Significant figures are the specific digits in a number that carry meaningful information about its precision. They tell us exactly how precise a measurement is by including all of the certain digits plus one final estimated digit dictated by the limits of the measuring tool.
Example
Imagine you measure a piece of wood with a regular ruler and get 14.3 centimeters. This number has 3 significant figures because you are sure about the 14 and the 3.
If you use a highly precise laser tool and get 14.325 centimeters, this number has 5 significant figures. It shows a much higher level of detail and accuracy.
Significant figures are essential in everyday life to maintain precision and prevent false precision in calculations.
Significant figures are used in everyday life when:
To determine which digits in a number are significant, follow the fundamental rules governing non-zero digits, trapped zeros, placeholders, and exact values.
All numbers from 1 to 9 always count as significant. Any zero that is trapped directly between non-zero digits also counts as significant.
Examples
Identify the significant figures in 437.
Solution: All three digits are non-zero numbers. They all count.
Result = 3 significant figures (4, 3, 7)
Identify the significant figures in 502.
Solution: The zero is trapped between 5 and 2. It counts as significant.
Result = 3 significant figures (5, 0, 2)
Zeros placed at the very front of a number never count as significant figures. They are only placeholders that show the scale of a decimal number.
Examples
Identify the significant figures in 0.008.
Solution: The zeros at the front do not count. Only the digit 8 counts.
Result = 1 significant figure (8)
Identify the significant figures in 0.0406.
Solution: The front zeros do not count. The zero between 4 and 6 counts because it is trapped.
Result = 3 significant figures (4, 0, 6)
Zeros at the very end of a whole number are ambiguous unless clarified by a visible decimal point or written in scientific notation. If a decimal point is visible, trailing zeros count because they signal measured precision.
Examples
Identify the significant figures in 6,500.
Solution: There is no visible decimal point. The two zeros at the end are assumed to be placeholders. Only 6 and 5 are significant.
Result = 2 significant figures (6, 5)
Identify the significant figures in 6,500. (with an explicit decimal point).
Solution: The presence of the terminal decimal point indicates that the zeros were precisely measured.
Result = 4 significant figures (6, 5, 0, 0)
Identify the significant figures in 3.20.
Solution: There is a visible decimal point. The trailing zero counts because it demonstrates precise measurement down to the hundredths place.
Result = 3 significant figures (3, 2, 0)
Numbers obtained from direct counting (not measured with an instrument) or from defined conversion factors are exact numbers. They possess an infinite number of significant figures and never limit the precision of a calculation.
Examples
When performing calculations, your final answer cannot show more precision than the values you started with. You must round your answers according to specific operational rules.
When adding or subtracting, track the number of decimal places (digits after the decimal point). Your final answer must match the exact same number of decimal places as the starting value with the fewest decimal places.
Examples
Calculate: 12.11 + 4.3
Solution:
12.11 has two decimal places.
4.3 has only one decimal place.
The raw mathematical sum is 16.41. We round this to one decimal place to match 4.3.
Result = 16.4
Calculate: 25.678 − 2.20
Solution:
25.678 has three decimal places.
2.20 has two decimal places.
The raw mathematical difference is 23.478. We round this to two decimal places to match 2.20.
Result = 23.48
When multiplying or dividing, look at the total number of significant figures in each starting number. Your final answer must contain the exact same number of total significant figures as the starting value with the fewest overall significant figures.
Examples
Calculate: 4.5 × 2.13
Solution:
4.5 has 2 significant figures.
2.13 has 3 significant figures.
The raw mathematical product is 9.585. We round this to 2 significant figures to match 4.5.
Result = 9.6
Calculate: 15.0 ÷ 3.0
Solution:
15.0 has 3 significant figures.
3.0 has 2 significant figures.
The raw mathematical quotient is 5. We must express it with 2 significant figures to match 3.0.
Result = 5.0
To achieve complete mastery over significant figures, you must know how to handle numbers that have no uncertainty, multi-step math problems, and numbers with ambiguous zeros.
When a problem combines multiple types of operations, you must follow the standard order of operations (PEMDAS). You must track the correct number of significant figures at each intermediate step, but you should only round the very final answer to avoid rounding errors.
Example Problem:
Calculate: (1.25 + 4.2) × 3.125
Solution:
Step 1: Perform the operation inside the parentheses first (Addition).
