Question

Given the numbers \(786,0.786\) and \(0.0786\) the number of significant figures for the three numbers is (1) 3,4 and 5 , respectively (2) 3,3 and 3 , respectively (3) 3,3 and 4, respectively (4) 3,4 and 4, respectively

Short answer

(2) 3, 3, and 3, respectively

Step-by-step solution

Identify Significant Figures in 786

The number 786 is a whole number with no leading zeros and no decimal point. Every digit is significant. Therefore, the number of significant figures in 786 is 3.

Identify Significant Figures in 0.786

The number 0.786 is a decimal. Leading zeros are not significant; only the non-zero digits are significant. Hence, the number of significant figures in 0.786 is 3.

Identify Significant Figures in 0.0786

The number 0.0786 is also a decimal. Leading zeros are not significant; only the non-zero digits are significant. Hence, the number of significant figures in 0.0786 is 3.

Match the Results to the Given Options

From the above steps, the number of significant figures are 3 for 786, 3 for 0.786, and 3 for 0.0786. This matches with option (2) which states 3, 3, and 3, respectively.

Key concepts

significant digits rules

Understanding significant digits is key for accuracy in scientific measurements. Significant digits, also known as significant figures, represent the meaningful numbers in a given value. These rules help identify which figures in a number are significant, ensuring precision in calculations and results.

There are a few simple rules to determine the number of significant digits:

• All non-zero digits are always significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant; they merely indicate the position of the decimal point.
• Trailing zeros in a number containing a decimal point are significant.

By following these rules, you can accurately determine the number of significant digits in any number, which is crucial for precise scientific calculations.

leading zeros

Leading zeros are the zeros that come before the first non-zero digit in a number. These zeros are not considered significant because they only serve to locate the decimal point. For example, in the number 0.0786, the leading zeros (the zeros before the 7) do not add any precision to the measurement.

To illustrate:

• In 0.786, the zero before 7 is a leading zero and is not significant. This number has three significant figures: 7, 8, and 6.
• In 0.0786, the zeros before 7 are leading zeros and are not significant. This number also has three significant figures: 7, 8, and 6.

Recognizing leading zeros and knowing they are not significant helps avoid overestimating the precision of a measurement.

non-zero digits

Non-zero digits play a crucial role in determining the significant figures of a number. Every non-zero digit in a number is always significant, as they indicate the precision of the measured value. Understanding this rule helps in counting the significant figures accurately.

Consider some examples:

• In the number 786, each digit (7, 8, and 6) is non-zero and, therefore, significant. Hence, it has 3 significant figures.
• For 0.786, the non-zero digits are 7, 8, and 6, making it 3 significant figures. The leading zero does not count.
• Similarly, in 0.0786, the digits 7, 8, and 6 are non-zero, and so it also has 3 significant figures, despite the leading zeros.

Knowing that non-zero digits are always significant helps ensure accurate representation and comparison of measurements.