Biology Chemistry Computer Engineering Electronics Mathematics Physics Science Home Introductory Notions Topics: Units & Standards Scientific Notation Significant Digits Precision Accuracy 1. Units and Standards There are 4 fundamental scientific measurements: Mass, Length, Time, and Electric Charge The following table summarizes their basic SI (International Metric System) units  Fundamental Quantity Mass Length, distance, displacement Time Charge SI Unit kg (kilogram) m (metre) s (second) C (coulomb) Symbol m  Δd t Q 2. Scalar vs. Vector Scalar quantities express only a quantity and a unit Vector quantities express a quantity, a unit and a direction Examples: Quantity Symbol Type Example  distance Δd scalar  Δd = 10 m displacement vector = 10 m [E]

3. Scientific notation
-very large or very small numbers are difficult to work with when written in common decimal notation
-it is possible to change the SI prefix so the number falls between 0.1 and 1000
-for example 237 000 000 mm can be expressed as 237 km or 0.000 000 895 kg can be expressed as 0.895 mg
-for situations where this is not possible the best method of dealing with this it the use of scientific notation
-scientific notation expresses a number between 1 and 10 x 10 n where n represents the number of places
 the decimal was moved from the right or left
-for example 124 500 000 km   = 1.245 x 10⁸ km

This value 1.245 x 10⁸ really represents 1.245 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
1. To multiple numbers in scientific notation, multiply the coefficients and add the exponents
example  (4.73 x 10⁵ m) (5.82 x 10⁷) = 27.5 x 10¹² m² = 2.75 x 10¹³ m²  to divide two numbers in scientific notation, subtract the exponents algebraically
example   (1.842 x 10⁶g) ÷ 1.0787 x 10²g/mol) = 1.707611 x 10⁴ mol = 1.708 x 10⁴ mol

2.    to add or subtract numbers in scientific notation, convert the numbers so they have the same exponent as
the number with the greatest power of ten. Once this is done add or subtract the coefficients but leave the
exponent alone.

Example:
(3.42 x 10⁶ m) + (8.53 x 10³ m) = (3.42 x 10⁶ m) + (0.00853 x 10⁶ m) = 3.42853 x 10⁶ m
= 3.43 x 10⁶ m

Exercise:
1.    Convert each value into scientific notation

a)    0.000 934

b)    7 983 000 000

c)    0.000 000 000 820 57

d)    496 x 10⁶

e)    0.000 06 x 10¹

f)      309 72 x 10^–8

2.    Add, subtract, multiply, or divide the following problems. Express answer in scientific notation
to the correct certainty.

a)    (3.21 x 10^–3 + (9.21 x 10²)

b)    (8.1 x 10³) + (9.21 x 10²)

c)    (1.010 1 x 10¹) – (4.823 x 10^–2)

d)    (1.209 x 10⁶) x (8.4 x 10⁷)

e)    (4.89 x 10^–4) ÷ (3.20 x 10^–2)

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