### 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 significant digits
•           - 19.4 has 3 significant digits
•           - 527.266 992 has 9 significant digits

2. All zeros that are located between two non-zero numbers are significant.

•           408 has 3 significant digits
•           25 076 has 5 significant digits

3.    Zeros that are located to the left of a measurement are not significant

•           0.0907 has 3 significant digits

4.    Zeros to the right of a measurement may or may not be significant.

•           22 700 may have 3 significant digits if the measurement is approximate

•           22 700 may have 5 significant digits if the measurement is exact

#### Rules for reporting significant digits in calculations

1.    Multiplying and dividing:
The value with the fewest number of significant digits, going into the calculation,
determines the number of significant digits in your answer.

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 significant digits in the question.

2.    Adding and subtracting:
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 Off
To get the appropriate number of significant digits for answers from calculations you do, it is necessary to round off 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 significant 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 significant 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 significant digits is 18.4

18.25 rounded to 3 significant 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 105 kg
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 – 86.25L
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