Calculation of Significant Digits for Addition, Subtraction, Multiplication, and Division

Significant digits (or significant figures) are a way of expressing the precision of a number. They are important in scientific calculations because they help communicate how accurate a measurement or calculation is. When performing operations with numbers that have different degrees of precision, it is important to know how to correctly calculate the significant digits of the result.

Rules for Significant Digits in Operations

Different rules apply to the calculation of significant digits when performing addition, subtraction, multiplication, and division. Below is a table summarizing these rules:

Operation	Rule for Significant Digits
Addition and Subtraction	The result should have the same number of decimal places as the number with the fewest decimal places.
Multiplication and Division	The result should have the same number of significant digits as the number with the fewest significant digits.

Addition and Subtraction

When adding or subtracting, the result should be rounded to the least precise decimal place of any number in the operation.

Example for Addition:

Let's add 12.345 and 0.67.

  12.345
+  0.67
-------
  13.015 (unrounded)

The number 0.67 has two decimal places, and 12.345 has three. The result should be rounded to two decimal places, which is the precision of the least precise number:

  13.015 → 13.02

Example for Subtraction:

Now, let's subtract 100.0 from 123.456.

  123.456
- 100.0
--------
   23.456 (unrounded)

The number 100.0 has one decimal place, so the result should be rounded to one decimal place:

  23.456 → 23.5

Multiplication and Division

For multiplication and division, the number of significant digits in the result should match the number in the operation with the fewest significant digits.

Example for Multiplication:

Multiply 4.56 (three significant digits) by 1.2 (two significant digits).

  4.56
× 1.2
------
  5.472 (unrounded)

The result must be rounded to two significant digits:

  5.472 → 5.5

Example for Division:

Divide 123 by 1.2345.

  123 ÷ 1.2345 = 99.635 (unrounded)

Since 123 has three significant digits and 1.2345 has five, the result should have three significant digits:

  99.635 → 99.6

Additional Points to Consider

• Zeros can be significant or non-significant depending on their position in the number. Leading zeros are not significant, while trailing zeros in a decimal number are significant.
• When a number ends in zeros but has no decimal point, the zeros may or may not be significant based on the context or if a bar is placed over the last significant zero.
• Exact numbers, such as those that come from definitions or counts (e.g., 100 centimeters in a meter), have an infinite number of significant digits and do not limit the precision in calculations.

Conclusion

Understanding the rules for significant digits in addition, subtraction, multiplication, and division is crucial for maintaining the correct level of precision in scientific calculations. Always consider the precision of the numbers you are working with and apply the appropriate rules to ensure that your results are as accurate as possible.