Order of Magnitude and Scientific Notation Calculator

Q: Is the order of magnitude of 8000 three or four?

Both, under the two definitions in use, and this is worth saying out loud rather than guessing. Written in scientific notation, 8000 is 8 × 10³, so the exponent — the floor of the logarithm — is 3. But 8000 is much nearer 10⁴ than 10³ on a logarithmic scale, so the nearest power of ten is 4. The split happens whenever the mantissa exceeds √10 = 3.1623. If you are writing the number down, use 3, because that is the exponent. If you are making an order-of-magnitude estimate or saying two quantities are of the same order, use 4, because that is the comparison being made.

Q: What is engineering notation, and when should I use it?

Scientific notation with the exponent restricted to multiples of three, so that it maps directly onto an SI prefix and the mantissa runs from 1 to 1000. 0.000 047 F is 47 × 10−⁶ F, which is 47 microfarads. Use it wherever the prefixes are how people speak — electronics, data rates, component values — and avoid it elsewhere, because the SI's prefix ladder moves in thousands and much of physics does not. The speed of light in engineering notation is 299.8 × 10⁶ m s−¹, or 299.8 megametres per second, which is correct and unreadable.

Q: How many SI prefixes are there, and what are the newest?

Twenty-four, running from quecto at 10−³⁰ to quetta at 10³⁰. The four newest are ronna, quetta, ronto and quecto, added by Resolution 3 of the 27th CGPM in 2022, which “decides to add to the list of SI prefixes to be used for multiples and submultiples of units the following prefixes” and names exactly those. They are the first new prefixes in over thirty years, so any table that stops at yotta and yocto is simply older than the resolution. Two details for writing them: the micro symbol is the Greek mu, not a lower-case u, and deca is the only prefix with a two-letter symbol, da.

Q: Why is the speed of light an exact number?

Because the metre is defined by fixing it. In the SI as revised in 2019, the speed of light in vacuum is one of the defining constants and is exactly 299 792 458 m s−¹ — the number has no uncertainty, and the metre is whatever length makes it true. That makes it a useful default on a page like this: all nine digits of the mantissa are exact, so the usual question of how many figures a measurement supports does not arise. Contrast the speed of sound in air, which is not a defined constant at all: it depends on temperature, and a figure such as 343 m s−¹ is something a question supplies rather than something that can be looked up as exact.

Order of magnitude has two definitions in use — the exponent in scientific notation, and the nearest power of ten — and they disagree for every number whose mantissa is above 3.16. This page gives both, the engineering form with its SI prefix, and how many decades apart two quantities are.

Last updated 11 Nov 2026

Order of magnitude, mantissa and exponent

Scientific, engineering and the nearest power of ten

The numberAny non-zero number. The default is the speed of light in vacuum in metres per second, which is 299 792 458 exactly — one of the SI’s defining constants, so every digit is a definition rather than a measurement and the number has no uncertainty at all. Zero has no order of magnitude, and the page refuses it rather than returning something made up.

A second number, to compare with the firstUsed only for the comparison rows: the ratio, and how many powers of ten separate the two. The default is a speed of sound in air of 343 m s−¹, which is a figure your question supplies rather than one this page asserts — the speed of sound depends on temperature and is not a defined constant. Put 1 here to read the first number’s own size in decades.

Which definition of order of magnitudeThese genuinely differ and nothing on the web arbitrates it. For 8 × 10³, the floor rule gives 3 because the number is written that way, while the nearest-power rule gives 4 because 8000 is closer to 10⁴ than to 10³. The two part company whenever the mantissa exceeds √10 = 3.1623. Say which you are using, especially in an estimation question where the whole answer is an order of magnitude. One implementation note, since it bites every program that does this: log₁₀(1000) comes back from binary floating point as 2.9999999999999996, so taking the floor of it naively gives 2. This page nudges the logarithm by a part in ten thousand million before flooring, which is right for all sixty-one powers of ten it can reach.

Significant figures to keep in the mantissaScientific notation is the only way to write a number whose precision is unambiguous, which is why NCERT’s rule about trailing zeros has an escape hatch in it: 1200 is two significant figures, and 1.200 × 10³ is four, because “the trailing zero(s) in a number with a decimal point are significant”. Choose the figures your measurement actually supports.

