Scientific notation writes a number as a × 10ⁿ, where the coefficient a is at least 1 and less than 10 and the exponent n is an integer. So 4,560,000 = 4.56 × 10⁶ and 0.000307 = 3.07 × 10⁻⁴. To multiply or divide, work on the coefficients and the powers of ten separately; to add or subtract, give both numbers the same power of ten first. Then check that the coefficient still sits between 1 and 10.
That last check, normalising, is where most marks are lost, so this guide treats it as its own step. The examples run from plain conversions to a chemistry mole problem and the time light takes to reach Earth from the Sun.
What Scientific Notation Means
a × 10ⁿ with 1 ≤ |a| < 10 and n an integer.n > 0: the number is 10 or more. n < 0: the number is between 0 and 1 (in size). n = 0: the number is already between 1 and 10.The power of ten carries the size and the coefficient carries the digits. That split is the whole point: 6.02 × 10²³ tells you at a glance that the number has 24 digits before the decimal point, which you would have to count by hand in 602,000,000,000,000,000,000,000.
Why insist on a coefficient from 1 to just under 10? It gives every number exactly one written form. Without the rule, 45 × 10², 4.5 × 10³ and 0.45 × 10⁴ would all be legal, and comparing two numbers would mean converting first. With it, the exponent alone tells you the order of magnitude. (Some books call the coefficient the mantissa or significand.)
Negative exponents follow the exponent rules: 10⁻ⁿ = 1/10ⁿ. So 10⁻³ = 1/1000 = 0.001, and 3.07 × 10⁻⁴ is 3.07 divided by 10,000. A negative number keeps its minus sign on the coefficient instead: −5,200 = −5.2 × 10³.
How to Convert a Number to Scientific Notation
- Place the decimal point after the first nonzero digit
That gives the coefficient. In 4,560,000 the first nonzero digit is 4, so the coefficient is 4.56 (trailing zeros that are only placeholders drop off).
- Count how many places the point moved
That count is the size of the exponent.
- Choose the sign
Moved left (the original number was big): positive exponent. Moved right (the original was smaller than 1): negative exponent. A quick sanity check: big numbers get positive exponents, small numbers get negative ones.
Write 4,560,000 and 0.000307 in scientific notation, then write 2.5 × 10⁻³ as an ordinary decimal.
4,560,000: the point moves 6 places left to sit after the 4, so 4.56 × 10⁶. Check: 4.56 × 1,000,000 = 4,560,000.
0.000307: the point moves 4 places right to sit after the 3, so 3.07 × 10⁻⁴. The zeros between the 3 and the 7 stay, because they are real digits.
2.5 × 10⁻³: a negative exponent of 3 means move the point 3 places left, filling with zeros: 0.0025.
4.56 × 10⁶, 3.07 × 10⁻⁴, 0.0025To compare two numbers in scientific notation, compare the exponents first and only look at the coefficients when the exponents match. 3.2 × 10⁻⁵ is bigger than 9.8 × 10⁻⁶, even though 9.8 is bigger than 3.2, because −5 > −6. Written out, they are 0.000032 and 0.0000098.
The Normalising Step Everyone Forgets
Every operation below can leave a coefficient outside the 1 to 10 range: 15 × 10⁻³ after a multiplication, 0.30 × 10⁻⁸ after a division. Both are equal to the right answer, but neither is in scientific notation, and many teachers and mark schemes won't accept them. The fix is one shift, and the rule is that the value must not change.
15 × 10⁻³ = 1.5 × 10⁻²Coefficient below 1: move the decimal point one place right and subtract 1 from the exponent.
0.30 × 10⁻⁸ = 3.0 × 10⁻⁹Repeat if one shift is not enough.
Write the coefficient change as a factor: 15 = 1.5 × 10¹. Then 15 × 10⁻³ = 1.5 × 10¹ × 10⁻³ = 1.5 × 10⁻². Making the coefficient 10 times smaller has to be paid for by making the power of ten 10 times bigger. If you remember only that the product of the two parts stays fixed, you can never shift the wrong way.
How to Multiply and Divide in Scientific Notation
Multiplication and division are the easy operations, because the commutative property lets you regroup: (a × 10ᵐ)(b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ and (a × 10ᵐ) / (b × 10ⁿ) = (a / b) × 10ᵐ⁻ⁿ. The exponent parts are the product and quotient rules for powers.
Find (3.0 × 10⁴)(5.0 × 10⁻⁷).
Coefficients: 3.0 × 5.0 = 15.
Exponents: 4 + (−7) = −3, so the raw result is 15 × 10⁻³.
Normalise: 15 is not below 10, so shift left and add 1: 1.5 × 10⁻².
Check in decimals: 30,000 × 0.0000005 = 0.015, and 1.5 × 10⁻² = 0.015.
