Orders of Magnitude
What is an Order of Magnitude?
At its heart, an order of magnitude is a way to compare numbers by looking at their size in powers of ten. It answers the question: "Is this thing about 10 times, 100 times, or 1,000 times bigger than that other thing?" Instead of getting bogged down by exact numbers, it helps us see the big picture.
Think of it like a ladder where each rung is ten times higher than the one below it. Jumping one rung means increasing by one order of magnitude, or a factor of 10. Jumping two rungs means an increase of two orders of magnitude, or a factor of $10 \times 10 = 100$.
Let's look at some simple examples:
- The number 50 can be written as $5.0 \times 10^1$. Its order of magnitude is $1$.
- The number 4,200 is about $4.2 \times 10^3$. Its order of magnitude is $3$.
- The number 0.08 is $8.0 \times 10^{-2}$. Its order of magnitude is $-2$.
Two numbers are said to be of the same order of magnitude if the larger one is less than ten times the smaller one. For instance, $15$ and $80$ are in the same order of magnitude ($10^1$), but $15$ and $200$ are not.
The Cosmic Ladder: From Atoms to Galaxies
The true power of orders of magnitude becomes clear when we look at the universe. The size of things we can observe ranges from the incredibly tiny, like subatomic particles, to the mind-bogglingly large, like the observable universe. Let's climb this cosmic ladder.
| Object | Size (meters) | Scientific Notation | Order of Magnitude |
|---|---|---|---|
| Proton (in an atomic nucleus) | 0.000000000000001 | $1 \times 10^{-15}$ m | -15 |
| Water Molecule | 0.0000000003 | $3 \times 10^{-10}$ m | -10 |
| Width of a Human Hair | 0.0001 | $1 \times 10^{-4}$ m | -4 |
| Average Human Height | 1.7 | $1.7 \times 10^{0}$ m | 0 |
| Height of Mount Everest | 8,849 | $8.8 \times 10^{3}$ m | 3 |
| Diameter of Earth | 12,700,000 | $1.27 \times 10^{7}$ m | 7 |
| Distance from Earth to Sun | 150,000,000,000 | $1.5 \times 10^{11}$ m | 11 |
| Diameter of the Milky Way Galaxy | 1,000,000,000,000,000,000,000 | $1 \times 10^{21}$ m | 21 |
Looking at this table, we can make incredible comparisons. The difference in orders of magnitude between a proton and a human is $0 - (-15) = 15$. This means a human is roughly $10^{15}$ (or 1,000,000,000,000,000) times larger than a proton! Similarly, the observable universe is about $10^{26}$ meters in diameter, which is $26 - (-15) = 41$ orders of magnitude larger than a proton. The number $10^{41}$ is so large it's almost impossible to comprehend.
Putting It Into Practice: Estimation and Fermi Problems
Orders of magnitude are not just for cosmic scales; they are incredibly useful for quick, everyday estimations. Physicist Enrico Fermi was famous for asking "Fermi problems" that required estimation using orders of magnitude. A classic example is: "How many piano tuners are there in Chicago?"
You wouldn't need an exact phone book to answer this. You could reason as follows:
- Estimate the population of Chicago: Let's say about $3 \times 10^6$ people (order of magnitude 6).
- Estimate households: If the average household has 2 people, that's $1.5 \times 10^6$ households (order of magnitude 6).
- Estimate pianos per household: Maybe 1 in 20 households has a piano. That gives $(1.5 \times 10^6) / 20 = 7.5 \times 10^4$ pianos (order of magnitude 4-5).
- Estimate tuning frequency: A piano might be tuned once a year.
- Estimate tunings per tuner: If a tuner does 2 pianos a day, 5 days a week, 50 weeks a year, that's $2 \times 5 \times 50 = 500$ tunings per year (order of magnitude 2-3).
- Calculate tuners: $(7.5 \times 10^4 \text{ pianos}) / (5 \times 10^2 \text{ tunings/tuner}) = 1.5 \times 10^2$ tuners.
So, our rough, order-of-magnitude estimate is about 150 piano tuners. The actual number might be 80 or 300, but we have correctly determined that the answer is in the hundreds, not the tens or thousands. This is the power of the method.
Common Mistakes and Important Questions
A: Yes, by definition, one order of magnitude is a factor of 10. A factor of 2 is about 0.3 orders of magnitude (since $\log_{10}(2) \approx 0.3$), and a factor of 20 is about 1.3 orders of magnitude ($20 = 2 \times 10^1$). The key is the power of ten.
A: This is a very common point of confusion. To find the order of magnitude, first write the number in scientific notation: $95 = 9.5 \times 10^1$. Since the multiplier (9.5) is greater than $\sqrt{10} \approx 3.16$, it's often rounded up, and the order of magnitude is considered $2$ for rough estimates. However, the more mathematically precise method is to take the base-10 logarithm: $\log_{10}(95) \approx 1.98$, whose integer part is 1. For consistency, we use the integer part of the logarithm, so the order of magnitude is $1$. In casual conversation, people might say "it's on the order of 100," but technically its order of magnitude is 1.
A: Absolutely. For numbers less than 1, the order of magnitude is negative. For example, the diameter of a hydrogen atom is about $1 \times 10^{-10}$ meters, so its order of magnitude is -10. This tells us it is 10 orders of magnitude smaller than a meter, meaning it is $10^{10}$ or ten billion times smaller.
The concept of orders of magnitude is a powerful lens through which to view the world. It allows us to tame unimaginably large and small numbers, making them manageable for comparison and understanding. From estimating the number of jellybeans in a jar to grasping the scale of the cosmos, this tool is fundamental to scientific literacy. By thinking in powers of ten, we develop a intuition for scale that is both practical and awe-inspiring, connecting the world of the infinitesimally small with the grandeur of the universe.
Footnote
[1] Logarithm: A quantity representing the power to which a fixed number (the base) must be raised to produce a given number. For base-10 logarithms, if $10^y = x$, then $\log_{10}(x) = y$. For example, $\log_{10}(1000) = 3$ because $10^3 = 1000$.
[2] Scientific Notation: A way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is written as the product of a number between 1 and 10 and a power of 10 (e.g., $3.4 \times 10^8$).