8Example

The speed of light, 299 792 458 m s⁻¹ exactly, with the floor definition and 3 significant figures, compared against a speed of sound of 343 m s⁻¹

One logarithm, and a choice of how to round it

x = m × 10ⁿ, 1 ≤ m < 10, n = ⌊log₁₀|x|⌋ · nearest power of ten = round(log₁₀|x|) · engineering: n′ = 3⌊n/3⌋

m: the mantissa or significand, always at least 1 and less than 10 in scientific notation. Its digits carry all the precision: the figures you write in m are the figures you are claiming, which is why scientific notation is the only unambiguous way to record a measurement
n = ⌊log⌋: the exponent, and the first definition of order of magnitude. It is the number of places the decimal point moves, and it is one less than the count of digits before the point
round(log): the second definition: the power of ten the number is nearest to on a logarithmic scale. It exceeds the first whenever m is above √10 = 3.1623, and it is the rule meant by “the same order of magnitude” and by an order-of-magnitude estimate
n′ = 3⌊n/3⌋: engineering notation, which forces the exponent to a multiple of three so that it maps onto an SI prefix. The mantissa then runs from 1 to 1000, and each step of three decades is one rung of the prefix ladder
log(x/y): the decades between two quantities, and the only honest way to compare things that differ by factors rather than amounts. A ratio of a million is six decades, which is two SI prefixes

Worked example

The speed of light, 299 792 458 m s⁻¹ exactly, with the floor definition and 3 significant figures, compared against a speed of sound of 343 m s⁻¹
TAKE THE LOGARITHM. log₁₀(299 792 458) = 8.476821. Everything on this page is a way of rounding that one number: floor it for the exponent, round it for the nearest power of ten, floor it in steps of three for engineering notation.
THE EXPONENT AND THE MANTISSA. The floor of 8.4768 is 8, so the number is 2.99792458 × 10⁸, and to three significant figures 2.998 × 10⁸ m s−¹. The nine digits of the mantissa here are unusual: the metre is defined by fixing this number, so every one of them is exact by definition rather than measured, and the usual question of how many figures to keep does not arise.
THE TWO DEFINITIONS, WHICH HAPPEN TO AGREE HERE. The nearest power of ten is round(8.4768) = 8 as well, because the mantissa 2.998 is below √10 = 3.1623. Had the number been 8 × 10⁸ instead, the floor rule would still say 8 while the nearest-power rule would say 9 — and that is a disagreement that changes an estimation answer by a factor of ten.
THE ENGINEERING FORM. The largest multiple of three at or below 8 is 6, so the engineering form is 299.79 × 10⁶, which would be 299.8 megametres per second. Nobody writes it that way, and that is the useful lesson: engineering notation exists to land on an SI prefix, which suits electronics and suits almost nothing in mechanics.
AND THE COMPARISON. 299 792 458 / 343 = 874 030, so log₁₀ of the ratio is 5.9415 — light is about six decades faster than sound, a factor of a million. That is the form an order-of-magnitude comparison should take, and it is also a warning about sums: a term six orders of magnitude smaller than another affects the seventh significant figure and can simply be dropped.

The twenty-four SI prefixes, as the SI Brochure now lists them

Four of these are newer than most textbooks: ronna, quetta, ronto and quecto were added by Resolution 3 of the 27th CGPM in 2022, which “decides to add to the list of SI prefixes to be used for multiples and submultiples of units the following prefixes” and lists exactly those four. They are the first new SI prefixes in over thirty years, so a table ending at yotta and yocto is older than the resolution rather than wrong. Two details are worth noting for writing: the micro symbol is the Greek letter mu and not a lower-case u, and deca is the only two-letter symbol in the list. The ladder moves in steps of a thousand above kilo and below milli, with hecto, deca, deci and centi the four survivors from the older decimal scheme — and centi is the one a school paper will use constantly, because of the centimetre.
Name	Symbol	Factor	Name	Symbol	Factor
quetta	Q	10³⁰	deci	d	10⁻¹
ronna	R	10²⁷	centi	c	10⁻²
yotta	Y	10²⁴	milli	m	10⁻³
zetta	Z	10²¹	micro	µ	10⁻⁶
exa	E	10¹⁸	nano	n	10⁻⁹
peta	P	10¹⁵	pico	p	10⁻¹²
tera	T	10¹²	femto	f	10⁻¹⁵
giga	G	10⁹	atto	a	10⁻¹⁸
mega	M	10⁶	zepto	z	10⁻²¹
kilo	k	10³	yocto	y	10⁻²⁴
hecto	h	10²	ronto	r	10⁻²⁷
deca	da	10¹	quecto	q	10⁻³⁰