1.5 × 10⁻²Find (2.4 × 10⁻³) / (8.0 × 10⁵).
Coefficients: 2.4 / 8.0 = 0.30.
Exponents: subtract the bottom from the top, −3 − 5 = −8, so the raw result is 0.30 × 10⁻⁸.
Normalise: 0.30 is below 1, so shift right and subtract 1: 3.0 × 10⁻⁹.
Size check: a tiny number divided by a big one should be very tiny, and 10⁻⁹ is.
3.0 × 10⁻⁹Powers and roots use the same idea. (2 × 10³)² = 2² × 10⁶ = 4 × 10⁶. For a square root, make the exponent even first: √(4.9 × 10⁷) = √(49 × 10⁶) = 7 × 10³. Taking √4.9 and √10⁷ separately works too, but it leaves you with 10³·⁵ and an awkward decimal.
How to Add and Subtract in Scientific Notation
Adding is different, and this is where the rules from multiplication cause trouble. You can only add coefficients that are counting the same thing, the same way you can add 4 hundreds and 3 hundreds but not 4 hundreds and 3 tens without converting. So: rewrite one number to match the other's power of ten, add or subtract the coefficients, keep the power, then normalise.
Find 4.2 × 10⁵ + 3.1 × 10⁴ and 6.0 × 10⁻³ − 5.5 × 10⁻³.
Match powers: 3.1 × 10⁴ = 0.31 × 10⁵ (coefficient shifted left, exponent up 1, value unchanged).
Add: 4.2 + 0.31 = 4.51, so the sum is 4.51 × 10⁵. Already normalised.
The subtraction already has matching powers: 6.0 − 5.5 = 0.5, giving 0.5 × 10⁻³.
Normalise: 0.5 is below 1, so 5 × 10⁻⁴.
Check in decimals: 420,000 + 31,000 = 451,000 and 0.0060 − 0.0055 = 0.0005.
4.51 × 10⁵ and 5 × 10⁻⁴Convert the smaller number up to the larger one's power, as above. The bigger number then keeps a normal coefficient and the sum usually needs no further shift.
Adding the exponents when adding the numbers: 4.2 × 10⁵ + 3.1 × 10⁴ = 7.3 × 10⁹. That is about 16,000 times too big. It happens because "add the exponents" is drilled for multiplication and the two operations look alike on paper. Ask what each operation does: multiplying two numbers combines their sizes, so the exponents add; adding two numbers barely changes the size of the bigger one, so the answer's exponent stays near the larger exponent.
Scientific Notation in Chemistry and Physics
Science is where scientific notation earns its keep, because the numbers are huge or tiny and carry units. Two constants come up constantly. The Avogadro constant is exactly 6.02214076 × 10²³ mol⁻¹ and the speed of light is exactly 299,792,458 m/s (NIST CODATA fundamental constants). The examples round them to 4 significant figures: 6.022 × 10²³ mol⁻¹ and 2.998 × 10⁸ m/s.
How many water molecules are in 0.250 mol of water?
Number of particles = moles × Avogadro constant: N = 0.250 mol × 6.022 × 10²³ mol⁻¹. The mol units cancel, leaving a pure count.
Write 0.250 as 2.50 × 10⁻¹. Coefficients: 2.50 × 6.022 = 15.055. Exponents: −1 + 23 = 22. Raw result: 15.055 × 10²².
Normalise: 1.5055 × 10²³.
Round to 3 significant figures, since 0.250 has 3: 1.51 × 10²³ molecules.
Size check: a quarter of a mole should be about a quarter of 6 × 10²³, which is 1.5 × 10²³.
N = 1.51 × 10²³ moleculesThe average Earth to Sun distance is one astronomical unit, defined as exactly 149,597,870,700 m (IAU 2012 Resolution B2, as published by the Paris Observatory). How long does light take to cover it?
Round the distance to 4 significant figures: d = 1.496 × 10¹¹ m. Use c = 2.998 × 10⁸ m/s.
Time = distance ÷ speed: t = (1.496 × 10¹¹ m) / (2.998 × 10⁸ m/s). The metres cancel and seconds remain.
Coefficients: 1.496 / 2.998 = 0.4990. Exponents: 11 − 8 = 3. Raw result: 0.4990 × 10³ s.
