Measurements always have uncertainties. If you measure the thickness of the cover of a hardbound book using an ordinary ruler, your measurement is reliable only to the nearest millimeter. Take for instance, if your result 3 mm, It would be wrong to state this result as 3.00 mm; given the limitations of the measuring device, you can’t tell whether the actual thickness is 3.00 mm, 2.85 mm, or 3.11 mm. But if you use a micrometer caliper, a device that measures distances reliably to the nearest 0.01 mm, the result will have more accuracy, say 2.91 mm. The distinction between these two measurements is in their uncertainty. The measurement using the micrometer caliper has a smaller uncertainty; it’s a more accurate measurement. The uncertainty is also called the error because it indicates the maximum difference there is likely to be between the measured value and the true value.
The uncertainty or error of a measured value depends on the measurement technique used. We often indicate the accuracy of a measured value—that is, how close it is likely to be to the true value—by writing the number, the symbol and a second number indicating the uncertainty of the measurement. If the diameter of a steel rod is given as 56.47 ± 0.02mm, this means that the true value is unlikely to be less than 56.45 mm or greater than 56.49 mm. In a commonly used shorthand notation, the number 1.347(56) means 1.347 ± 0.056. The numbers in parentheses show the uncertainty in the final digits of the main number. We can also express accuracy in terms of the maximum likely fractional error or percent error (also called fractional uncertainty and percent uncertainty). A resistor labeled probably “47 ohms ± 10%” has a true resistance that differs from 47 ohms by no more than 10% of 47 ohms—that is, by about 5 ohms. The resistance is probably between 42 and 52 ohms. For the diameter of the steel rod given above, the fractional error is or about 0.0004; the percent error is or about 0.04%. Even small percent errors can sometimes be very significant.
In many cases the uncertainty of a number is not stated explicitly. Instead, the uncertainty is indicated by the number of meaningful digits, or significant figures, in the measured value. Given the thickness of the cover of a book as 2.91 mm, which has three significant figures, this means that the first two digits are known to be correct, while the third digit is uncertain. The last digit is in the hundredths place, so the uncertainty is about 0.01 mm. Two values with the same number of significant figures may have different uncertainties; a distance given as 137 km also has three significant figures, but the uncertainty is about 1 km.
When numbers that have uncertainties are used to compute other numbers, the computed numbers are also uncertain. When numbers are multiplied or divided, the number of significant figures in the result can be no greater than in the factor with the fewest significant figures. For example, 3.1416 × 2.34 × 0.58 = 4.3.
When we add and subtract numbers, it’s the location of the decimal point that matters, not the number of significant figures. For example, 123.62 + 8.9 = 132.5. Although 123.62 have an uncertainty of about 0.01, 8.9 have an uncertainty of about 0.1. So their sum has an uncertainty of about 0.1 and should be written as 132.5, not 132.52.
As an application of these ideas, suppose you want to verify the value of pie, the ratio of the circumference of a circle to its diameter. The true value of this ratio to ten digits is 3.141592654. To test this, you draw a large circle and measure its circumference and diameter to the nearest millimeter, supposing you obtain the values 424 mm and 135 mm as the circumference and diameter respectively, and you punch these into your calculator and obtain the quotient (424 mm)/(135mm) = 3.140740741 . This may seem to disagree with the true value but keep in mind that each of your measurements has three significant figures, so your measured value of can have only three significant figures. It should be stated simply as 3.14. Within the limit of three significant figures, your value does agree with the true value.
Even if you do the arithmetic with a calculator that displays ten digits, it would be wrong to give a ten-digit answer because it misrepresents the accuracy of the results. Always round your final answer to keep only the correct number of significant figures or, in doubtful cases, one more at most.
When calculations are made with very large or very small numbers, we can show significant figures much more easily by using scientific notation, sometimes called powers-of-10 notation. The distance from the earth to the moon is about 384,000,000 m, but writing the number in this form doesn’t indicate the number of significant figures. Instead, we move the decimal point eight places to the left (corresponding to dividing by 108) and multiply by 108 that is, 384,000,000 m = 3.84 × 108 m. In this form, it is clear that we have three significant figures. The number 4.00 × 10-7 also has three significant figures, even though two of them are zeros. Note that in scientific notation the usual practice is to express the quantity as a number between 1 and 10 multiplied by the appropriate power of 10. When an integer or a fraction occurs in a general equation, we treat that number as having no uncertainty at all. For example, in the equation “(Vx)^2 = (Vox)^2 + 2ax(x – xo)”, the coefficient 2 is exactly 2. We can consider this coefficient as having an infinite number of significant figures (2.000000. . .). The same is true of the exponent 2 in (Vx)^2 and (Vox)^2.
Finally, it should be noted that precision is not the same as accuracy. A cheap digital watch that gives the time as 10:35:17 A.M. is very precise (the time is given to the second), but if the watch runs several minutes slow, then this value isn’t very accurate. On the other hand, a grandfather clock might be very accurate (that is, displays the correct time), but if the clock has no second hand, it isn’t very precise. A high-quality measurement is both precise and accurate.

DRILL
The rest energy E of an object with rest mass m is given by Einstein’s famous equation E = m.c^2, where c is the speed of light in vacuum. Find E for an electron for which (to three significant Figures) m = 9.11 × 10^(-31) kg. The SI unit for E is the joule (J); 1J = 1 kg.m^2/s^2 and c = 2.99792458 × 10^8 m/s.

SOLUTION
The target variable is the energy E. We are given the value of the mass, m, and that of the speed of light in vacuum, c.
Substituting the values of m and c into Einstein’s equation, we find
E = (9.11 × 10^(-31) kg) (2.99792458 × 10^( m/s)2
= 8.187659678 × 10-14 kg.m^2/s^2.
Since the value of m was given to only three significant figures, we must round this to
E = 8.19 × 10^(-14) kg.m^2/s^2 = 8.19 × 10^(-14) J.

Source:
http://www.edufrisk..com.ng/2016/01/uncertainty-and-significant-figures.html