Orders of Magnitude
- When a number is expressed in an order of 10, this is an order of magnitude
- For example, if a number is described as 3 × 108 then that number is actually 3 × 100 000 000
- The order of magnitude of 3 × 108 is just 108
- When the number is greater than 5, round up to the next order of magnitude
- For example, the order of magnitude of 6 × 108 is 109
- A quantity is an order of magnitude larger than another quantity if it is about ten times larger
- Similarly, two orders of magnitude would be 100 times larger, or 102
- In physics, orders of magnitude can be very large or very small
- When estimating values, it’s best to give the estimate of an order of magnitude to the nearest power of 10
- For example, the diameter of the Milky Way is approximately 1 000 000 000 000 000 000 000 m
- It is inconvenient to write this many zeros, so it’s best to use scientific notation as follows:
1 000 000 000 000 000 000 000 = 1 × 1021 m
- The order of magnitude is 1021
- Orders of magnitude make it easier to compare the relative sizes of objects
- For example, a quantity with an order of magnitude of 106 is 10 000 times larger than a quantity with a magnitude of 102
Approximation & Estimation
- To estimate is to obtain an approximate value
- For very large or small quantities, using orders of magnitudes to estimate calculations is a valid approach
- Estimation is typically done to the nearest order of magnitude
Estimating Physical Quantities Table
| Object of Interest | Approximate Length (m) | Order of Magnitude (m) |
| distance to the edge of the observable Universe | 4.40 × 1026 | 1026 |
| distance from Earth to Neptune | 4.5 × 1012 | 1012 |
| distance from London to Cape Town | 9.7 × 106 | 107 |
| length of a human | 1.7 | 100 |
| length of an ant | 9 × 10−4 | 10−3 |
| length of a bacteria cell | 2 × 10−6 | 10−6 |
Worked example
Estimate the order of magnitude of the following:
(a) The temperature of an oven (in Kelvin)
(b) The volume of the Earth (in m3)
(c) The number of seconds in a person's life if they live to be 95 years old
Answer:
(a) The temperature of an oven
Step 1: Identify the approximate temperature of an oven
- A conventional oven works at ∼200 °C which is 473 K
Step 2: Identify the order of magnitude
- Since this could be written as 4.73 × 102 K
- The order of magnitude is ∼102
(b) The volume of the Earth
Step 1: Identify the approximate radius of the Earth
- The radius of the Earth is ∼6.4 × 106 m
Step 2: Use the radius to calculate the volume
- The volume of a sphere is equal to:
V = 4/3 π r3
V = 4/3 × π × (6.4 × 106)3
V = 1.1 × 1021 m3
Step 3: Identify the order of magnitude
- The order of magnitude is ∼1021
(c) The number of seconds in 95 years
Step 1: Find the number of seconds in a single year
1 year = 365 days with 24 hours each with 60 minutes with 60 seconds
365 × 24 × 60 × 60 = 31 536 000 seconds in a year
Step 2: Find the number of seconds in 95 years
95 × 31 536 000 = 283 824 000 seconds
- This is approximately 2.84 × 108 seconds
- Therefore the order of magnitude is ∼108
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
Worked example
Express 4 600 000 in scientific notation.
Answer:
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
| Prefix | Abbreviation | Value |
| peta | P | 1015 |
| tera | T | 1012 |
| giga | G | 109 |
| mega | M | 106 |
| kilo | k | 103 |
| hecto | h | 102 |
| deca | da | 101 |
| deci | d | 10−1 |
|
centi |
c | 10−2 |
|
milli |
m | 10−3 |
|
micro |
μ | 10−6 |
|
nano |
n | 10−9 |
|
pico |
p | 10−12 |
|
femto |
f | 10−15 |
Worked example
What is 3.6 Mm + 2700 km, in m?
Answer:
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
- 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
- Rule 1: In an integer, all digits count as significant if the last digit is non-zero
- Combinations of numbers must always be to the smallest number significant figures
Worked example
What is the solution to this problem to the correct number of significant figures: 18 × 384?
Answer:
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
Exam Tip
When studying IB DP Physics, it is recommended to state your answer on a single line explicitly (if possible) with all necessary details to ensure the examiners can mark correctly and for best practice.
You are expected to know metric multipliers for your exams. Make sure you become familiar with them in order to avoid any mistakes.