Significant figures, also known as significant digits, are a set of digits in a numerical value that conveys the precision or certainty of a measurement. They are important in scientific and mathematical contexts to represent the reliability of a measurement or calculation. The rules for determining significant figures are as follows:

1. Non-zero digits: All non-zero digits are always considered significant. For example, in the number 123.45, there are five significant figures (1, 2, 3, 4, and 5).

2. Leading zeros: Leading zeros (zeros to the left of the first non-zero digit) are not considered significant. For example, in the number 0.00456, only the digits 4, 5, and 6 are significant, resulting in three significant figures.

3. Captive zeros: Zeros between significant digits are considered significant. For example, in the number 10.07, there are four significant figures (1, 0, 0, and 7).

4. Trailing zeros: Trailing zeros in a decimal number are considered significant. For example, in the number 50.0, there are three significant figures (5, 0, and 0). However, in a whole number, trailing zeros may or may not be significant. For example, the number 500 may have one, two, or three significant figures depending on the context and how the number was obtained.

5. Exact numbers: Exact numbers, such as counting numbers (e.g., 1, 2, 3) and defined constants (e.g., the speed of light, Avogadro’s number), are considered to have an infinite number of significant figures.

Here are some examples of numbers with their respective significant figures:

1. 123.45 – This number has five significant figures (1, 2, 3, 4, and 5).

2. 0.00456 – This number has three significant figures (4, 5, and 6). Leading zeros are not counted.

3. 10.07 – This number has four significant figures (1, 0, 0, and 7). Captive zeros between significant figures are counted.

4. 50.0 – This number has three significant figures (5, 0, and 0). Trailing zeros in a decimal number are counted.

5. 500 – The number of significant figures in 500 can vary depending on context:

– If 500 is a count of items (e.g., 500 apples), it is considered to have one significant figure.

– If 500 is a measurement with precision (e.g., 500.0 mL), it has four significant figures.

6. 1000 – Like 500, the number of significant figures in 1000 can vary based on context:

– If 1000 is a count of items (e.g., 1000 people), it is considered to have one significant figure.

– If 1000 is a measurement with precision (e.g., 1000.0 meters), it has five significant figures.

7. 7.00 – This number has three significant figures (7, 0, and 0). Trailing zeros in a decimal number are counted when precision is implied.

The rules for handling significant figures vary slightly depending on whether you are performing addition and subtraction or multiplication and division. Here’s how you should handle significant figures in each case:

**Addition and Subtraction:** When adding or subtracting numbers, you need to consider the number of decimal places, rather than significant figures. The rule is to round the result to the least number of decimal places present in the original numbers being added or subtracted. Here’s how to do it:

**Count the decimal places:**Look at the numbers you’re adding or subtracting and identify the one with the fewest decimal places.**Round the result:**After performing the addition or subtraction, round the result to match the least number of decimal places you identified in step 1.

Example: Suppose you want to add 3.45 and 1.2.

- 3.45 has two decimal places.
- 1.2 has one decimal place.

So, when you add them, you should round the result to one decimal place: 3.45 + 1.2 = 4.65 rounded to 4.7.

**Multiplication and Division:** When multiplying or dividing numbers, you should pay attention to the number of significant figures in each number. The rule is to round the result to match the fewest number of significant figures in the original numbers. Here’s how to do it:

**Count the significant figures:**Determine the number of significant figures in each of the numbers you are multiplying or dividing.**Round the result:**After performing the multiplication or division, round the result to match the fewest number of significant figures from step 1.

Example: Suppose you want to multiply 2.5 and 3.14.

- 2.5 has two significant figures.
- 3.14 has three significant figures.

So, when you multiply them, you should round the result to two significant figures: 2.5 x 3.14 = 7.85 rounded to 7.9.

In summary, when adding or subtracting, consider decimal places and round to the least number of decimal places. When multiplying or dividing, consider significant figures and round to the least number of significant figures. These rules help ensure that your calculated results reflect the precision of the original numbers used in the calculation.

