Scientific notation is the shorthand scientists, engineers, and calculators use to write very large and very small numbers without drowning in zeros. Instead of writing 300,000,000, you write 3 × 10⁸. Instead of writing 0.00045, you write 4.5 × 10⁻⁴. This calculator converts any decimal you enter into normalized scientific notation, E-notation, and engineering notation, and it can also multiply two numbers that are already written in scientific form.
The rule is simple: every number x is rewritten as a × 10ⁿ, where the coefficient a is at least 1 but less than 10, and n is an integer. The speed of light, 300,000,000 meters per second, becomes 3 × 10⁸ m/s. The mass of an electron, 0.000000000000000000000000000911 kg, becomes 9.11 × 10⁻³¹ kg. Drop a number into the box and every form updates instantly.
How to Use
- Type the number you want to rewrite into the Enter a Number box. Positive, negative, and decimal values all work. Try 150,000,000 to see a familiar distance-to-Sun value, or 0.00045 for a tiny lab measurement.
- Set Significant Figures to control how many digits appear in the coefficient. The default of 3 matches most chemistry and physics lab reports; use 2 for rough estimates and 4 or more for engineering precision.
- Leave the Operation selector on Convert to turn one decimal into scientific notation. Switch it to Multiply when you already have two numbers in the form a × 10ⁿ and want their product.
- In Multiply mode, fill in Coefficient A, Exponent A, Coefficient B, and Exponent B. For example, to multiply (2 × 10³) by (3 × 10⁴), enter 2 and 3 for A, 3 and 4 for B.
- Read the result card: the top line shows normalized scientific notation, followed by E-notation (the compact form calculators and spreadsheets use, like 3e8), engineering notation (exponents always a multiple of 3, matching SI prefixes such as kilo, mega, and micro), and the original decimal echo.
- Toggle the precision up and down to watch the coefficient round while the exponent stays put — rounding only changes a, never n.
The Normalization Rule
A number is in normalized scientific notation exactly when the absolute value of the coefficient is at least 1 and strictly less than 10. To normalize any number, count how many places you must move the decimal point until exactly one non-zero digit remains to its left:
- Move the decimal to the left for large numbers — each left move adds +1 to the exponent. 150,000,000 becomes 1.5 when you move the point 8 places left, so the result is 1.5 × 10⁸.
- Move the decimal to the right for small numbers — each right move adds −1 to the exponent. 0.00045 becomes 4.5 when you move the point 4 places right, so the result is 4.5 × 10⁻⁴.
- Negative numbers keep their sign on the coefficient. −150,000,000 is −1.5 × 10⁸, not 1.5 × 10⁻⁸.
- Zero is a special case. There is no non-zero digit to anchor the decimal, so by convention 0 is written 0 × 10⁰.
Mathematically, the exponent is the floor of log₁₀ of the absolute value, and the coefficient is the original number divided by 10 raised to that exponent. That is the same transformation the Python format function applies when you ask for the 'e' presentation type, which is why this tool's output agrees with scientific calculators and spreadsheet cells.
Why Three Notations?
- Scientific notation (a × 10ⁿ) is the textbook form used on exams and in journals. It maximizes information in the coefficient.
- E-notation (aen) drops the caret and is the form computers, spreadsheets, and programming languages print.
3.00e8means the same thing as 3.00 × 10⁸. - Engineering notation restricts the exponent to multiples of 3 so it lines up with SI prefixes. 300,000,000 becomes 300 × 10⁶, which reads naturally as "300 mega" — the same "mega" in megabytes, megahertz, and megawatts.
When you compare numbers written in scientific notation, compare the exponents first. A 1 × 10¹² dollar economy is a trillion-dollar economy; even 9.9 × 10¹¹ is smaller, because 11 is less than 12. That rule is what makes order-of-magnitude estimates possible.
