Calculation of Addition, Subtraction, Multiplication, and Division with Significant Figures (Simplified Rules)

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• Addition and subtraction (addition, division) including significant figures
  • Simplified rules

    
    

    （１）Adding and subtracting calculations

    (A) If the x of 10^X multiplied by each number is different, first match the 10^X. At that time, the number of significant digits of each number should not change.

    (B) Find the last significant digit of each number.

    (C) Find the highest significant digit in (B). (10's, 1's, first decimal place,,,,etc.)

    (D) Round (round off) the significant digits of the calculation result to the place found in (C).

     

    【Examples of how to determine significant figures by addition and subtraction calculations ① through ③】

     

    ①                          5.302  –       1.2        = (4.102)

    (A) x(=0) in ×10^X is the same

    (B) The last significant digit　：Third decimal place　　First decimal place

    (C) Last significant digit of largest significant digit：First decimal place

    (D) Round the significant digits of the calculation result to the first decimal place.

                      (4.102) = 4.1       (2 significant digits)

     

    ②                   530.2      – 1.23×10²

    (A) Match the x in ×10X because they are different. (Aligned to x10⁰)

                             530.2    –    123    = (407.2)

    (B) The last significant digit　：First decimal place　　First place

    (C) Last significant digit of largest significant digit：First place

    (D) Round the significant digits of the calculation result to the first place.

                      (407.2) = 470       (3 significant digits)

      

    ③                   5.3×10^-6      – 1.2856×10^-4 

    (A) Match the x in ×10X because they are different. (Aligned to x10^-4)

                      　0.053×10^-4  – 1.2856×10^-4  = (-1.2326×10^-4)

    (B) The last significant digit　：Third decimal place　Fourth decimal place

    (C) Last significant digit of largest significant digit：Third decimal place

    (D) Round the significant digits of the calculation result to the third decimal place.

                         (-1.2326×10^-4) = -1.233×10^-4       （4 significant digits）
    
  • 2)　Multiplication and division

    
    

    (A) Find the number of significant digits in each number

    (B) Find the least number of significant digits in (A)

    (C) Match the number of significant digits in the calculation result with the one found in (B).

     

    【Examples of how to determine significant figures by multiplication and division calculations ① - ②】

     

    ①                   0.123 ×　12.34　　÷1000 = (0.00151782)

    (A) Number of significant digits：（3 digits）　（4 digits）　（4 digits）

    (B) Least number of digits：3 digits

    (C)  (0.00151782) = 0.00152（3 digits） = 1.52×10^-3     （3 digits）

     

     

    ②                   0.123 ×　12.34　　÷1.0×10³ = (0.00151782)

    (A) Number of significant digits：（3 digits）　（4 digits）　（2 digits）

    (B) Least number of digits：2 digits

    (C)  (0.00151782) = 0.0015（2 digits） = 1.5×10^-3  （2 digits）

     

    
    

    (3)Calculation of addition, multiplication and division

    　First, the addition and subtraction are calculated to determine the number of significant digits of the number. Then calculate the multiplication and division.

    
    

  • Supplement: Rounding of numbers (round off and bankers' rounding)
    

    　In the example above, the numbers were rounded to the last place where the significant digits are aligned. With rounding, 1 to 4 are rounded down (rounding range 4) and 5 to 9 are rounded up (rounding range 5), so the probability of rounding up is higher. When processing large amounts of data, it is sometimes necessary to compensate for this difference in probability. In such cases, there is the concept of bankers' rounding. This is not discussed in this text, but if you are interested, please look it up.
    
    

    （Note that computer spreadsheet software automatically rounds numbers to the nearest significant digit when making them appear smaller.）