Exact numbers are quantities that are known with complete certainty and are not the result of a measurement. They have an infinite number of significant figures because they are not subject to the uncertainties associated with measurement instruments. Exact numbers are typically constants, integers, or numbers that can be determined precisely by counting or definition. Here are some examples of exact numbers:

**Counting Numbers:**Any whole number that results from counting objects is an exact number. For example, if you count the number of books on a shelf and find there are 10 books, the number 10 is an exact number with an infinite number of significant figures.**Defined Constants:**Constants that have a precise value defined in physics or mathematics are considered exact numbers. For example:- The speed of light in a vacuum, denoted as “c,” is exactly 299,792,458 meters per second.
- Avogadro’s number, denoted as “Nₐ,” is exactly 6.02214076 × 10^23 particles per mole.

**Exact Ratios:**Ratios that can be determined exactly, such as the ratio of the circumference of a circle to its diameter (π), are considered exact numbers. The value of π is approximately 3.14159265359, but it is an exact number when used in its exact form.**Integer Values:**Any whole number, positive or negative, is considered an exact number. For example, -7, 0, and 42 are all exact numbers.

In calculations involving exact numbers, you don’t need to worry about significant figures or rounding because they are known with certainty. However, when you perform calculations that combine exact numbers with measured quantities (numbers with significant figures), you should apply the rules for significant figures as mentioned in a previous response to ensure that the precision of your final result is correctly represented.

Significant figures (or significant digits) are crucial in scientific and engineering contexts because they convey the precision and accuracy of measured or calculated values. Here are the significant aspects of significant figures, along with examples to illustrate their significance:

**Precision and Accuracy**: Significant figures help communicate the precision of a measurement or calculation. They indicate how well a value is known or measured. A higher number of significant figures implies greater precision.- Example 1: If you measure the length of an object as 12.34 cm, the four significant figures suggest a high degree of precision, indicating that you are confident in the measurement’s accuracy.
- Example 2: Measuring the same length as 12 cm implies less precision because there is only one significant figure. This could mean you measured with a less precise tool or rounded the measurement.

**Error Estimation**: The number of significant figures also provides a way to estimate the uncertainty or error in a measurement. The last significant figure is considered uncertain, and its range is typically expressed with a ± symbol.- Example 3: If you measure the temperature as 25.6°C, it suggests that the true temperature could be anywhere within the range of 25.5°C to 25.7°C. The ±0.1°C represents the measurement’s uncertainty.

**Consistency in Calculations**: When performing mathematical operations (addition, subtraction, multiplication, division) involving measurements, it’s essential to round the result to the appropriate number of significant figures to maintain consistency and prevent overestimation of precision or underestimation of error.- Example 4: If you add 12.34 g and 6.2 g, the sum should be reported as 18.54 g (rounded to two significant figures) to reflect the precision of the original measurements.
- Example 5: When dividing 3.45 m by 1.2 s, the result should be 2.9 m/s (rounded to two significant figures) to match the least number of significant figures in the original values.

**Scientific Notation**: Significant figures are often used with scientific notation to express very large or very small numbers accurately and concisely.- Example 6: The speed of light in a vacuum is approximately 299,792,458 m/s. In scientific notation, this is written as 2.9979 x 10^8 m/s, indicating four significant figures.

**Avoiding Misrepresentation**: Using the correct number of significant figures prevents misrepresentation of data or results. Reporting too many significant figures can imply unwarranted precision, while too few can hide important information.- Example 7: If you report the result of a measurement as 12.3456789 cm, it suggests a precision that may not be achievable with the measuring instrument, potentially misleading others.

In summary, significant figures are essential in science and engineering for quantifying precision, estimating error, ensuring consistency in calculations, expressing very large or small numbers, and preventing the misrepresentation of data. They provide a standardized way to communicate the reliability of measurements and calculations within these fields.