Order of Magnitude Calculator
Enter any number to see its order of magnitude - the power of 10 that describes its scale - together with its full scientific notation form and the coefficient. Use the comparison mode to find how many orders of magnitude separate two quantities, which is useful for understanding scale differences in science, engineering, and everyday estimation.
Formula
Worked example
For x = 1,370: log10(1370) = 3.1367..., floor(3.1367) = 3, so n = 3. Coefficient b = 1370 / 10^3 = 1.37. Scientific notation: 1.37 x 10^3. For x = 0.082: log10(0.082) = -1.086, floor(-1.086) = -2, so n = -2. Coefficient b = 0.082 / 10^-2 = 8.2. Scientific notation: 8.2 x 10^-2.
What is an order of magnitude?
An order of magnitude is the approximate scale of a number expressed as a power of 10. It answers the question "how big is this, roughly?" without caring about the precise digits. Two numbers share the same order of magnitude if they sit within the same power-of-10 band, for example both 1,100 and 9,800 have order of magnitude 3 because they both lie between 10^3 (1,000) and 10^4 (10,000). The formal definition is: for any non-zero number x, the order of magnitude n is the integer such that x = b x 10^n, where 1 is less than or equal to the absolute value of b and b is less than 10. Equivalently, n = floor(log10(|x|)), where floor means rounding down to the nearest integer.
How to find the order of magnitude
The quickest manual method is to write the number in scientific notation and read off the exponent. For a whole number, count the digits minus one: 1,370 has four digits, so its order of magnitude is 3. For a decimal less than 1, count the leading zeros after the decimal point and add 1, then negate: 0.082 has one leading zero, giving -1 - 1 = -2. For numbers right at a power-of-10 boundary the distinction can be subtle: 999 has order of magnitude 2 (floor of log10(999) = 2.9997... = 2), while 1,000 has order of magnitude 3 (log10(1000) = 3 exactly). This calculator uses the standard floor method, which is consistent with scientific notation and matches most scientific usage.
Comparing two numbers by order of magnitude
Scientists use order-of-magnitude comparisons constantly because many quantities of interest span enormous ranges. The mass of a proton (~1.67 x 10^-27 kg) and the mass of Earth (~5.97 x 10^24 kg) differ by 51 orders of magnitude - that is, Earth is roughly 10^51 times more massive. Similarly, a human hair (~70 micrometres = 7 x 10^-5 m) is about 7 orders of magnitude wider than a DNA strand (~2 nanometres = 2 x 10^-9 m). Switch this calculator to compare mode, enter any two quantities, and it will report how many powers of 10 separate them and their exact ratio.
Orders of magnitude in science and engineering
Every SI prefix corresponds to exactly one or more orders of magnitude: milli- is 10^-3, micro- is 10^-6, nano- is 10^-9. Engineers use order-of-magnitude estimates - sometimes called Fermi estimates after physicist Enrico Fermi - to sanity-check calculations quickly before committing to detailed analysis. If a precise answer is 3.8 x 10^6 but a rough estimate gives 10^7, the one-order-of-magnitude difference is a red flag worth investigating. Conversely, if the estimate and the answer agree to within one order of magnitude, confidence is high that the calculation is on the right track.
Common quantities by order of magnitude
| Order | Power of 10 | SI Prefix | Example Quantity |
|---|---|---|---|
| -15 | 10^-15 | femto- (f) | Diameter of a proton (~1.7 fm) |
| -12 | 10^-12 | pico- (p) | Width of a DNA base pair (~0.3 nm = 3 x 10^-10 m) |
| -9 | 10^-9 | nano- (n) | Width of a DNA double helix (~2 nm) |
| -6 | 10^-6 | micro- (u) | Typical bacterium length (~1-10 um) |
| -3 | 10^-3 | milli- (m) | Thickness of a credit card (~0.76 mm) |
| 0 | 10^0 = 1 | (none) | Height of a newborn baby (~0.5 m) |
| 3 | 10^3 | kilo- (k) | Length of a marathon (~42 km = 4.2 x 10^4 m) |
| 6 | 10^6 | mega- (M) | Diameter of Earth (~1.27 x 10^7 m) |
| 9 | 10^9 | giga- (G) | Distance from Earth to Moon (~3.84 x 10^8 m) |
| 12 | 10^12 | tera- (T) | Distance from Earth to Sun (~1.5 x 10^11 m) |
| 15 | 10^15 | peta- (P) | One light-year (~9.46 x 10^15 m) |
| 24 | 10^24 | yotta- (Y) | Mass of Earth (~5.97 x 10^24 kg) |
A selection of well-known quantities grouped by their order of magnitude, from the subatomic to the cosmic.
Frequently asked questions
What is the order of magnitude of 0?
Zero has no order of magnitude. The order of magnitude is defined as floor(log10(|x|)), and log10(0) is negative infinity. When you enter 0, this calculator returns no result.
Can the order of magnitude be negative?
Yes. Any number smaller in absolute value than 1 has a negative order of magnitude. For example, 0.082 = 8.2 x 10^-2 has order of magnitude -2, and a proton radius of about 8.5 x 10^-16 m has order of magnitude -16. Negative orders of magnitude describe very small quantities.
Is order of magnitude the same as scientific notation?
They are related but not identical. Scientific notation is the full representation b x 10^n where 1 is at most |b| and |b| is less than 10. The order of magnitude is just the exponent n - the single integer that tells you the scale. So 1.37 x 10^3 is the scientific notation of 1,370, and 3 is its order of magnitude.
What does "two orders of magnitude larger" mean?
It means roughly 100 times larger (10^2 = 100). One order of magnitude is a factor of ~10, two orders is ~100, three orders is ~1,000, and so on. "Roughly" applies because the coefficient b can be anything from 1 to just under 10, so two numbers that are each at order of magnitude 3 could actually differ by nearly a factor of 10.
Why does this calculator use floor instead of rounding to the nearest integer?
The floor method is the standard definition used in scientific notation: a = b x 10^n requires 1 is at most |b| and |b| is less than 10, which forces n = floor(log10(|a|)). Some references discuss a "nearest-power" convention that rounds 3.162... (square root of 10) upward rather than down, but that convention is less common in textbooks and scientific literature. This calculator uses the floor definition to stay consistent with scientific notation.
How many orders of magnitude are there between a nanometre and a metre?
Nine. A nanometre is 10^-9 m and a metre is 10^0 = 1 m, so the difference in orders of magnitude is 0 - (-9) = 9. This means a metre is exactly 10^9 (one billion) nanometres.
What is a Fermi estimate?
A Fermi estimate (named after physicist Enrico Fermi) is an order-of-magnitude approximation made from first principles with little or no data. A classic example: "How many piano tuners are there in Chicago?" You estimate Chicago population (~3 million), fraction owning pianos (~1 in 20 = 150,000 pianos), how often each is tuned (~once a year), how long a tuning takes (~2 hours), and how many hours a tuner works per year (~2,000). That gives roughly 150 tuners - and the real number is indeed in the hundreds. The point is not to be exact but to land within one or two orders of magnitude.