Examples
Example 1 — The speed of light. Enter 300,000,000 with 3 significant figures. The calculator moves the decimal eight places left, giving 3.00 × 10⁸, or 3.00e8. Engineering notation reports 300.00 × 10⁶, which you can read as 300 mega-meters per second.
Example 2 — A small measurement. Enter 0.00045 with 2 significant figures. The decimal moves four places right, producing 4.5 × 10⁻⁴, or 4.5e-4. Engineering notation drops the exponent to −6 and shows 450.0 × 10⁻⁶, matching the micro prefix.
Example 3 — Multiplying in scientific form. Set Operation to Multiply, then enter A = 2 × 10³ and B = 3 × 10⁴. Multiply the coefficients (2 × 3 = 6) and add the exponents (3 + 4 = 7), giving 6.0 × 10⁷. If the product of the coefficients were 25 instead, you would re-normalize: 25 × 10⁷ becomes 2.5 × 10⁸.
Example 4 — Money at scale. The US national debt is measured in the tens of trillions of dollars. A figure like $34,000,000,000,000 rewrites as 3.4 × 10¹³ dollars, and a household budget of $60,000 rewrites as 6.0 × 10⁴. Comparing 10¹³ to 10⁴ immediately shows the national debt is about a billion times larger than a typical household income, without counting zeros.
Example 5 — A negative value. Enter −150,000,000 with 3 significant figures. The sign stays on the coefficient: −1.50 × 10⁸. Negative exponents mean "smaller than one," not "negative number" — keep those two ideas separate.
FAQ
How do you convert a decimal to scientific notation?
Move the decimal point until exactly one non-zero digit remains to its left. The number of places you moved it becomes the exponent: left for large numbers (positive exponent), right for small numbers (negative exponent). For 150,000,000, moving the point 8 places left gives 1.5 × 10⁸.
What is the difference between scientific notation and E-notation?
They are the same number written differently. Scientific notation uses a caret and a superscript, like 3 × 10⁸. E-notation replaces that with the letter e, like 3e8, and is what calculators, spreadsheets, and programming languages print. Both mean "multiply the coefficient by 10 raised to the power after the e."
How do you multiply two numbers in scientific notation?
Multiply the coefficients together, then add the exponents. For (2 × 10³) × (3 × 10⁴), the coefficients give 2 × 3 = 6 and the exponents give 3 + 4 = 7, so the answer is 6 × 10⁷. If the new coefficient is 10 or larger, move the decimal one more place and add 1 to the exponent to renormalize.
What does engineering notation mean?
Engineering notation is a variant of scientific notation where the exponent is always a multiple of 3. That lines the coefficient up with SI prefixes: 10³ is kilo, 10⁶ is mega, 10⁻³ is milli, and 10⁻⁶ is micro. For example, 0.00045 becomes 450 × 10⁻⁶, which reads directly as 450 micrometers.
How do you write 0 in scientific notation?
By convention, zero is written 0 × 10⁰. Because there is no non-zero digit to anchor the decimal point, there is no way to normalize it the way other numbers are normalized. Most calculators simply display 0.
Do negative numbers work in scientific notation?
Yes. The sign stays with the coefficient, and the exponent behaves exactly as it does for positive numbers. −150,000,000 is −1.5 × 10⁸. A negative exponent (like 10⁻⁴) means the number is smaller than one; do not confuse that with a negative coefficient.
How many significant figures should I use?
In science and engineering, significant figures reflect the precision of your measurement. School labs commonly use 3; published physics constants often use 4 to 6; rough estimates can use 1 or 2. Changing the precision rounds the coefficient but never changes the exponent.
Disclaimer
This scientific notation calculator is provided for informational purposes only and does not constitute mathematical, scientific, engineering, or academic advice. Outputs are derived from the standard normalization rule x = a × 10ⁿ with 1 ≤ |a| < 10; rounding, floating-point edge cases, and zero are handled by the conventions described above. Always double-check your exponent direction — left for large numbers, right for small ones — before submitting homework, lab reports, or engineering calculations.