Math
Order of Magnitude Calculator
Enter any number, including scientific notation such as 6.02e23 or 6.02 × 10^23, to find its order of magnitude: the exponent of ten in its scientific form.
The result also shows the two powers of ten the number sits between, the log-10 position, and the nearest power of ten under the rounding convention. Switch to compare mode to find how many orders of magnitude separate two quantities.
Any real number. Scientific notation is accepted as 6.02e23 or 6.02 × 10^23.
Try an example
Result
Order of magnitude
—
- Scientific notation
- —
- Between powers of ten
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- log10 of the size
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- Nearest power of ten
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- Comparison
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Study path
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What the order of magnitude calculator solves
The order of The length or size of a vector, ignoring which way it points. of a number is the power of ten that describes its size. Write the number in Writing a number as one digit, a decimal part, and a power of ten., m × 10^n with 1 ≤ m < 10; the The small raised number saying how many times to multiply the base by itself. n is the order of magnitude. Avogadro's number 6.02 × 10^23 has order 23, a proton mass 1.67 × 10^-27 kg has order -27, and 250 = 2.5 × 10^2 has order 2. Two quantities are 'an order of magnitude apart' when their exponents differ by 1, meaning roughly a factor of ten.
| Number | Scientific notation | Order of magnitude |
|---|---|---|
| 7 | 7 × 10^0 | 0 |
| 250 | 2.5 × 10^2 | 2 |
| 0.0034 | 3.4 × 10^-3 | -3 |
| 6.02 × 10^23 | 6.02 × 10^23 | 23 |
| 1,000,000 | 1 × 10^6 | 6 |
| 999,999 | 9.99999 × 10^5 | 5 |
Floor convention vs nearest power of ten
Most classes use the scientific-notation exponent, which is the floor of log10. Some physics estimates instead round to the nearest power of ten on a log scale: because √10 ≈ 3.16 is the halfway point between 10^n and 10^(n+1) on a log scale, a mantissa above 3.16 rounds up. Under that convention 6.02 × 10^23 rounds to 10^24. The calculator reports both and warns when they differ.
Comparing two numbers
Subtract the orders: the Sun's mass (2 × 10^30 kg) and Earth's mass (6 × 10^24 kg) differ by 30 - 24 = 6 orders of magnitude, about a million times. The exact log-scale gap, log10(a/b), is shown too, since 2/6 shaves a bit off the round figure (the true ratio is 3.3 × 10^5, so 5.5 orders).
How to use it
- Type the number in ordinary or scientific notation (6.02e23 and 6.02 × 10^23 both work).
- Read the order of magnitude, the scientific notation, and the two powers of ten the number sits between.
- Choose Compare two numbers and enter a second value to see the difference in orders and the exact ratio.
- Check the warning if the mantissa is above 3.16; your course may want the rounded-up order.
How to read the answer
The order of magnitude n means the number is at least 10^n and less than 10^(n+1). A larger order means a bigger number, and each step of 1 in the order is a factor of ten. In compare mode, a difference of k orders means one number is roughly 10^k times the other; the exact log-scale gap refines that to a decimal. Zero has no order of magnitude because log10(0) is undefined.
Common mistakes and edge cases
- Counting digits instead of taking the exponent: 250 has three digits but order of magnitude 2.
- Forgetting the negative exponent for small numbers: 0.0034 has order -3, not 3.
- Treating 'one order of magnitude' as a difference of 10 rather than a factor of 10.
- Mixing conventions: the floor exponent of 6.02 × 10^23 is 23, but the nearest power of ten is 10^24. State which one you are using.
- Asking for the order of magnitude of 0; it is undefined.
Worked examples
Avogadro's number
6.02 × 10^23 has order 23.
Order of magnitude
23
Small number
0.0034 = 3.4 × 10^-3.
Order of magnitude
-3
Just below a power of ten
999,999 is still order 5.
Order of magnitude
5
Negative number
The sign is ignored: -250 has order 2.
Order of magnitude
2
Compare Sun and Earth masses
2e30 vs 6e24: 6 orders apart.
Order of magnitude
30
Compare proton and electron masses
1.67e-27 vs 9.11e-31: 4 orders.
Order of magnitude
-27
Zero
Undefined.
Order of magnitude
Error
Frequently asked questions
What is an order of magnitude?+
The power of ten that describes a number's size: the exponent n when the number is written as m × 10^n with 1 ≤ m < 10. It is a rough scale, so 250 and 900 share order of magnitude 2.
How do I find the order of magnitude of a number?+
Write it in scientific notation and read the exponent, or compute ⌊log10 |x|⌋. For 45,000 = 4.5 × 10^4 the order is 4.
What does 'two orders of magnitude' mean?+
A factor of about 100 (10^2). Three orders of magnitude is about 1,000 times, and so on.
Is the order of magnitude of 6.02 × 10^23 equal to 23 or 24?+
By the scientific-notation exponent it is 23. Under the nearest-power-of-ten convention (rounding log10 = 23.78) it is 24. Textbooks usually use 23; some physics estimation problems use 24.
What is the order of magnitude of a number less than 1?+
Negative. 0.05 = 5 × 10^-2 has order -2; 0.0034 has order -3.
Does zero have an order of magnitude?+
No. log10(0) is undefined and 0 is not between any two powers of ten.
How many orders of magnitude apart are two numbers?+
Subtract their exponents, or compute log10 of the ratio. 5 × 10^8 and 2 × 10^5 are 8 - 5 = 3 orders apart; the exact log gap is log10(2500) ≈ 3.4.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026