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State the number of significant figures in the following: (a) \(0.007 \mathrm{~m}^{2}\) (b) \(2.64 \times 10^{24} \mathrm{~kg}\) (c) \(0.2370 \mathrm{~g} \mathrm{~cm}^{-3}\) (d) \(6.320 \mathrm{~J}\) (e) \(6.032 \mathrm{~N} \mathrm{~m}^{-2}\) (f) \(0.0006032 \mathrm{~m}^{2}\)

Short Answer

Expert verified
(a) 1, (b) 3, (c) 4, (d) 4, (e) 4, (f) 4 significant figures.

Step by step solution

01

Identify Non-Zero Digits

For each number, identify all the non-zero digits, as they are always significant.
02

Count of Non-Zero and Zeroes Between Non-Zero Digits

In the numbers, count all the non-zero digits and any zeroes between them. These are significant figures.
03

Determine Leading, Trailing, and Captive Zeroes

Identify leading zeroes before the first non-zero digit, which are not significant. Determine captive zeroes (between non-zero digits), which are significant. Identify trailing zeroes in a decimal, which count as significant.
04

Apply Rules to Each Example

Apply the rules above to each number: - (a) 0.007 m虏: Only 7 is significant. - (b) 2.64 x 10虏鈦 kg: 2, 6, and 4 are significant. - (c) 0.2370 g cm鈦宦: 2, 3, 7, and the trailing 0 are significant. - (d) 6.320 J: 6, 3, 2, and the trailing 0 are significant. - (e) 6.032 N m鈦宦: 6, 0, 3, and 2 are significant. - (f) 0.0006032 m虏: 6, 0, 3, and 2 are significant.
05

Count Significant Figures

Count the number of significant figures in each example: - (a) 1 significant figure - (b) 3 significant figures - (c) 4 significant figures - (d) 4 significant figures - (e) 4 significant figures - (f) 4 significant figures

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Non-Zero Digits
Non-zero digits are the numbers from 1 through 9. These digits are always considered significant because they give essential information about the measure and the precision of a number. For instance, in the number 2.64, all three digits (2, 6, and 4) are non-zero, and each contributes to the quantity's magnitude and accuracy. Non-zero digits are straightforward; whenever you see them, count them as significant. In equations or scientific notation, like the number 2.64 x 10虏鈦 kg, the digits before the multiplication sign are always non-zero and significant. This allows scientists and students alike to measure with precision and consistency, critical for ensuring accuracy in scientific calculations.
Leading Zeroes
Leading zeroes are the zeroes that come before the first non-zero digit in a number. In measurements, these zeroes are not counted as significant figures because they do not affect the accuracy of the number. They are placeholders that help to locate the decimal point correctly. For instance, in 0.007 m虏, the zeroes before the 7 do not add any precision and are thus ignored when counting significant figures, leaving only the 7 itself as significant. Remembering this simple rule helps in avoiding errors when dealing with very small numbers.
Trailing Zeroes
Trailing zeroes are zeroes at the end of a number. Recognizing if these zeroes are significant or not depends on the presence of a decimal point. If a decimal point is visible, then the trailing zeroes are considered significant because they indicate precision in measurement. For instance, in the number 6.320 J, the trailing zero is significant as it follows a decimal point, indicating precise measurement up to the thousandths place. However, if you had a number like 3200 without a decimal point, the zeroes may not be counted as significant unless otherwise indicated, such as in scientific notation.
Captive Zeroes
Captive or "sandwich" zeroes are zeros placed between non-zero digits. These zeroes are always considered significant because they are an integral part of the precision of a number. They reveal more about the scale and accuracy of the measurement. For example, in the number 6.032 N m鈦宦, the zero between 6 and 3 is counted as a significant figure. This zero is important as it impacts the precise interpretation of the number intended by the measurement. Always count captive zeroes; they play a crucial role in expressing measurement accuracy.

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