Arithmetic
Scientific Notation Calculator
Convert a number to scientific notation (a × 10ⁿ), expand it back to standard form, or do arithmetic in notation — with the coefficient, exponent, and every step shown live as you type.
Number to convert
a × 10ⁿAny decimal — large, small, or negative. Commas are ignored.
Scientific notation
6.022 × 1023
Coefficient 6.022 · Exponent 23
E-notation
6.022e23
Engineering
602.2e21
Try an example
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The basics
What is scientific notation?
Scientific notation writes any number as a single significant digit before the decimal, times a power of ten. It separates the significant figures from the scale, so a number that would sprawl across the page becomes a tidy pair: a coefficient and an exponent.
a × 10ⁿ, 1 ≤ |a| < 10
The coefficient
The coefficient (also called the mantissa) carries the meaningful digits. It always sits between one and ten, so every number has exactly one scientific form.
6022… → a = 6.022
n = places the point moved
The exponent
The exponent records the scale — how many powers of ten to multiply by. Positive for big numbers, negative for small ones.
moved 23 left → n = 23
Method
How to convert by hand
Three steps turn any number into a × 10ⁿ.
- 1
Find the new decimal spot
Place the decimal point just after the first non-zero digit. That gives a coefficient between 1 and 10.
6022000… → 6.022 - 2
Count the places you moved
Count how many places the point travelled. Left moves give a positive exponent; right moves give a negative one.
moved 23 left → n = 23 - 3
Write coefficient × 10ⁿ
Combine the two: the coefficient times ten raised to that exponent. Drop trailing zeros from the coefficient.
6.022 × 10²³
Worked examples
Arithmetic in scientific notation
The same coefficient-and-exponent split makes multiplication, division, and even addition mechanical. Each tab computes its own working.
Given 3 × 10⁸ × 2 × 10⁶
Rule: Multiply coefficients, add exponents
Combine: (3 × 2) × 10^(8 + 6) = 6 × 10¹⁴
= 6 × 10¹⁴
The rules
Operating on powers of ten
Four rules cover every operation. Multiply and divide act on the exponents directly; add and subtract need a shared exponent first.
(a·10ᵐ)(b·10ⁿ) = ab·10ᵐ⁺ⁿ
Multiplication
Multiply the coefficients and add the exponents. Re-normalise if the product leaves the 1–10 range.
(a·10ᵐ)/(b·10ⁿ) = (a/b)·10ᵐ⁻ⁿ
Division
Divide the coefficients and subtract the exponents — the mirror image of multiplication.
match n, then a ± b
Addition & subtraction
You can only add like powers, so shift one number until both exponents match, then add or subtract the coefficients.
(a·10ⁿ)ᵏ = aᵏ·10ⁿᵏ
Powers & roots
Raise the coefficient to the power and multiply the exponent by it. Roots divide the exponent — make it a whole number first.
Where it shows up
Why it matters
Scientific notation is the working language of any field that deals with extreme scales.
6.022 × 10²³
Chemistry
A mole holds Avogadro's number of particles — a figure with twenty-four digits that notation tames to four.
9.461 × 10¹⁵ m
Astronomy
Distances between stars are measured in light-years; expanded into metres they become unreadable without notation.
9.109 × 10⁻³¹ kg
Physics
Particle masses and atomic radii are tiny negative-exponent numbers that notation keeps precise and legible.
Quick chart
What is each power of ten?
Every common power of ten with its expanded value, everyday name, and SI prefix. Filter by exponent (try 6), by name (try million), or by prefix (try milli).
Power of ten · expanded value · name · SI prefix
| Power | Value | Name | SI prefix |
|---|---|---|---|
| 10¹² | 1,000,000,000,000 | trillion | tera (T) |
| 10⁹ | 1,000,000,000 | billion | giga (G) |
| 10⁶ | 1,000,000 | million | mega (M) |
| 10³ | 1,000 | thousand | kilo (k) |
| 10² | 100 | hundred | hecto (h) |
| 10¹ | 10 | ten | deca (da) |
| 10⁰ | 1 | one | — |
| 10⁻¹ | 0.1 | tenth | deci (d) |
| 10⁻² | 0.01 | hundredth | centi (c) |
| 10⁻³ | 0.001 | thousandth | milli (m) |
| 10⁻⁶ | 0.000001 | millionth | micro (µ) |
| 10⁻⁹ | 0.000000001 | billionth | nano (n) |
| 10⁻¹² | 0.000000000001 | trillionth | pico (p) |
Reference
Famous numbers in notation
A few constants and everyday magnitudes, shown both ways — proof of how much notation compresses.
