What this calculator does
An order of magnitude is the power of ten a number sits on. 4,300 has an order of magnitude of 3, because it lies between 10³ and 10⁴. The purpose is to strip a number down to its rough size, so that two quantities can be compared at a glance without the leading digits getting in the way.
Describing something as two orders of magnitude larger means roughly a hundred times larger, since each order is a factor of ten. This calculator takes the floor of the base-ten logarithm, which is the standard convention and worth being clear about: every number from 1,000 up to just under 10,000 has order 3, no matter where in that span it falls. The order is the power of ten a number sits on, not the power it happens to be closest to.
The formula
The base-ten logarithm of the absolute value is taken and rounded down to the nearest whole number. That floor is the order of magnitude. Dividing the original number by ten raised to that power gives the leading coefficient, always at least 1 and always below 10, which together with the power is the number written in scientific notation.
| Term | Meaning |
|---|---|
| Order of magnitude | The floor of log₁₀ of the absolute value, so the exponent of the power of ten the number sits on. |
| Leading coefficient | What is left after dividing out that power of ten, a value from 1 up to just below 10. |
| Scientific notation | The leading coefficient multiplied by a power of ten, the standard way of writing very large and very small numbers. |
| Floor | Rounding down to the nearest whole number, which is why 9.99 and 1.01 share the same order. |
The inputs explained
| Field | What to enter |
|---|---|
| Number | The number to examine. Negative values are handled by taking the absolute value first, and zero has no order of magnitude because the logarithm of zero is undefined. |
When to use it
Comparing two quantities at a glance
When the exact figures are not the point, the difference in orders is. Saying one thing is three orders of magnitude larger than another conveys the relationship immediately, where two long numbers side by side would not.
Sanity-checking a calculation
A quick order-of-magnitude estimate catches the errors that matter most, the ones where a result comes out a thousand times too large. If the estimate and the computed answer sit on different powers of ten, something has gone wrong.
Writing a number in scientific notation
The order is the exponent, and what remains after dividing it out is the coefficient, so finding the order is the first step of converting any number to scientific notation by hand.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the order of magnitude of numbers at different scales?
The digits 4.3 placed at a range of scales, from thousandths up to billions.
| Number | Order of magnitude | Power of ten it sits on | In scientific notation |
|---|---|---|---|
| 0.004 | -3 | 0.001 | 4.000 × 10^-3 |
| 4.3 | 0 | 1 | 4.300 × 10^0 |
| 43 | 1 | 10 | 4.300 × 10^1 |
| 4,300 | 3 | 1,000 | 4.300 × 10^3 |
| 430,000 | 5 | 100,000 | 4.300 × 10^5 |
| 4,300,000,000 | 9 | 1,000,000,000 | 4.300 × 10^9 |
Questions
Is the order of magnitude rounded up or down?
Down, always. The convention is the floor of the base-ten logarithm, so 9,999 and 1,000 both have order 3. Some informal usage rounds to the nearest power of ten instead, which would put 9,999 at order 4, so it is worth checking which convention a source is using.
What does an order of magnitude larger mean?
Ten times larger. Two orders is a hundred times, three orders is a thousand times, and so on. The phrase is used when the rough scale of a difference matters more than its exact size.
Can a number have a negative order of magnitude?
Yes, and every number below 1 does. 0.004 sits between 10⁻³ and 10⁻², so its order is −3. The same floor rule applies, just on the other side of zero.
What is the order of magnitude of zero?
It has none. The order comes from a logarithm, and the logarithm of zero is undefined, so the question has no answer rather than an answer of 0.
To write a number in scientific notation in full, see the scientific notation converter. For the engineering variant that restricts exponents to multiples of three, see the engineering notation calculator.