Precision, Significant Figures and Exponents
Precision of a measurement
Measurement  Precision  Comment 

4  1s  
4.1  0.1s  
4.01  0.01s  
4.010  0.001s  The trailing zero tells us that this was measured to the 0.001s 
41  1s  
4,127  1s  
4,010  10s  This may only have been measured to the nearest 10 
4,100  100s  This may only have been measured to the nearest 100 
401  1s  
0.1  0.1s  
0.010  0.001s  The trailing zero tells us that this was measured to the 0.001s. 
Conclusion: Trailing zeros affect precision only if they are after the decimal, otherwise, one considers the right most nonzero.
Significant figures in a measurement
Number  Significant Figures 
Comment 

4  1  
4.0  2  This was measured to the 0.1s 
4.00  3  This was measured to the 0.01s 
40  1  This may only have been measured to the nearest 10 
41  2  
4,100  2  This may only have been measured to the nearest 100 
4.100  4  This was measured to the 0.001s 
4.10  3  This was measured to the 0.01s 
4.1  2  
401  3 
Zeros within the number are a part of the measurement 
0401  3  Leading zeros before the decimal are silly 
4,010  3  This may only have been measured to the nearest 10 
040,100  3  This may only have been measured to the nearest 100. The leading zero is silly. 
0.1  1  The leading zero only tells us where to place the decimal 
0.010  2  This was measured to the 0.001s. The leading zeros only tell us where to place the decimal 
120.00  5  This was measured to the 0.01s, and so the other zeros are within the number. 
Conclusion: Trailing zeros after the number are significant, only if the last trailing zero is after the decimal. Leading zeros are never significant. Zeros within the other significant digits count.
Scientific Notation and Exponents
Number  Scientific notation  Calculator Exponent form 
Significant figures 
Comment 

100  1 x 10^{2}  1 e 2  1  
1,000  1 x 10^{3}  1 e 3  1  
10,000  1 x 10^{4}  1 e 4  1  
0.1  1 x 10^{1}  1 e 1  1  
0.01  1 x 10^{2}  1 e 2  1  
0.001  1 x 10^{3}  1 e 3  1  
0.0001  1 x 10^{4}  1 e 4  1  
1  1 x 10^{0}  1 e 0  1  
10  1 x 10^{1}  1 e 1  1  
12.49  1.249 x 10^{1}  1.249 e 1  4  
123,000  1.23 x 10^{5}  1.23 e 5  3  
0.000123  1.23 x 10^{4}  1.23 e 4  3  
0.010  1.0 x 10^{2}  1.0 e 2  2  Measured to the 0.001s so the trailing zero is significant 
Note: Scientific notation is always a number between 0 and 10 multiplied by 10 to the power of something, eg 2.54 x 10^{3}
Conclusion: The exponent is as big as the number of times you had to move the decimal to get the number between 0 and 10. Significant figures are easy to determine once in Scientific Notation  you just count up the digits in the "mantissa". (The mantissa is the 1.23 part of 1.23 x 10^{2}. The exponent is the ^{2 } part of 1.23 x 10^{2}.)
Why we ignore zeros in certain cases
Consider 12mm. This can also be written as 0.012m, since there are 1,000mm in 1m. Now obviously, 12mm has 2 significant figures. So, since 0.012m is the same thing as 12mm, then 0.012m should also have 2 significant figures.
How about 341m? Clearly that has 3 significant figures. But, 341m is the same as 34100cm, since there are 100cm in 1m, and so 34100cm should also have 3 significant figures.
So can you see that zeros used to determine the magnitude of the number mean nothing in terms of significant figures? In short, just put the measurement in scientific notation, and you'll see how many significant figures there are.
The only time zeros mean anything in terms of significant figures is when they are a part of the number, like 403, or when they are at the end of the number and after the decimal, like 21.60, in which case they help explain how precise the measurement was.
Rounding
Number  Rounded to nearest unit 
Comment 

