How to write a number in scientific notation
Scientific notation writes a number as a × 10n. The coefficient a runs from 1 up to, but not including, 10. The exponent n is a whole number, and it counts how far the decimal point moved.
A big number: 299,792,458
Load sample fills the box with this one, the speed of light in metres per second.
- First non-zero digit: 2.
- Put the point after the 2: 2.99792458.
- The point started after the final 8 and moved 8 places left.
- Left hops give a positive exponent: 2.99792458 × 108.
A small number: 0.00045
The same steps run the other way once the point has to travel right.
- First non-zero digit: 4.
- Put the point after the 4: 4.5.
- The point moved 4 places right.
- Right hops give a negative exponent: 4.5 × 10−4.
Left and right are easy to mix up under exam pressure, so check the size instead. A number of 10 or more takes a positive exponent. A number between 0 and 1 takes a negative one. Anything from 1 to 9.99 takes 100.
A negative exponent never makes the number negative. 4.5 × 10−4 is a small positive number. Only a minus in front of the coefficient, as in −4.5 × 10−4, puts the number below zero.
To convert back to a decimal, reverse the walk. A positive exponent moves the point right, and zeros fill any empty places. 6.02 × 1023 uses up its two decimals, then needs 21 zeros.
One number, five spellings
Each row below holds the same value. The difference is who expects to read which.
| Form | Written as | Where you meet this form |
|---|---|---|
| Plain decimal | 125,000 | Invoices, reports, everyday writing |
| Scientific | 1.25 × 105 | Textbooks, journals, exam papers |
| E notation | 1.25e+5 | Calculators, spreadsheets, code, CSV exports |
| Engineering | 125 × 103 | Datasheets and circuit work, since the exponent is always a multiple of 3 |
| SI prefix | 125 k | Units on the job: 125 kW, 125 km |
Watch the phrase "standard form". In UK schools, standard form means scientific notation, so 1.25 × 105 is the expected answer. In many US classrooms, standard form means the plain number, 125,000. Check your textbook's own example before you answer a question worded this way.
You will also see the coefficient called the mantissa. Plenty of calculators and websites use the word, but mantissa has an older job in maths as the fractional part of a logarithm. Teachers usually prefer coefficient, and computer scientists say significand. This page says coefficient.
What 1.23E+05 means on a screen
The E stands for "times ten to the power of". The letter has no link to Euler's number, 2.718. So 1.23E+05 is 123,000, and 1.23E-05 is 0.0000123. Each kind of software switches to this form at a different point:
- Excel. A General cell flips to E notation at 12 digits. The bigger trap is storage. Excel keeps 15 significant digits, so a 16-digit ID typed as a number loses the tail for good: 1234567890123456 turns into 1234567890123450. Paste
1.23457E+15above and you get 1,234,570,000,000,000, because the missing digits no longer exist. Format the column as Text before you import long IDs. - JavaScript. Numbers print in full until 1021, then become
1e+21. Small ones switch at1e-7. - Python. The switch comes at
1e+16and1e-05. Python pads the exponent to two digits, and the box above reads both styles. - Hand calculators. The EE or EXP key already means × 10. Keying 6.02 × 10 EE 23 gives 6.02 × 1024, ten times too big. Type 6.02 EE 23.
- PDFs. Copying a formula often flattens the superscript, so 6.02 × 1023 pastes as "6.02 x 1023". The box above stops and asks instead of multiplying by 1023.
Arithmetic in scientific notation
Multiplying and dividing get easier in this notation. Adding and subtracting get one extra step.
| Operation | Rule | Example |
|---|---|---|
| Multiply | Multiply coefficients, add exponents | (3.00 × 108) × (5.00 × 102) = 15.0 × 1010 = 1.50 × 1011 |
| Divide | Divide coefficients, subtract exponents | (8.0 × 106) ÷ (2.0 × 103) = 4.0 × 103 |
| Add | Match exponents, add coefficients | 3.2 × 104 + 0.15 × 104 = 3.35 × 104 |
| Subtract | Match exponents, subtract coefficients | 5.0 × 106 − 0.30 × 106 = 4.7 × 106 |
Load sample on the Calculate tab runs the top row. Light covers 3.00 × 108 metres a second, and sunlight takes about 500 seconds to reach Earth. The product, 1.50 × 1011 metres, is the distance to the Sun.
Look at the middle of that row. 15.0 × 1010 holds the right value in the wrong shape, because 15.0 sits above 10. Move the point one place left, add 1 to the exponent, and the form is correct. Skipping this step is the most common way to lose a mark on a right answer.
Calculate prints two answers because lab work and pure maths want different ones. 3.2 × 104 + 1.5 × 103 is exactly 3.35 × 104. Treat those as measurements, though, and 3.2 × 104 is only known to the nearest thousand, so the reported sum is 3.4 × 104. If your assignment grades significant figures, copy the second one.
The zeros scientific notation settles
Does 1500 have two significant figures, three or four? The plain number gives no way to tell. A scale reading to the nearest gram and a rough guess both print the same four characters. Scientific notation removes the doubt:
This page has to pick a reading, so trailing zeros in a plain whole number count as placeholders. 1500 becomes 1.5 × 103, and an amber note flags the call. If you measured every digit, set the Significant figures menu to 4, or type 1500. with a final point.
Zeros after a decimal point always stay. 0.00450 keeps its last zero, as 4.50 × 10−3. The leading zeros were never significant, which the significant figures counter breaks down digit by digit. Rounding can also carry: 9.96 to two figures is 1.0 × 101, so the exponent changes along with the digits.
Real measurements to try
Sorted smallest to largest, spanning 55 powers of ten. Press Load on any row to see the full breakdown. The red blood cell width is a typical figure, and the Earth's mass is a measured estimate. The rest are published constants, rounded here.
| Quantity | Value | Try |
|---|---|---|
| Electron mass | 9.109 × 10−31 kg | |
| Electron charge | 1.602 × 10−19 C | |
| Bohr radius of hydrogen | 5.29 × 10−11 m | |
| Red blood cell width | 7 × 10−6 m | |
| Speed of light | 2.998 × 108 m/s | |
| Light year | 9.461 × 1015 m | |
| Avogadro's number | 6.022 × 1023 per mole | |
| Earth's mass | 5.972 × 1024 kg |
Prefixes to read off engineering notation
Each step in engineering notation lands on a named prefix, which is why datasheets use it.
| Power | Prefix | Symbol | In words (US short scale) |
|---|---|---|---|
| 1012 | tera | T | trillion |
| 109 | giga | G | billion |
| 106 | mega | M | million |
| 103 | kilo | k | thousand |
| 10−3 | milli | m | thousandth |
| 10−6 | micro | µ | millionth |
| 10−9 | nano | n | billionth |
| 10−12 | pico | p | trillionth |
What this converter leaves out
- Exponents stop at 9,999 in either direction. The plain number row stops at 1,000 places, and the other rows carry on past that.
- A dot is the only decimal point. "1,5" gets rejected, not guessed at. Commas work as thousands separators only.
- No fractions or expressions. 1/3, sqrt(2) and 2^10 need the scientific calculator first.
- Division stops at 20 significant digits and says so. Other exact answers show their first 40 digits.
- Halves round up, away from zero. Some lab manuals round halves to the nearest even digit, so 2.45 at two figures gives 2.5 here and 2.4 under that rule.
- Base 10 only. For binary or hex, use the number base converter.
- The SI row knows nothing about units. The kilogram is the awkward one: 5.972 × 1024 kg is 5.972 Yg, since the gram carries the prefix.