Big and small numbers made readable
The distance light travels in a year, the mass of an atom, the number of bytes in a data centre: some numbers have so many zeros that they are hard to read and easy to miscount. Scientific notation, also called standard form, writes them as a number between 1 and 10 times a power of 10, so 0.000045 becomes 4.5 × 10⁻⁵.
This scientific notation calculator converts any number into scientific, E and engineering notation, works exactly digit by digit so long numbers keep every figure, turns a × 10ⁿ back into a plain number, and multiplies, divides, adds or subtracts numbers written this way.
How to use it
- To convert, type the number. Commas are fine, and you can type E notation such as 6.02e23.
- Optionally set the significant figures to round to.
- To go back, choose Back to a plain number and enter the number in front and the power of 10.
- To calculate, choose Calculate and type two numbers in any of these forms.
How scientific notation works
number = a × 10ⁿ, with 1 ≤ a < 10
n = places the decimal point moves: left is positive, right is negative
E notation: 4.5e-5 means 4.5 × 10⁻⁵
engineering notation: the power is a multiple of 3 (10³, 10⁶, 10⁻⁶)Multiplying is easy in this form: multiply the front numbers and add the powers. Dividing subtracts the powers. Adding and subtracting first need the powers lined up.
Worked example: 0.000045
- Move the decimal point 5 places to the right to get 4.5
- Moving right gives a negative power: 4.5 × 10⁻⁵
- In E notation: 4.5e-5
- In engineering notation: 45 × 10⁻⁶ (45 micro units)
So 0.000045 is 4.5 × 10⁻⁵. In the other direction, 3.2 × 10⁻⁵ is 0.000032, and 6.02 × 10²³ is 602,000,000,000,000,000,000,000.
Examples worth knowing
| Plain number | Scientific notation | E notation |
|---|---|---|
| 1,500,000 | 1.5 × 10⁶ | 1.5e+6 |
| 0.000045 | 4.5 × 10⁻⁵ | 4.5e-5 |
| 300,000,000 | 3 × 10⁸ | 3e+8 |
| 0.001 | 1 × 10⁻³ | 1e-3 |
Calculators and spreadsheets often show E notation when a number is too long for the screen, so 3e+8 on a display means 3 × 10⁸, or three hundred million.
Significant figures in scientific notation
Scientific notation makes significant figures clear. Writing 1.50 × 10³ says three figures were measured, while 1.5 × 10³ says two, even though both equal 1,500 in plain form. In a plain whole number such as 1,500, trailing zeros are ambiguous, so the calculator counts only the figures it can be sure of.
Related tools
- For powers and roots in a full expression, use the scientific calculator.
- To trim figures first, the rounding calculator handles significant figures.
- Prefixes such as kilo and milli appear on the Ohm’s law calculator.
Frequently asked questions
How do I write a number in scientific notation?
Move the decimal point until one non-zero digit sits in front of it, then count how many places you moved. Moving left gives a positive power of 10 and moving right gives a negative one. So 52,000 becomes 5.2 × 10⁴, and 0.0071 becomes 7.1 × 10⁻³.
What does e mean in a number like 3e8?
The e stands for exponent, meaning times ten to the power of. So 3e8 is 3 × 10⁸, or 300,000,000, and 4.5e-5 is 4.5 × 10⁻⁵, or 0.000045. Calculators, spreadsheets and programming languages use it when a number is too long to show in full.
How do I multiply numbers in scientific notation?
Multiply the front numbers and add the powers of 10, then tidy up so the front number is between 1 and 10. For example, (3 × 10⁸) × (2 × 10⁻³) = 6 × 10⁵. Choose Calculate above, type both numbers and pick multiply.
What is the difference between scientific and engineering notation?
Scientific notation keeps one digit before the decimal point. Engineering notation lets the front number run from 1 to 999 so that the power of 10 is always a multiple of 3, matching prefixes such as kilo, mega and micro. So 45 × 10⁻⁶ is 45 micro units.
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