What will be 0.008594, correct to three significant figures, be equal to?
0.00859
Understanding Significant Figures and Rounding
The question asks us to round the number 0.008594 to three significant figures. Let's first understand what significant figures are, especially in a decimal number less than 1.
In a decimal number less than 1, leading zeros (zeros before the first non-zero digit) are not considered significant. The significant figures start from the first non-zero digit.
For the number 0.008594:
- The zeros before '8' (0.00) are leading zeros and are not significant.
- The first non-zero digit is 8. This is the first significant figure.
- The digits following the first significant figure are also significant.
So, the significant figures in 0.008594 are 8, 5, 9, and 4. There are a total of four significant figures.
Rounding to Three Significant Figures
To round 0.008594 to three significant figures, we need to identify the third significant figure and look at the digit immediately following it (the fourth significant figure).
The significant figures are in order:
- First significant figure: 8
- Second significant figure: 5
- Third significant figure: 9
- Fourth significant figure: 4
We want to round to the third significant figure, which is 9. We look at the fourth significant figure, which is 4.
The rule for rounding is:
- If the digit after the required place (in this case, the fourth significant figure) is 5 or greater, we round up the digit at the required place.
- If the digit after the required place is less than 5, we keep the digit at the required place as it is.
In our case, the fourth significant figure is 4, which is less than 5. Therefore, the third significant figure (9) remains unchanged.
We keep the first three significant figures (8, 5, 9) and replace any subsequent digits with zeros if they are before the decimal point, or simply drop them if they are after the decimal point, ensuring the place value of the significant figures is maintained.
So, 0.008594 rounded to three significant figures is 0.00859.
Comparing with Options
Let's examine the given options:
- Option 1: 0.00859
The significant figures are 8, 5, and 9. There are exactly three significant figures. This matches our calculated result. - Option 2: 0.009
The only significant figure is 9. This is one significant figure, not three. - Option 3: 0.00860
The significant figures are 8, 6, and 0 (the trailing zero after the decimal point is significant). There are three significant figures. This would be the result if the fourth significant figure was 5 or greater, causing the 9 to round up. - Option 4: 0.008
The only significant figure is 8. This is one significant figure, not three.
Based on the rounding rules and the number of significant figures, 0.008594 corrected to three significant figures is 0.00859.
The final answer is 0.00859.
| Number | Significant Figures | Number of Significant Figures | Rounded to 3 S.F. |
|---|---|---|---|
| 0.008594 | 8, 5, 9, 4 | 4 | 0.00859 |
| 0.00859 | 8, 5, 9 | 3 | - |
| 0.009 | 9 | 1 | - |
| 0.00860 | 8, 6, 0 | 3 | - |
| 0.008 | 8 | 1 | - |
Revision Table: Key Concepts
| Concept | Description | Example (using 0.008594) |
|---|---|---|
| Significant Figures | Digits in a number that carry meaningful contributions to its measurement resolution. | 8, 5, 9, 4 are significant. Leading zeros (0.00) are not. |
| Rounding | Reducing the number of significant figures while keeping the value close to the original. | Changing 0.008594 to 0.00859. |
| Rounding Rule (< 5) | If the digit after the desired place is less than 5, the digit at the desired place remains unchanged. | The 4th significant figure is 4 (< 5), so the 3rd significant figure (9) stays 9. |
Additional Information: Significant Figures Rules
Here are some general rules for identifying significant figures:
- Non-zero digits: All non-zero digits are significant. (e.g., 123.45 has 5 significant figures).
- Zeros between non-zero digits: Zeros located between non-zero digits are significant. (e.g., 1002 has 4 significant figures).
- Leading zeros: Zeros to the left of the first non-zero digit are not significant. They only indicate the position of the decimal point. (e.g., 0.00123 has 3 significant figures).
- Trailing zeros (with decimal point): Zeros to the right of a non-zero digit and at the end of a number are significant if the number contains a decimal point. (e.g., 12.300 has 5 significant figures; 120.0 has 4 significant figures).
- Trailing zeros (without decimal point): Zeros at the end of a number that does not have a decimal point may or may not be significant. This is often ambiguous. (e.g., 1200 could have 2, 3, or 4 significant figures. Scientific notation helps clarify this, e.g., <code>1.2 x 10<sup>3</sup></code> (2 s.f.), <code>1.20 x 10<sup>3</sup></code> (3 s.f.), <code>1.200 x 10<sup>3</sup></code> (4 s.f.)).
Understanding these rules is crucial for correctly determining the precision of a number and performing calculations involving significant figures.
Important Questions from Decimals
The value of \(\frac{1}{4} + \frac{{[{{(20.35)}^2} - {{(8.35)}^2}] \times 0.0175}}{{{{(1.05)}^2} + (1.05)(27.65)}}\) is:
The value of \(0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7 \) is:
The value of \(\frac{48.3\times[(4.95)^2+4.95\times13.25]}{[(12.55)^2-(5.65)^2]\times19.8} \) is:
Find the value of (1.6) 3 - (0.9) 3 - (0.7) 3.
What is the value of x, if \(5\left( {1 - \frac{x}{5}} \right) - (5 - x) - \frac{1}{{200}}{\rm{of (20 - x) = 0}}{\rm{.08}}\) ?