Normalise: 4.990 × 10² s, which is 499.0 s. Divide by 60 for minutes: 499.0 / 60 = 8.317, about 8.3 minutes.
t = 4.990 × 10² s ≈ 8.3 minPhysics problems often hand you a prefix instead of a power. The SI prefixes are powers of ten in disguise (NIST metric SI prefixes), so a wavelength of 450 nm is 450 × 10⁻⁹ m = 4.50 × 10⁻⁷ m. Engineering notation is the close cousin that only uses exponents divisible by 3, matching the prefixes: 450 × 10⁻⁹ m is already in engineering form.
| Prefix | Symbol | Power of ten | Example |
|---|---|---|---|
| giga | G | 10⁹ | 2.4 GHz = 2.4 × 10⁹ Hz |
| mega | M | 10⁶ | 3.5 MJ = 3.5 × 10⁶ J |
| kilo | k | 10³ | 12 km = 1.2 × 10⁴ m |
| milli | m | 10⁻³ | 25 mL = 2.5 × 10⁻² L |
| micro | µ | 10⁻⁶ | 40 µs = 4.0 × 10⁻⁵ s |
| nano | n | 10⁻⁹ | 450 nm = 4.50 × 10⁻⁷ m |
For more constant-heavy algebra with units, the kinematic equations guide and the chemical equations guide use the same unit-cancelling habit.
Significant Figures in Scientific Notation
Every digit in the coefficient is significant, and that is the second reason scientists use the notation. Written as 1,500, a measurement could have 2, 3 or 4 significant figures. Written as 1.5 × 10³, 1.50 × 10³ or 1.500 × 10³, there is no doubt. So don't drop a trailing zero from a coefficient: 1.50 × 10³ and 1.5 × 10³ are equal in value but say different things about the measurement.
For the rounding in a calculation, the usual rule pair is: multiply or divide, keep the fewest significant figures of the inputs (Example 5 kept 3); add or subtract, keep the fewest decimal places once the numbers share a power of ten (in Example 4, 4.2 + 0.31 gives 4.51, which strict rounding to one decimal place would report as 4.5 × 10⁵). Follow whichever convention your course uses, and round only at the end, not after each step.
Scientific Notation on a Calculator
Most scientific and graphing calculators have an EE or EXP key that means "times ten to the". To enter 6.022 × 10²³, press 6.022, EE, 23. The display, and spreadsheets, show results in E notation: 6.022E23 means 6.022 × 10²³, and 3E-9 means 3 × 10⁻⁹. Spreadsheets pad the exponent, so 3.25E+08 is 3.25 × 10⁸.
Typing 6.022 × 10 EE 23. The EE key already supplies the 10, so this enters 6.022 × 10 × 10²³ = 6.022 × 10²⁴, ten times too big. The opposite slip happens without EE: typing 1.496 × 10 ^ 11 ÷ 2.998 × 10 ^ 8 divides only by 2.998 and then multiplies by 10⁸, giving about 4.99 × 10¹⁸ instead of 499. Put brackets around each number in scientific notation, or use EE.
How to Check Your Answer
- Is it normalised?
The coefficient must be at least 1 and less than 10. If not, shift it and adjust the exponent the opposite way.
- Estimate with powers of ten
Round each coefficient to one digit and redo the problem in your head. For Example 6,
(1.5 × 10¹¹) / (3 × 10⁸) = 0.5 × 10³ = 500, close to 499. - Run the units
In science problems, cancel the units the same way you combine the powers. If metres divided by metres per second does not leave seconds, the setup is wrong before any arithmetic.
- Convert back once
For short numbers, write the answer as a decimal and see whether it matches a decimal calculation, as in Examples 2 and 4.
If you've worked a problem by hand and the answer disagrees with the back of the book, the Solver AI app on iPhone and iPad can scan the printed or handwritten problem and show a step-by-step solution, so you can find the step where your exponent changed. Use it after your own attempt, and on tests follow your exam's calculator rules.
Practice Problems
Write every answer in normalised scientific notation.
- Write
0.0000082in scientific notation. - Find
(6.0 × 10⁸)(4.0 × 10⁻³). - Find
(1.2 × 10⁻⁴) / (4.0 × 10⁻⁹). - Find
7.5 × 10⁻³ + 2.0 × 10⁻⁴. - Using atomic weights H = 1.008 and O = 15.999 (CIAAW abridged standard atomic weights), water's molar mass is
18.015 g/mol. Find the mass of one water molecule in kilograms, to 3 significant figures.
1. 8.2 × 10⁻⁶ (point moves 6 places right).
2. 24 × 10⁵ = 2.4 × 10⁶.
3. 0.30 × 10⁵ = 3.0 × 10⁴ (the exponent is −4 − (−9) = 5).
4. 7.5 × 10⁻³ + 0.20 × 10⁻³ = 7.7 × 10⁻³.
5. Mass per molecule = 18.015 g/mol ÷ (6.022 × 10²³ mol⁻¹) = 2.9915 × 10⁻²³ g, or 2.99 × 10⁻²³ g. Since 1 g = 10⁻³ kg, that is 2.99 × 10⁻²⁶ kg.