| Quantity | Scientific notation |
|---|---|
| Avogadro's number (particles per mole) | 6.022 × 10²³ |
| One light-year, in metres | 9.461 × 10¹⁵ |
| Earth–Sun distance, in metres | 1.496 × 10¹¹ |
| Human population (approx.) | 7.5 × 10⁹ |
| Speed of light, in m/s | 3 × 10⁸ |
| One millimetre, in metres | 1 × 10⁻³ |
| One micron (red-blood-cell scale), in metres | 1 × 10⁻⁶ |
| Bohr radius (atom scale), in metres | 5.29 × 10⁻¹¹ |
| Electron mass, in kilograms | 9.109 × 10⁻³¹ |
Worth knowing
Common mistakes to avoid
Most scientific-notation slips come from the same handful of habits. Watch for these.
a must be < 10
Coefficient out of range
Writing 12 × 10³ isn't normalised — shift it to 1.2 × 10⁴. The coefficient always stays between one and ten.
left = +, right = −
Flipping the exponent sign
Moving the decimal left makes the exponent positive; moving right makes it negative. Mixing these up is the most common error.
match before you add
Adding unlike powers
You can't add coefficients until the exponents match. Multiply and divide skip this, but addition and subtraction never do.
10⁰ = 1
Forgetting zero and one
Ten to the zero is one, not zero. And the number zero has no normalised form — it is simply written as 0.
Going deeper
Significant figures & precision
One quiet advantage of notation: it makes precision unambiguous. The digits in the coefficient are the significant figures — no trailing-zero guesswork.
2.50 × 10³ → 3 sig figs
Counting precision
In standard notation, 2,500 could carry two, three, or four significant figures. The coefficient settles it: every digit you write counts.
round a, keep n
Rounding cleanly
To round to a given precision, round the coefficient and leave the exponent alone. The magnitude never changes when you trim digits.
FAQ
Common questions
Scientific notation — also called standard form — writes a number as a coefficient between 1 and 10 multiplied by a power of ten, for example 6.022 × 10²³. It keeps very large and very small numbers compact and makes their magnitude obvious at a glance.
Move the decimal point until exactly one non-zero digit sits to its left, giving a coefficient with 1 ≤ |a| < 10. The number of places you moved is the exponent: moving left makes it positive, moving right makes it negative. Write the result as a × 10ⁿ.
Multiply the coefficients and add the exponents. For (3 × 10⁸) × (2 × 10⁶) you get 6 × 10¹⁴. If the new coefficient lands outside 1–10, shift the decimal and adjust the exponent to re-normalise it.
First make the exponents match by shifting one coefficient, then add or subtract the coefficients and keep the shared power of ten. For 6.5 × 10⁴ + 2 × 10³, rewrite the second as 0.2 × 10⁴, giving 6.7 × 10⁴.
E notation is how calculators and computers display scientific notation, where “E” means “times ten to the power of”. So 1.2E3 is 1.2 × 10³ = 1,200, and 1.2E−3 is 1.2 × 10⁻³ = 0.0012. The number after E is the exponent.
Both use a coefficient times a power of ten. Scientific notation keeps the coefficient between 1 and 10, while engineering notation restricts the exponent to multiples of three (so it lines up with SI prefixes like kilo, mega, milli). 47,000 is 4.7 × 10⁴ in scientific and 47 × 10³ in engineering notation.
A negative exponent means a number smaller than one. The exponent tells you how many places to move the decimal to the left, padding with leading zeros. So 5.29 × 10⁻¹¹ expands to 0.0000000000529.
Yes. “Standard form” is the British and exam-board name for the same idea used in US and scientific contexts as “scientific notation”: a number written as a × 10ⁿ with 1 ≤ a < 10.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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