Order of Magnitude in Physics
The order of magnitude (odm) of a number refers to the power of 10 that is closest to it. To find the odm, convert the number into scientific notation, expressed as \( a \times 10^n \), where \( a \) is a number between 1 and 10, and \( n \) is the power of 10.
In practice, determining the order of magnitude follows a simple rule:
- If \( a \) is less than 5, the odm is \( 10^n \);
- If \( a \) is 5 or greater, round up by adding 1 to the exponent, making the odm \( 10^{n+1} \).
This simplification provides a quick sense of a number's "size" without requiring the exact value.
The order of magnitude is a useful concept for grasping the scale of a measurement, especially for very large or very small values, allowing you to understand its approximate range or "power of ten." This approach enables rapid estimation and aids in comparing vastly different numbers.
Order of magnitude applies across various fields, from physics to economics, helping to communicate and compare widely differing quantities.
Example
Here are some practical examples to illustrate the concept:
| Number | Scientific Notation | Odm | Explanation |
|---|---|---|---|
| 360 | 3.6·102 | 102 | 360 in scientific notation is \( 3.6 \times 10^2 \). Since 3.6 is less than 5, its order of magnitude is \( 10^2 \). |
| 9200 | 9.2·103 | 104 | 9200 in scientific notation is \( 9.2 \times 10^3 \). As 9.2 is greater than 5, round up, making the order of magnitude \( 10^4 \). |
| 0.00042 | 4.2·10-4 | 10-4 | 0.00042 in scientific notation is \( 4.2 \times 10^{-4} \). With 4.2 being less than 5, the odm is \( 10^{-4} \). |
| 580 | 5.8·102 | 103 | 580 in scientific notation is \( 5.8 \times 10^2 \). Since 5.8 exceeds 5, round up to \( 10^3 \) as its odm. |
| 0.0072 | 7.2·10-3 | 10-2 | 0.0072 in scientific notation is \( 7.2 \times 10^{-3} \). Since 7.2 is above 5, increase the exponent by 1, making the odm \( 10^{-2} \). |
| 12500 | 1.25·104 | 104 | 12500 in scientific notation is \( 1.25 \times 10^4 \). As 1.25 is less than 5, the odm remains \( 10^4 \). |
| 0.000056 | 5.6·10-5 | 10-4 | 0.000056 in scientific notation is \( 5.6 \times 10^{-5} \). Since 5.6 is more than 5, add 1 to the exponent, so the odm is \( 10^{-4} \). |
| 475 | 4.75·102 | 102 | 475 in scientific notation is \( 4.75 \times 10^2 \). As 4.75 is under 5, its odm is \( 10^2 \). |
| 21000 | 2.1·104 | 104 | 21000 in scientific notation is \( 2.1 \times 10^4 \). As 2.1 is below 5, the odm remains \( 10^4 \). |
| 680000 | 6.8·105 | 106 | 680000 in scientific notation is \( 6.8 \times 10^5 \). Since 6.8 is above 5, increase the exponent by 1, so the odm is \( 10^6 \). |
These examples demonstrate how odm provides a quick sense of how large or small certain quantities are.
An Exception: Numbers Between 5 and 5.5
There is a “gray area” for numbers between 5 and 5.5, where, by convention, the higher power of ten is chosen, even if it’s not the closest.
For instance, take 510. In scientific notation, it can be written as $$ 5.1 \times 10^2 $$. According to the rule, since the decimal is 5 or greater, the order of magnitude is \( 10^3 \). However, if you check the distances of \( 510 \) from \( 10^2 \) and \( 10^3 \), you’ll see that the closest power is actually \( 10^2 \).
- \( |10^2-510| = |100-510| = 410 \)
- \( |10^3-510| = |1000-510| = 490 \)
On the number line, 510 is indeed closer to 100 (\( 10^2 \)) than to 1000 (\( 10^3 \)).
This discrepancy only occurs for values of \( a \) between 5 and 5.5, and is typically resolved on a case-by-case basis.
In these cases, conventionally, the higher power of 10 is still chosen, even if it’s not the closest.
