1.1.2 Using Scientific Notation

Scientific Notation

• In physics, measured quantities cover a large range from the very large to the very small
• Scientific notation is a form that is based on powers of 10
• The scientific form must have one digit in front of the decimal place
  • Any remaining digits remain behind the decimal place 
  • The magnitude of the value comes from multiplying by 10ⁿ where n is called 'the power'
  • This power is positive when representing large numbers or negative when representing small numbers

Step 1: Write the convention for scientific notation

• • To convert into scientific notation, only one digit may remain in front of the decimal point
    • Therefore, the scientific notation must be 4.6 × 10ⁿ

  • The value of n is determined by the number of decimal places that must be moved to return to the original number (i.e. 4 600 000)

Step 2: Identify the number of digits after the 4

• • In this case, that number is +6

Step 3: Write the final answer in scientific notation

• • The solution is: 4.6 × 10⁶

Metric Multipliers

• When dealing with magnitudes of 10, there are metric names for many common quantities
• These are known as metric multipliers and they change the size of the quantity they are applied to
  • They are represented by prefixes that go in front of the measurement

• Some common examples that are well-known include
  • kilometres, km (× 10³)
  • centimetres, cm (× 10^–2)
  • milligrams, mg (× 10^–3)

• Metric multipliers are represented by a single letter symbol such as centi- (c) or Giga- (G)
  • These letters go in front of the quantity of interest
  • For example, centimetres (cm) or Gigawatts (GW)

Common Metric Multipliers Table

Step 1: Check which metric multipliers are in this problem

• • M represents Mega- which is × 10⁶ (not milli- which is small m!)
  • k represents kilo- which is a multiplier of × 10³

Step 2: Apply these multipliers to get both quantities to be metres

3.6 × 10⁶ m  +  2.7 × 10⁶ m

Step 3: Write the final answer in units of metres

6.3 × 10⁶ m

• Significant figures are the digits that accurately represent a given quantity
• Significant figures describe the precision with which a quantity is known
  • If a quantity has more significant figures then more precise information is known about that quantity

Rules for Significant Figures

• Not all digits that a number may show are significant
• In order to know how many digits in a quantity are significant, these rules can be followed
  • Rule 1: In an integer, all digits count as significant if the last digit is non-zero
    • Example: 702 has 3 significant figures

  • Rule 2: Zeros at the end of an integer do not count as significant
    • Example: 705,000 has 3 significant figures

  • Rule 3: Zeros in front of an integer do not count as significant
    • Example: 0.002309 has 4 significant figures

  • Rule 4: Zeros at the end of a number less than zero count as significant, but those in front do not.
    • Example: 0.0020300 has 5 significant figures

  • Rule 5: Zeros after a decimal point are also significant figures.
    • Example: 70.0 has 3 significant figures

• Combinations of numbers must always be to the smallest number significant figures

Step 1: Identify the smallest number of significant figures

• • 18 has only 2 significant figures, while 384 has 3 significant figures
  • Therefore, the final answer should be to 2 significant figures

Step 2: Do the calculation with the maximum number of digits

18 × 384 = 6912

Step 3: Round to the final answer to 2 significant figures

6.9 × 10³