Significant Digits

• -all measurements involve uncertainty.
• -the certainty of any measurement is communicated by the number of significant digits in the measurement
• -significant digits include the digits you are certain about, and a final, uncertain digit you estimate
• -the digits you record from a measurement are termed significant digits
Rules for determining significant digits
1. All non-zero numbers are significant
- 7.886 has 4 sig. digits
- 19.4 has 3 sig. digits
- 527.266 992 has 9 sig. digits

2. All zeros that are located between two non-zero numbers are significant.
- 408 has 3 sig. digits
- 25 076 has 5 sig. digits

3.    Zeros that are located to the left of a measurement are not significant
- 0.0907 has 3 sig. digits

4.    Zeros to the right of a measurement may or may not be significant.
- 22 700 may have 3 sig. digits if the measurement is approximate
- 22 700 may have 5 sig. digits if the measurement is exact

Rules for reporting significant digits in calculations

1.    Multiplying and dividing:

The value with the fewest number of sig. digits, going into the calculation,

77.8 km/h x 5.9015 h = 459.136 7 km = 459 km the answer must have 3
significant digits to express the same certainty as the value, 77.8 km/h,
with the fewest sig. digits in the question.

The value with the fewest number of decimal places, going into the calculation,
determines the number of decimal places that you should report in your answer.

12.5 g + 145.67 g + 70.5456 g + 3.001 g = 231.7166 g = 231.7 g

3.    Rounding
To get the appropriate number of sig. digits for answers from calculations you do, it is necessary to round your answers.

-         If your answer ends in a number that is greater than 5 increase the preceding digit by 1.
2.346 rounded to 3 sig. digits is 2.35
-    If your answer ends in a number that is less than 5 leave the preceding number unchanged.
2.343 rounded to 3 sig. digits is 2.34
-    If your answer ends with 5, increase the preceding number by 1 if it is odd, leave the preceding
number unchanged if it is even.
18.35 rounded to 3 sig. digits is 18.4
18.25 rounded to 3 sig. digits is 18.2

Exercises:

1.    Determine the number of significant digits in the following measurements.
a)    32.07 m
b)    0.0047 g
c)    5 x 10kg
d)    6400 s
e)    204.0 cm
f)      0.000 001 µm
g)    0.156 345 g
h)    10.0 cm

2.    Express each answer using the correct number of significant digits.
a)    55.671g + 45.78
b)    1.9 mm + 0.62 mm
c)    87.9478 L – 86.25 L
d)    0.350 mL + 1.70 mL + 1.019 mL
e)    5.841 g x 6.03 g
f)      17.51g ÷ 2.2 cm3
g)    23 457.12 cm x 45.341 cm