Where the two definitions of order of magnitude part company

The boundary is the mantissa √10 = 3.1623, because that is where a number stops being closer to its own power of ten and starts being closer to the next one. Above it the two definitions differ by one, which in an order-of-magnitude question is a factor of ten in the answer — so a Fermi estimate should say which rule it used. The last row shows the same thing for a small number and is the case most likely to be got wrong: 0.00008 is 8 × 10−⁵, so the floor rule gives −5, while the nearest power of ten is 10−⁴. Neither is incorrect; they are different questions, and only one of them is the exponent you write down.
Number	Scientific notation	Floor of log₁₀	Round of log₁₀	Agree?
1 000	1 × 10³	3	3	yes
2 000	2 × 10³	3	3	yes — mantissa below 3.1623
3 162	3.162 × 10³	3	3	the boundary: √10 × 10³
3 200	3.2 × 10³	3	4	no
8 000	8 × 10³	3	4	no
9 999	9.999 × 10³	3	4	no
299 792 458	2.998 × 10⁸	8	8	yes
0.000 08	8 × 10−⁵	−5	−4	no

Two definitions, one logarithm, and the prefix ladder that only moves in threes

Scientific notation does two jobs, and the second is the one that matters in a laboratory. The obvious job is compactness. The useful job is that it makes precision unambiguous: NCERT’s rule that “the terminal or trailing zero(s) in a number without a decimal point are not significant” means 1200 carries two significant figures, while 1.200 × 10³ carries four because “the trailing zero(s) in a number with a decimal point are significant”. The digits are identical in ordinary notation and the claim about precision is completely different, and only scientific notation records which claim you are making.

Order of magnitude has two definitions in common use, and they disagree more often than they agree. The first is the exponent in scientific notation, the floor of the base-ten logarithm: 8000 is 8 × 10³, so its order of magnitude is 3. The second is the nearest power of ten, the round of the logarithm: 8000 is much closer to 10⁴, so its order of magnitude is 4. They part company whenever the mantissa exceeds √10 = 3.1623, which is a little over half the numbers you meet. Neither is wrong — the first answers “what exponent do I write?” and the second “which power of ten is this?” — but in an estimation problem, where the answer IS an order of magnitude, the choice moves it by a factor of ten and should be stated.

The nearest-power definition is also what “the same order of magnitude” means. Two quantities are of the same order when their ratio is within a factor of about three, which is to say the logarithm of their ratio is under half a decade. That is the arithmetic behind a Fermi estimate, and it is why the honest way to compare two very different quantities is in decades rather than percentages. The speed of light is not a hundred million per cent faster than sound; it is about six decades faster, a factor of a million.

Engineering notation is the third form, and it exists for the prefix ladder. It forces the exponent to a multiple of three so the number lands on an SI prefix, with the mantissa running from 1 to 1000 rather than 1 to 10. Above kilo and below milli the prefixes move only in thousands, up to the ronna and quetta added by the 27th CGPM in 2022 and down to ronto and quecto. The four survivors of the older decimal scheme — hecto, deca, deci and centi — are the exceptions, and centi is the one a school paper uses constantly because of the centimetre. Engineering notation suits electronics, where every component value is a prefixed number, and suits almost nothing in mechanics: 299.8 megametres per second is correct and nobody has ever written it.

Frequently asked questions

Is the order of magnitude of 8000 three or four?

Both, under the two definitions in use, and this is worth saying out loud rather than guessing. Written in scientific notation, 8000 is 8 × 10³, so the exponent — the floor of the logarithm — is 3. But 8000 is much nearer 10⁴ than 10³ on a logarithmic scale, so the nearest power of ten is 4. The split happens whenever the mantissa exceeds √10 = 3.1623. If you are writing the number down, use 3, because that is the exponent. If you are making an order-of-magnitude estimate or saying two quantities are of the same order, use 4, because that is the comparison being made.

