Determine the number of significant digits in each number, and write the specific significant digits.
405000
0.0098
39.999999
13.00
80,000,089
55,430.00
0.000033
620.03080
The Correct Answer and Explanation is:
Sure! Let’s analyze each number to determine the number of significant digits and list the specific significant digits.
1. 405000
- Significant digits: 3 (4, 0, 5)
- Explanation: The trailing zeros in this number are ambiguous without additional formatting (like a decimal point). Typically, if there is no decimal point, trailing zeros are not considered significant. So, only 4, 0, and 5 are significant here.
2. 0.0098
- Significant digits: 2 (9, 8)
- Explanation: Leading zeros are never significant; they only indicate the position of the decimal point. So only the digits 9 and 8 are significant.
3. 39.999999
- Significant digits: 8 (3, 9, 9, 9, 9, 9, 9, 9)
- Explanation: All digits in a decimal number are significant, including trailing zeros. Here, 8 digits are significant.
4. 13.00
- Significant digits: 4 (1, 3, 0, 0)
- Explanation: The zeros after the decimal point are significant because they indicate precision.
5. 80,000,089
- Significant digits: 8 (8, 0, 0, 0, 0, 0, 8, 9)
- Explanation: All non-zero digits and any zeros between non-zero digits are significant. Here, the zeros between 8 and 8 are significant as they occur between significant digits. Trailing zeros would be significant only if there was a decimal point, but here zeros are between 8’s.
6. 55,430.00
- Significant digits: 7 (5, 5, 4, 3, 0, 0, 0)
- Explanation: The zeros after the decimal point are significant, indicating precision. All digits are significant.
7. 0.000033
- Significant digits: 2 (3, 3)
- Explanation: Leading zeros are not significant; only the digits 3 and 3 are significant.
8. 620.03080
- Significant digits: 8 (6, 2, 0, 0, 3, 0, 8, 0)
- Explanation: All digits including zeros between non-zero digits and trailing zeros after a decimal point are significant.
Summary Table:
| Number | Significant Digits | Digits |
|---|---|---|
| 405000 | 3 | 4, 0, 5 |
| 0.0098 | 2 | 9, 8 |
| 39.999999 | 8 | 3, 9, 9, 9, 9, 9, 9, 9 |
| 13.00 | 4 | 1, 3, 0, 0 |
| 80,000,089 | 8 | 8, 0, 0, 0, 0, 0, 8, 9 |
| 55,430.00 | 7 | 5, 5, 4, 3, 0, 0, 0 |
| 0.000033 | 2 | 3, 3 |
| 620.03080 | 8 | 6, 2, 0, 0, 3, 0, 8, 0 |
Explanation (300+ words):
The concept of significant digits (or significant figures) is fundamental in science and mathematics to express the precision of a measured or calculated number. Significant digits include all the digits in a number that contribute to its accuracy, including all non-zero digits, zeros between significant digits, and trailing zeros in decimal numbers.
- Non-zero digits are always significant. For example, digits like 4, 5, 9, and 3 in numbers are always counted.
- Leading zeros, which are zeros before any non-zero digit, serve only as placeholders to locate the decimal point and are not significant. For example, in 0.0098, the zeros before 9 are not significant, so only two significant digits exist (9 and 8).
- Captive zeros (zeros between non-zero digits) are significant. For example, in 80,000,089, all zeros between the first 8 and the last digits are significant because they are sandwiched between significant digits.
- Trailing zeros have special rules:
- If trailing zeros are after a decimal point, they are significant because they show precision. For example, 13.00 has four significant digits, including the zeros.
- If trailing zeros are before a decimal point (e.g., 405000 without a decimal), those zeros are often not significant unless otherwise specified by scientific notation or a decimal point.
- Decimal points make trailing zeros significant. For example, 55,430.00 contains trailing zeros after the decimal point, so they are significant, making a total of 7 significant digits.
Understanding significant digits ensures accuracy in scientific calculations and conveys the precision of measurements. It helps avoid overstating precision and allows consistent communication of data reliability.