KAHIBARO
Table of Contents
Comparing Quantities Using Orders of Magnitude
When working in physics, many quantities differ by huge factors. The mass of an electron and the mass of the Earth, the size of an atom and the size of a galaxy, or the power of a pocket calculator and a power station cannot be compared easily using ordinary numbers. Orders of magnitude provide a simple way to describe and compare such very large or very small quantities by focusing on powers of ten instead of exact values.
In this chapter, the focus is on what it means for two quantities to differ by a certain number of orders of magnitude, how to estimate and express orders of magnitude using scientific notation, and how physicists use order of magnitude thinking to make rough, useful approximations without detailed calculation.
Definition of Order of Magnitude
To understand orders of magnitude, it helps to think in terms of powers of ten. Any positive number $x$ can be written in scientific notation as
$$
x = a \times 10^n
$$
where $1 \le a < 10$ and $n$ is an integer. The exponent $n$ gives a basic idea of the “size” of the number.
Informally, the order of magnitude of a quantity is the power of ten that is closest to the quantity. Two numbers differ by one order of magnitude if one is about ten times larger than the other, by two orders of magnitude if one is about $10^2 = 100$ times larger, and so on.
A difference of one order of magnitude corresponds to a factor of about $10$.
A difference of $k$ orders of magnitude corresponds to a factor of about $10^k$.
For example, a number around $10^3$ has order of magnitude $10^3$, and a number around $10^{-6}$ has order of magnitude $10^{-6}$. The order of magnitude does not usually include the exact leading digit. Instead it focuses on the exponent that roughly describes the scale of the number.
Physicists often say that a quantity $A$ is “of order $10^n$” and write this symbolically as
$$
A \sim 10^n
$$
to mean that $A$ is within about a factor of $10$ of $10^n$.
Orders of Magnitude and Scientific Notation
Scientific notation, which you will study in a separate chapter, provides a natural way to talk about orders of magnitude. The exponent in scientific notation usually points directly to the order of magnitude.
Consider the following examples:
| Quantity | Scientific notation | Approximate order of magnitude |
|---|---|---|
| Speed of light $c \approx 3.0\times 10^8\ \text{m/s}$ | $3.0 \times 10^8$ | $10^8$ |
| Radius of Earth $\approx 6.4\times 10^6\ \text{m}$ | $6.4 \times 10^6$ | $10^7$ |
| Proton mass $\approx 1.7\times 10^{-27}\ \text{kg}$ | $1.7 \times 10^{-27}$ | $10^{-27}$ |
| Electron charge $\approx 1.6\times 10^{-19}\ \text{C}$ | $1.6 \times 10^{-19}$ | $10^{-19}$ |
Sometimes a number sits between two powers of ten, and we must decide which order of magnitude is more appropriate. For instance, $6.4\times 10^6$ is closer to $10^7$ than to $10^6$, so it is often said to be of order $10^7$. On the other hand, $2.1\times 10^6$ is closer to $10^6$, so it is of order $10^6$.
A simple rule of thumb is to look at the leading digit. If the number is written as $a \times 10^n$ with $1 \le a < 10$, then:
- if $1 \le a < 3.2$, the order of magnitude is usually taken as $10^n$,
- if $3.2 \le a < 10$, the order of magnitude is often taken as $10^{n+1}$.
This is not an exact rule of mathematics, but a practical convention. Different authors may draw the boundary at slightly different leading digits, but in all cases the goal is to capture the scale, not the precise value.
Comparing Quantities by Orders of Magnitude
Orders of magnitude let us compare quantities using rough size ratios rather than precise factors. If two values differ by only a factor of 2 or 3, they are often said to be of the same order of magnitude. If they differ by a factor of 10, 100, or 1000, they differ by one, two, or three orders of magnitude.
Suppose the diameter of a human hair is about $10^{-4}\ \text{m}$ and the diameter of an atom is about $10^{-10}\ \text{m}$. The ratio of these diameters is
$$
\frac{10^{-4}}{10^{-10}} = 10^{6}.
$$
So a hair is about $10^6$ times wider than an atom. We say that a hair is six orders of magnitude larger in diameter than an atom.
Similarly, the radius of the observable universe is about $10^{26}\ \text{m}$, while the radius of a proton is about $10^{-15}\ \text{m}$. The ratio is
$$
\frac{10^{26}}{10^{-15}} = 10^{41}.
$$
The universe is roughly $10^{41}$ times larger in radius than a proton, so they differ by 41 orders of magnitude.