1.25 + 4.2 = 5.45
Analysis: 1.25 has two decimal places, and 4.2 has one decimal place. The result must be limited to one decimal place (the tenths position). This means the value is significant up to the 4 (5.45). We keep the trailing 5 for now to prevent rounding errors but remember that this number effectively has only 2 significant figures.
Step 2: Perform the multiplication using the intermediate value.
5.45 × 3.125 = 17.03125
Analysis: Apply the multiplication rule. Our first number (5.45) has 2 significant figures. Our second number (3.125) has 4 significant figures. The final answer must be limited to the lowest total count, which is 2 significant figures.
Step 3: Round the final value.
Round 17.03125 to 2 significant figures. The digit to drop is 0, so the 7 remains unchanged.
Final Answer = 17
A number like 6,500 is mathematically ambiguous because it is unclear if the zeros were estimated or precisely measured. Writing numbers in Scientific Notation clears up this confusion immediately. In scientific notation, every single digit written in the base number is 100% significant.
How Scientific Notation Changes Count:
Example 1
State the total number of significant figures in the following values:
A) 0.00504
B) 32,000
C) 150.0
solution :
For A (0.00504): The front zeros do not count. The zero inside counts because it is trapped between 5 and 4. This gives a total of 3 significant figures.
Result A = 3 significant figures
For B (32,000): There is no visible decimal point, so the trailing zeros at the end do not count. Only 3 and 2 are significant.
Result B = 2 significant figures
For C (150.0): There is a visible decimal point, which makes all trailing zeros significant. This gives a total of 4 significant figures.
Result C = 4 significant figures
Example 2
Round the number 42,783 to:
A) 2 significant figures
B) 3 significant figures
solution :
For A: The first two significant digits are 4 and 2. The next digit is 7. Since 7 is 5 or more, round the 2 up to 3. Change the remaining positions into placeholder zeros.
Result A = 43,000
For B: The first three significant digits are 4, 2, and 7. The next digit is 8. Since 8 is 5 or more, round the 7 up to 8. Change the remaining positions into placeholder zeros.
Result B = 42,800
Example 3
Calculate: 143.2 − 12.15. Write your answer using the correct rules for significant figures.
Solution:
Step 1: Perform the normal subtraction operation.
143.2 − 12.15 = 131.05
Step 2: Apply the addition and subtraction rule. Check the decimal places of the starting numbers. 143.2 has only one decimal place, while 12.15 has two decimal places. The answer must be rounded to match the fewest decimal places (one decimal place).
Step 3: Round 131.05 to one decimal place. The digit to drop is 5. Since the digit is 5 or greater, round the preceding digit (0) up by 1.
Final Answer = 131.1
Example 4
Calculate: 3.24 × 2.0. Write your answer using the correct rules for significant figures.
solution :
Step 1: Perform the normal multiplication operation.
3.24 × 2.0 = 6.48
Step 2: Apply the multiplication and division rule. Check the total significant figures of the starting numbers. 3.24 has 3 significant figures. 2.0 has 2 significant figures because the trailing zero after the decimal point counts. The answer must match the fewest total significant figures (2 significant figures).
Step 3: Round 6.48 to 2 total significant figures. The digit to drop is 8, so round the 4 up to 5.
Final Answer = 6.5
Example 5 (Word Problem)
A rectangular room measures 12.4 meters long and 5.0 meters wide. Calculate the area of the room using the correct rules for significant figures.
Solution:
Step 1: Identify that the area of a rectangle is calculated by multiplying length by width.
Area = 12.4 × 5.0
Step 2: Perform the normal multiplication calculation.
12.4 × 5.0 = 62
Step 3: Apply the significant figures rule for multiplication. 12.4 has 3 significant figures, and 5.0 has 2 significant figures (the trailing zero counts because of the decimal point). The final answer must match the lowest count, which is exactly 2 significant figures.
The calculated result 62 already contains exactly two significant figures, so no additional rounding is required.
Final Answer = 62 square meters
Attempt the following Questions:
(1.) State the total number of significant figures in the number 0.04080.
(2.) Round the number 0.003526 to 2 significant figures.
(3.) Calculate the answer using correct significant figures: 24.56 + 3.2
(4.) Calculate the answer using correct significant figures: 15.4 × 2.0
(5.) Divide 8.42 by 2.0 and write the answer using correct significant figures.
Take Note:
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