point are also significant. For instance, in 2.50, the zero is significant.

Conversely, leading zeros (those at the beginning of a number) are not deemed significant as they merely indicate the decimal place. For example, in the number 0.0041, the leading zeros are not significant.

Trailing Zeros

Trailing zeros can be a bit confusing, but with the right understanding, it becomes simpler to identify which of them are significant.

Trailing zeros are zeros that appear at the end of a number. Their significance depends on whether or not they are placed after a decimal point.

• In a number with a decimal point, trailing zeros are considered significant because they indicate the precision of the number. For example, in 4.0100, both trailing zeros are significant.
• In a whole number without a decimal point, trailing zeros may or may not be significant depending on the context. For example, in the number 5000, the trailing zeros may be significant or just placeholders.

To avoid ambiguity in such cases, scientific notation is often used to clarify the level of precision.

Decimal Points

Decimal points play a key role in determining which zeros are significant. Understanding their impact can clear up many misconceptions.

Here’s a simple guide:

• Any number that includes a decimal point allows all digits after the decimal point to be significant if they affect the number's precision.
• For example, the number 0.0410 has three significant figures: 4, 1, and the trailing zero, because the zero is after the decimal point.
• Conversely, leading zeros do not count as significant; they only indicate the position of the decimal point, as in the number 0.0041.

Knowing whether a number has a decimal point and where that decimal point is located will help you identify significant figures correctly.

Significant Figures

Significant figures (or significant digits) are the digits in a number that contribute to its precision. Here's what you need to know for identifying them:

• All non-zero digits are always significant. For example, the number 123.45 has five significant figures: 1, 2, 3, 4, and 5.
• Any zeros between significant digits are significant. In the number 102.03, all five digits are significant, including the zeros.
• Trailing zeros in a number with a decimal point are significant. For instance, 0.00400 has three significant figures: 4, 0, and 0.

Remember, the concept of significant figures allows scientists and mathematicians to communicate the precision of a measurement effectively. It’s essential to count and identify these figures accurately in any calculation.