The number of orders of magnitude between two positive quantities $A$ and $B$ is essentially the difference of the exponents when they are written in scientific notation. More precisely, consider the base 10 logarithm, written as $\log_{10}$. Then
$$
\text{Number of orders of magnitude} \approx \left|\log_{10} A - \log_{10} B\right|.
$$
If $A = 10^a$ and $B = 10^b$ exactly, then the number of orders of magnitude between them is exactly $|a - b|$.
Two quantities are typically considered to be of the same order of magnitude if their ratio is between about $0.1$ and $10$.
If the ratio of two quantities is about $10^k$, they differ by $k$ orders of magnitude.
Estimating Orders of Magnitude
In physics, you often do not know exact numbers, or you may not need them. It is frequently enough to know the order of magnitude, which means knowing the exponent in the power of ten that describes the scale of a quantity.
To estimate the order of magnitude of a number, you can:
- Express it roughly in scientific notation.
- Decide which power of ten it is closest to.
For example, suppose a certain city has a population of about 2.7 million. This is $2.7 \times 10^{6}$ people. The order of magnitude is $10^{6}$, because $2.7$ is closer to $1$ than to $10$. If another city had 95 million people, or $9.5 \times 10^{7}$ people, it would be convenient to say that its population is of order $10^{8}$.
As another example, the charge of an electron is about $1.6 \times 10^{-19}\ \text{C}$. If you have $10^{19}$ electrons, their total charge would be about
$$
Q \approx 10^{19} \times 1.6 \times 10^{-19}\ \text{C} \approx 1.6\ \text{C}.
$$
This total charge is of order $10^0$ coulombs, because $1.6$ is between $1$ and $10$ and closer to $10^0$ than any other power of ten.
Estimation of orders of magnitude encourages you to ignore unnecessary precision and focus on the exponent. This is especially useful when dealing with complex problems where exact computations would be time consuming or impossible without a calculator.
Examples from Physics Across Many Scales
The usefulness of orders of magnitude becomes clear when you look at the huge range of physical quantities that appear in physics. The following table shows approximate orders of magnitude for some common lengths:
| Object or scale | Approximate length (m) | Order of magnitude |
|---|---|---|
| Size of observable universe | $10^{26}$ | $10^{26}$ |
| Distance Earth to Sun | $10^{11}$ | $10^{11}$ |
| Radius of Earth | $10^{7}$ | $10^{7}$ |
| Height of a person | $10^{0}$ | $10^{0}$ |
| Thickness of a sheet of paper | $10^{-4}$ | $10^{-4}$ |
| Wavelength of visible light | $10^{-7}$ | $10^{-7}$ |
| Size of a virus | $10^{-8}$ | $10^{-8}$ |
| Size of an atom | $10^{-10}$ | $10^{-10}$ |
| Size of an atomic nucleus | $10^{-15}$ | $10^{-15}$ |
In the same way, masses cover a wide range:
| Object | Approximate mass (kg) | Order of magnitude |
|---|---|---|
| Mass of observable universe (matter) | $10^{53}$ (very rough) | $10^{53}$ |
| Mass of Sun | $10^{30}$ | $10^{30}$ |
| Mass of Earth | $10^{24}$ | $10^{24}$ |
| Mass of a human | $10^{2}$ | $10^{2}$ |
| Mass of a grain of sand | $10^{-6}$ | $10^{-6}$ |
| Mass of a human cell | $10^{-12}$ | $10^{-12}$ |
| Mass of a bacterium | $10^{-15}$ | $10^{-15}$ |
| Mass of a proton | $10^{-27}$ | $10^{-27}$ |
| Mass of an electron | $10^{-30}$ | $10^{-30}$ |
Here, the mass of the Sun is roughly $10^{28}$ times the mass of a typical human, so they differ by 28 orders of magnitude. This type of comparison quickly tells you how insignificant or significant a certain quantity is on a particular scale.
Orders of magnitude also appear in time scales. The age of the universe is about $10^{17}$ seconds, while a nuclear reaction may happen in about $10^{-22}$ seconds. Between these two extremes there are roughly $10^{39}$ in ratio, meaning a difference of 39 orders of magnitude.
Approximate Equality and Order of Magnitude Notation
In more advanced physics texts, you will often see a variety of symbols used to indicate approximate relationships, many of them related to orders of magnitude.
