Significant Figures

Significant figures refer to the certain digits and the estimated digit of a measurement.

To determine the significant figures, here are some rules to be considered:

  1. All nonzero digits are significant.
  2. All digits between the nonzero digits are significant.
  3. Zeroes to the left of the significant digits are not significant.
  4. Zeroes to the right of significant digits are significant.

Example 1:

The height of the basketball player is 6.50 feet. How many significant figures are in this value?

Answer: From rule numbers 1 and 4, there are 3 significant figures in the value given as the height of the player.

Example 2:

How many significant figures are there in the value of the mass of an object which is 0.005 kg.?

Answer: According to the rules of significant figures, all zeros to the left of the significant digit are not significant, thus, the value 0.005 kg has only 1 significant figure.

Significant figures are also used to show the uncertainty of some of the physical quantity measurements which are not shown clearly in the given value. These uncertainties are just indicated by the number of meaningful digits or the significant figures. For example, the mass of the apple is 1.57 g. The value has three significant figures. The first two digits of the apple’s mass is known to be correct while the last digit is uncertain.

In addition and subtraction of scientific data like measurements, the number of significant figures is determined by the given number with the largest uncertainty or the one with the fewest digits to the right of the decimal point.

Example 3:

What is the total mass of the two baskets of apples measured 5.84 kg and 3.7962 kg each, respectively.

Solution: \(5.84\;kg+3.7962\;kg=9.6362\;kg=9.64\;kg\).

The final answer is expressed only as 9.64 kg. Eventhough the other given value is more precise, the other has only two digits after the decimal points, thus, it can only give a precise measurement until the hundredths place giving us a clue to express the final answer with only three significant figures.

In multiplying and dividing numbers using significant figures, the product or quotient may have no more significant figures than the given number with the fewest significant figure.

Example 4:

A garden is measured with a length of 10.4 m and width of 4.932 m. What is the area of the garden?

Solution: The area of the garden is solved as \(A=l \times w=(10.4\;m)(4.932\;m)=51.2928\;m^2=51.3\;m^2\). The final answer has only 3 significant figures since based on the given values, one has 3 significant figures and the other value has 4 significant figures, and the one with the least number of significant figure must be followed for the final answer.

In scientific notation, the rules on significant figures only apply on the digit term. The exponential term only indicates the number of moves of the decimal point.

Example 5.

How many significant figures are there in \(3.450 \times 10^3 \)?

Answer: All nonzero digits are significant and zeros to the right of significant digits are also significant, thus, there are 4 significant figures.

Example 6.

How will you express 6,800 with three significant figures?

Answer: We can write the number in scientific notation and applying the rules 1 and 4, we have \(6.80 \times 10^3\).

Example 7.

The number 0.0009 can be written in scientific notation with 4 significant figures as?

Answer: \(9.000 \times 10^{-4}\), since nonzero digits are significant and the zeros to the right of the significant digit are also significant.