12.0  12  
12.1  12  
12.4  12  
12.5  13  
12.9  13  
12.49  12  only consider the 4, not the 9 
19.9  20  
19.45  19  
19.1  19 
Conclusion: if the digit to the right of the last desired digit is less than 5, round down, otherwise round up.
Addition and subtraction
Precision  
12.231  0.001s  
The least precise number  57.10  +  0.01s 
Calculated answer  69.331  
Corrected to least precision (0.01s)  69.33  0.01s 
Precision  
231.12  + 
0.01s 

The least precise number  100  100s  
Calculated answer  331.12  0.01s  
Corrected to least precision (100s)  300  100s 
Conclusion: Round to the precision of the least precise measurement
Multiplication and division
Significant 

12.231  x  5  
The least significant figures  57.10  4  
Calculated answer  698.390,1  7  
Rounded up to the least number of significant figures (4)  698.4  4 
Significant 

The least significant figures  12  x  2 
57.101  5  
Calculated answer  685.212  6  
Rounded up to the least number of significant figures (2)  690  2 
Conclusion: Round to the number of significant figures in the measurement with the least significant figures
Numbers that are not measurements
A piece of string is 12.50 inches long. How long is it when you fold it in half?
12.5 x 0.5 = 6.25.
"Half" is not a measurement, it is a pure number that is perfectly accurate. So it's significant figures do not affect the issue. The measurement with the least significant figures is 12.50, since it is the only measurement. 12.50 has 4 significant figures, so too must the answer have that many.
So the answer should be reported as 6.250
Why do I still get a different answer to the book?
Sometimes students understand significant figures, but find that they still get the answers wrong. This may be due to a difference in the way that the student and the book calculate the result, particularly when the calculation can be done in multiple steps, as the book tends to do them.
For example:
What is the volume of a cylinder with radius 2cm and height 7 cm?
Method 1
Volume of a cylinder = Area of the circular base times height
Area of the circular base = Πr^{2 } (What is Π? Click here to find out)
= Π x (2)^{2 }cm^{2}
= 12.56637061.. cm^{2}
But since the radius of 2cm was measured to 1 significant figure, the answer too must be reported as 1 significant figure:
Area of the circular base = 10 cm^{2}
Now we multiply this area by the height of 7cm to get the volume of the cylinder:
Volume of the cylinder = 10 cm^{2} x 7 cm
= 70 cm^{3 }(1)
Method 2
On the other hand, what if we did the calculation in one step without calculating the area as an intermediate result?
Volume of a cylinder = Πr^{2}h
= Π x (2)^{2 }x 7 cm^{3}
= 87.9645943... cm^{3}
But since the radius of 2cm was measured to 1 significant figure, and the height to 1 significant figure, the answer too must be reported as 1 significant figure:
Volume of a cylinder = 90 cm^{3} (2)
Now, look at the two different results from (1) and (2). They are very different numbers. The formulae used in the two methods are correct. The problem comes in method 1, when we round an intermediate result. This rounding of rounded numbers makes the final result inaccurate.
Method 3
If we did method 1 again, this time without rounding the intermediate result we get a better answer:
Area of the circular base = Πr^{2}
= Π x (2)^{2 }cm^{2}
= 12.56637061.. cm^{2}
Now we multiply this by the height of 7cm to get the Volume of the cylinder:
Volume of the cylinder = 12.56637061.. cm^{2} x 7 cm
= 87.9645943... cm^{3}
But since the radius of 2cm was measured to 1 significant figure, and the height to 1 significant figure, the answer too must be reported as 1 significant figure:
Volume of a cylinder = 90 cm^{3} (3)
And this result agrees with method (2)
You be the judge. If the answer book says 70 cm^{3 }and you got 90 cm^{3} by doing the calculation in one step, how can you be judged wrong? Who is to say that you cannot do the calculation all in one step? Appeal for grace from the person marking your test!
Practice makes perfect
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