One common notation is:
- $A \approx B$ means $A$ is approximately equal to $B$, often within a few percent.
- $A \sim B$ often means $A$ is of the same order of magnitude as $B$. This usually means that $A/B$ lies between about $0.1$ and $10$.
When you read $A \sim B$ in a physics context, it commonly means that $A$ and $B$ are of the same order of magnitude, so their ratio is between roughly $0.1$ and $10$.
Another frequent phrase is “$A$ is negligible compared to $B$.” In order of magnitude terms, this often means that $A$ is several orders of magnitude smaller than $B$. For example, if $A \sim 10^{-6}$ and $B \sim 10^{0}$, then $A/B \sim 10^{-6}$, so $A$ may be treated as essentially zero relative to $B$ in many calculations.
Using order of magnitude notation in this way helps physicists simplify equations and focus on the dominant effects in a problem while ignoring contributions that are too small to matter in practice.
Fermi Problems and Back-of-the-Envelope Estimates
A famous type of exercise that relies on orders of magnitude is the Fermi problem, named after physicist Enrico Fermi. These problems ask you to estimate a quantity that seems impossible to calculate directly, such as “How many piano tuners are there in a large city?” or “How many grains of sand are there on a beach?”
To solve such problems, you make simple assumptions and rough estimates for quantities you can imagine, then combine them by multiplication or division. The goal is not to obtain an exact answer but to determine the correct order of magnitude.
For example, imagine you want to estimate the number of air molecules in a typical room. You might estimate:
- the room volume as about $50\ \text{m}^3$,
- the number density of air molecules as roughly $10^{25}$ molecules per cubic meter.
Then the total number of molecules is about
$$
N \sim 50 \times 10^{25} \approx 5 \times 10^{26}.
$$
This suggests that the number of molecules in the room is of order $10^{27}$. This estimate can be made quickly, without detailed data or a calculator, and it is good enough for many reasoning tasks in physics.
Fermi problems train you to think in terms of orders of magnitude and develop a physical intuition for what is reasonable or unreasonable. Being able to judge whether a given answer is in the right range is often more important than having precise digits.
Logarithmic Scales and Orders of Magnitude
Many physical quantities span so many orders of magnitude that it is convenient to represent them on logarithmic scales. On such a scale, equal distances on a graph correspond to equal changes in the exponent, that is, to equal changes in order of magnitude.
For example, in seismology, earthquake strength is often described using a logarithmic scale. An increase of 1 unit on the scale corresponds to about ten times more energy released. Similarly, the astronomical magnitude scale for stars, the decibel scale for sound intensity, and some scales for measuring acidity or alkalinity rely heavily on logarithmic relationships. In all these cases, the important physical differences are in orders of magnitude rather than simple linear factors.
Although the mathematical details of logarithmic scales belong to more advanced discussions, it is useful even at a beginner level to recognize that such scales are designed precisely to handle wide ranges of orders of magnitude in a compact way.
Practical Use of Orders of Magnitude in Problem Solving
In physics problem solving, orders of magnitude are used to simplify and to guide reasoning in several ways:
First, they help you decide which terms in an equation can be ignored. If one term is several orders of magnitude smaller than another, it often has negligible influence on the result. For example, in a calculation of the total mass of the Earth and a small object, the mass of the object is so small compared with the Earth that adding it changes the total by far less than one part in $10^{20}$. For nearly all purposes, you can ignore the small object.
Second, orders of magnitude help you check whether an answer is plausible. If you compute the time it takes for light to reach the Moon and obtain a value of $10^9$ seconds, you can immediately see that something is wrong, because you know the distance to the Moon is about $10^8$ meters and the speed of light is about $10^8\ \text{m/s}$, so the time should be of order $10^0$ seconds.
Third, orders of magnitude make it possible to compare very different physical processes on a common scale. For example, you might compare the energy released in a chemical reaction (often around $10^3$ to $10^6$ joules per kilogram) with that in a nuclear reaction (around $10^{13}$ joules per kilogram). The nuclear process releases about $10^{7}$ times more energy per kilogram, a difference of seven orders of magnitude. This simple comparison immediately shows why nuclear fuels are so much more energy dense than chemical fuels.
Overall, the concept of orders of magnitude is not only a way of expressing very large or very small numbers but also a way of thinking about physics problems in terms of scales, relative sizes, and the dominance of certain effects over others. This way of thinking will appear again and again throughout the study of physics.