Chapter 2: Problem 5
How many significant figures do these numbers have? a) 765,890 b) 765,890.0 c) \(1.2000 \times 10^{5}\) d) 0.0005060
Short Answer
Expert verified
a) 5, b) 7, c) 5, d) 4
Step by step solution
01
Identify Significant Figures in Whole Numbers
For number a) 765,890: A whole number with no decimal point. Significant figures include all non-zero digits which are 765 and 89. The trailing zero in this context does not count as a significant figure. So, the number has 5 significant figures.
02
Include Trailing Zeros After Decimal Points
For number b) 765,890.0: The presence of a decimal point indicates that trailing zeros are significant. Therefore, every digit in 765,890.0 is a significant figure. The number has 7 significant figures.
03
Consider All Non-zero Digits, and Leading or Trailing Zeros
For number c) \(1.2000 \times 10^{5}\): All non-zero digits are significant. The trailing zeros in the decimal are also significant because they come after a decimal point. There are 5 significant figures in 1.2000.
04
Count Significant Zeros in Decimal Numbers
For number d) 0.0005060: Leading zeros are not significant, but any zeros between non-zero digits or following the decimal are significant. Thus, 5, 0, 6, and the final 0 are significant figures. This number has 4 significant figures.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decimal Numbers Significance
When dealing with decimal numbers, understanding which digits are significant is crucial.
All non-zero digits in a number are always significant because they represent actual measurements.
In a decimal number, zeros can often be tricky to evaluate for significance.
If zeros appear after the decimal point and follow a non-zero number, those zeros are significant.
For instance, in the number 23.400, the zeros are significant, providing a level of precision in the measurement to the tenths, hundredths, and beyond.
However, if you have zeros in front of a decimal digit, they are not significant.
To summarize, when analyzing decimal numbers for significant figures:
All non-zero digits in a number are always significant because they represent actual measurements.
In a decimal number, zeros can often be tricky to evaluate for significance.
If zeros appear after the decimal point and follow a non-zero number, those zeros are significant.
For instance, in the number 23.400, the zeros are significant, providing a level of precision in the measurement to the tenths, hundredths, and beyond.
However, if you have zeros in front of a decimal digit, they are not significant.
To summarize, when analyzing decimal numbers for significant figures:
- All non-zero digits are significant.
- Zeros following a decimal point and after a non-zero digit are significant.
Trailing Zeros Significance
Trailing zeros can sometimes be a source of confusion when determining significant figures.
These zeros occur at the end of a number and can have different significances depending on their position relative to a decimal point.
If a number contains trailing zeros and there is a decimal point present, all trailing zeros are significant.
This is because the decimal indicates a precise measurement to that specific place value.
For instance, the number 450.00 has five significant figures because the zeros after the decimal point add to the precision.
When a number does not have a decimal point, trailing zeros do not count as significant.
For example, in the whole number 765,890, without a decimal, the last zero is not significant as it doesn't specify precision beyond the non-zero digits.
In summary, remember:
These zeros occur at the end of a number and can have different significances depending on their position relative to a decimal point.
If a number contains trailing zeros and there is a decimal point present, all trailing zeros are significant.
This is because the decimal indicates a precise measurement to that specific place value.
For instance, the number 450.00 has five significant figures because the zeros after the decimal point add to the precision.
When a number does not have a decimal point, trailing zeros do not count as significant.
For example, in the whole number 765,890, without a decimal, the last zero is not significant as it doesn't specify precision beyond the non-zero digits.
In summary, remember:
- Trailing zeros are significant if they come after a decimal point.
- Without a decimal point, trailing zeros are generally not significant.
Scientific Notation Significance
Scientific notation is a method used to express very large or very small numbers in a more compact form.
Understanding significant figures in scientific notation is very straightforward.
In scientific notation, the digits found in the coefficient (the part before "× 10") are all significant.
For example, in the number \(1.2000 \times 10^{5}\), you look at the coefficient 1.2000.
Here, all five digits, including the zeros, are significant because they come after a decimal and indicate precision.
Scientific notation is particularly helpful in ensuring that all significant figures are clearly presented regardless of the magnitude of the number.
Key points when considering scientific notation:
Understanding significant figures in scientific notation is very straightforward.
In scientific notation, the digits found in the coefficient (the part before "× 10") are all significant.
For example, in the number \(1.2000 \times 10^{5}\), you look at the coefficient 1.2000.
Here, all five digits, including the zeros, are significant because they come after a decimal and indicate precision.
Scientific notation is particularly helpful in ensuring that all significant figures are clearly presented regardless of the magnitude of the number.
Key points when considering scientific notation:
- All non-zero digits in the coefficient are significant.
- Any zeros in the coefficient are significant if they clarify precision (e.g., zeros following a decimal).
Leading Zeros Insignificance
When discussing significant figures, leading zeros are a common area of misunderstanding.
Leading zeros are zeros that appear before all non-zero digits in a decimal number.
These zeros are not considered significant because they only serve as placeholders and denote the scale of the number.
For instance, in 0.0005060, the first three zeros are leading zeros and are not significant.
They do not contribute to the measurement's accuracy or precision.
Only the zeros after the non-zero digits, such as the ones between 5 and 6 and the zero at the end in this number, are significant as these indicate measured precision.
To ensure no confusion with leading zeros, remember:
Leading zeros are zeros that appear before all non-zero digits in a decimal number.
These zeros are not considered significant because they only serve as placeholders and denote the scale of the number.
For instance, in 0.0005060, the first three zeros are leading zeros and are not significant.
They do not contribute to the measurement's accuracy or precision.
Only the zeros after the non-zero digits, such as the ones between 5 and 6 and the zero at the end in this number, are significant as these indicate measured precision.
To ensure no confusion with leading zeros, remember:
- Leading zeros before any non-zero digit are not significant.
- Zeros between significant figures or after a decimal are significant.