What is engineering notation, and when should I use it?

Scientific notation with the exponent restricted to multiples of three, so that it maps directly onto an SI prefix and the mantissa runs from 1 to 1000. 0.000 047 F is 47 × 10−⁶ F, which is 47 microfarads. Use it wherever the prefixes are how people speak — electronics, data rates, component values — and avoid it elsewhere, because the SI’s prefix ladder moves in thousands and much of physics does not. The speed of light in engineering notation is 299.8 × 10⁶ m s−¹, or 299.8 megametres per second, which is correct and unreadable.

How many SI prefixes are there, and what are the newest?

Twenty-four, running from quecto at 10−³⁰ to quetta at 10³⁰. The four newest are ronna, quetta, ronto and quecto, added by Resolution 3 of the 27th CGPM in 2022, which “decides to add to the list of SI prefixes to be used for multiples and submultiples of units the following prefixes” and names exactly those. They are the first new prefixes in over thirty years, so any table that stops at yotta and yocto is simply older than the resolution. Two details for writing them: the micro symbol is the Greek mu, not a lower-case u, and deca is the only prefix with a two-letter symbol, da.

Why is the speed of light an exact number?

Because the metre is defined by fixing it. In the SI as revised in 2019, the speed of light in vacuum is one of the defining constants and is exactly 299 792 458 m s−¹ — the number has no uncertainty, and the metre is whatever length makes it true. That makes it a useful default on a page like this: all nine digits of the mantissa are exact, so the usual question of how many figures a measurement supports does not arise. Contrast the speed of sound in air, which is not a defined constant at all: it depends on temperature, and a figure such as 343 m s−¹ is something a question supplies rather than something that can be looked up as exact.

Related calculators

References

BIPM, The International System of Units (SI), 9th edition 2019 (the SI Brochure), base-unit and prefix tables read from bipm.org on 11 November 2026. The seven base quantities and units: time (second, s), length (metre, m), mass (kilogram, kg), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol), luminous intensity (candela, cd). The prefix table now spans sixty decades, quecto (q, 10⁻³⁰) to quetta (Q, 10³⁰).
Resolution 3 of the 27th CGPM (2022), On the extension of the range of SI prefixes (bipm.org, read 11 November 2026). It “decides to add to the list of SI prefixes to be used for multiples and submultiples of units the following prefixes”: ronna (R, 10²⁷), ronto (r, 10⁻²⁷), quetta (Q, 10³⁰), quecto (q, 10⁻³⁰). The official version is the French text. These are the first new SI prefixes in over thirty years, so a table printed before 2022 stops at yotta and yocto and is incomplete rather than wrong.
NCERT, Physics: Textbook for Class XI, Part I, Chapter 1, “Units and Measurements” (keph101.pdf, read from ncert.nic.in on 11 November 2026). The source of every significant-figure and rounding rule here, quoted not paraphrased: “All the non-zero digits are significant”; “All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all”; “The terminal or trailing zero(s) in a number without a decimal point are not significant”; “The trailing zero(s) in a number with a decimal point are significant”. Also 1.3.1 (×/÷ keeps the fewest significant figures, +/− the fewest decimal places), 1.3.2 (at a dropped five, “if the preceding digit is even, the insignificant digit is simply dropped” and “if it is odd, the preceding digit is raised by 1”), 1.5 dimensional formulae and 1.6.1 homogeneity. Worth recording what this rationalised edition does NOT contain: no section on errors in measurement and none on combination of errors — that phrase appears once inside an example in 1.3.3, and vernier callipers and the screw gauge appear only in the exercises.
CBSE, Senior School Curriculum 2025-26: Physics (Code 042), Classes XI-XII (cbseacademic.nic.in, read 11 November 2026). Unit I lists “significant figures, Determining the uncertainty in result” and “Dimensions of physical quantities, dimensional analysis and its applications”. The Class XI practical list is where the instrument pages come from: Section A experiment 1 measures “diameter of a small spherical/cylindrical body” and the “internal diameter and depth of a given beaker/calorimeter using Vernier Callipers”, experiment 2 the “diameter of a given wire and thickness of a given sheet using screw gauge”, and Activity 1 is “To make a paper scale of given least count, e.g., 0.2cm, 0.5 cm”. Cited, not endorsed.