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Calculate with scientific notation
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About this scientific notation calculator
Scientific notation writes a number as a coefficient times a power of ten, such as 4.5 × 106 for 4,500,000 or 3.2 × 10−4 for 0.00032. This calculator does two jobs. It converts any number you type into scientific notation, E notation and engineering notation, and it adds, subtracts, multiplies and divides numbers in scientific notation, showing each step. It accepts plain decimals, E notation such as 6.02e23, and the written form 3.2×10^-5. Everything runs in your browser.
How to use the scientific notation calculator
- To convert, type a number in the first box and click Convert. You get the scientific form, E notation, engineering notation, the full decimal form when it is not too long, and a plain-language note on how far the decimal point moved.
- To round, pick a number of significant figures. Leave it on As typed to keep exactly the digits you entered.
- To calculate, enter A and B, choose the operation and the rounding, and click Calculate. The steps and the answer in scientific and E notation appear underneath.
What counts as scientific notation
A number is in standard scientific notation when it is written as a × 10n, where the coefficient a is at least 1 and less than 10 (in absolute value) and n is a whole number. So 4.5 × 106 is in scientific notation, but 45 × 105 and 0.45 × 107 are not, even though they have the same value.
| Name | Example for 4,500,000 | Rule |
|---|---|---|
| Scientific notation | 4.5 × 106 | Coefficient from 1 up to (not including) 10 |
| E notation | 4.5e6 | Same value on calculators, spreadsheets and code. The e stands for × 10 to the power |
| Engineering notation | 4.5 × 106 | Exponent is a multiple of 3 (3, 6, 9, −3, −6 and so on) |
| Standard (decimal) form | 4,500,000 | No exponent |
For 0.00032 the three forms differ: scientific is 3.2 × 10−4, but engineering notation is 320 × 10−6 because the exponent has to be a multiple of 3. Engineering notation lines up with metric prefixes such as milli (10−3), micro (10−6), kilo (103) and mega (106).
How to convert a number to scientific notation
- Large numbers. Move the decimal point to the left until one nonzero digit is in front of it. The number of places you moved is a positive exponent. 4,500,000 becomes 4.5 with the point moved 6 places left, so 4.5 × 106.
- Small numbers. Move the decimal point to the right until one nonzero digit is in front of it. The number of places is a negative exponent. 0.00032 becomes 3.2 after moving 4 places right, so 3.2 × 10−4.
- Going back. A positive exponent moves the point right, a negative exponent moves it left, by that many places.
Trailing zeros in a whole number such as 1500 are treated as placeholders, so it becomes 1.5 × 103. If the zeros are measured, write a decimal point (1500.) or type 1.500e3 to keep them. This matches the usual significant figure rules; for more on those, see the Sig Fig Calculator.
Rules for calculating in scientific notation
- Multiply: multiply the coefficients and add the exponents. (3 × 104)(2 × 105) = 6 × 109. If the coefficient ends up 10 or more, shift it: (4 × 103)(5 × 106) = 20 × 109 = 2 × 1010.
- Divide: divide the coefficients and subtract the exponents. (6 × 108) ÷ (3 × 103) = 2 × 105.
- Add or subtract: first rewrite both numbers with the same exponent, then add or subtract the coefficients and keep the exponent. 2.5 × 103 + 4.0 × 102 = 2.5 × 103 + 0.40 × 103 = 2.9 × 103.
Worked example: the default numbers
The converter starts with 0.00032. Moving the decimal point 4 places to the right gives 3.2, so the number is 3.2 × 10−4, or 3.2e-4 in E notation. In engineering notation it is 320 × 10−6. The calculator starts with A = 3.2 × 104 and B = 1.5 × 10−2. Multiplying the coefficients gives 3.2 × 1.5 = 4.8 and adding the exponents gives 4 + (−2) = 2, so the product is 4.8 × 102, which is 480. You can check it: 32,000 × 0.015 = 480.
Real numbers written in scientific notation
| Quantity | Decimal form | Scientific notation |
|---|---|---|
| Speed of light in a vacuum (m/s) | 299,792,458 | 2.99792458 × 108 |
| Avogadro constant (per mole) | 602,214,076,000,000,000,000,000 | 6.02214076 × 1023 |
| Electron mass (kg), about | 0.00000000000000000000000000000091094 | 9.1094 × 10−31 |
Common mistakes
- Leaving the coefficient outside the 1 to 10 range, as in 45 × 105. The value is right, but it is not in standard scientific notation.
- Getting the sign of the exponent backwards. Numbers smaller than 1 have negative exponents.
- Adding exponents when adding numbers. Exponents add only when you multiply.
- Typing 6.02 × 10^23 into a calculator as 6.02 × 10 ^ 23 with keys that follow the order of operations differently. Use the EE or EXP key, or E notation, so the number is treated as one value.
For significant figures, the Sig Fig Calculator counts and rounds them, and the Unit Converter handles metric prefixes. For percent work, try the Percentage Calculator.
Frequently asked questions
Move the decimal point until exactly one nonzero digit is in front of it, then multiply by 10 raised to the number of places you moved. Moving left gives a positive exponent and moving right gives a negative one. 4,500,000 is 4.5 x 10^6 and 0.00032 is 3.2 x 10^-4.
E notation is a plain-text way to write scientific notation. The letter e means "times 10 to the power of", so 6.02e23 is 6.02 x 10^23 and 3.2e-4 is 3.2 x 10^-4. Calculators, spreadsheets and programming languages use it because it needs no superscript.
Multiply the coefficients and add the exponents. (3 x 10^4) x (2 x 10^5) = 6 x 10^9. If the new coefficient is 10 or more, divide it by 10 and add 1 to the exponent, so 20 x 10^9 becomes 2 x 10^10.
Rewrite both numbers with the same power of ten, usually the larger one, then add or subtract the coefficients and keep that power. 2.5 x 10^3 + 4.0 x 10^2 is 2.5 x 10^3 + 0.40 x 10^3 = 2.9 x 10^3. Exponents only add when you multiply.
In scientific notation the coefficient is from 1 up to 10 and the exponent can be any whole number. In engineering notation the exponent is always a multiple of 3, so the coefficient can run from 1 up to 1000. For example, 0.00032 is 3.2 x 10^-4 in scientific notation and 320 x 10^-6 in engineering notation.
It follows significant figure conventions. Trailing zeros in a whole number with no decimal point, such as 1500, are treated as placeholders, giving 1.5 x 10^3. Write 1500. or 1.500e3 to keep them as significant digits, or pick a number of significant figures in the dropdown.