Convert, E-Notation & Arithmetic
Last reviewed: April 2026
Scientific notation expresses very large or very small numbers in the form a × 10ⁿ, where a is between 1 and 10 and n is an integer. For example, 93,000,000 (miles to the Sun) = 9.3 × 10⁷, and 0.000001 (one micrometer) = 1 × 10⁻⁶. This calculator converts between standard and scientific notation and performs arithmetic (add, subtract, multiply, divide) in scientific notation.
To multiply: multiply the coefficients and add the exponents. (3 × 10⁴)(2 × 10³) = 6 × 10⁷. To divide: divide the coefficients and subtract the exponents. To add or subtract: convert to the same exponent first. Scientific notation is essential in physics, chemistry, astronomy, and engineering where values span many orders of magnitude. For number base conversions, see our Number Base Converter.
| Standard Form | Scientific Notation | Context |
|---|---|---|
| 0.000001 | 1 × 10⁻⁶ | Micrometer |
| 0.01 | 1 × 10⁻² | Centimeter |
| 300,000,000 | 3 × 10⁸ | Speed of light (m/s) |
| 6,022,000,000,000,000,000,000,000 | 6.022 × 10²³ | Avogadro number |
| 93,000,000 | 9.3 × 10⁷ | Earth-Sun distance (miles) |
Scientific notation is a standardized way of expressing very large or very small numbers as a coefficient between 1 and 10 multiplied by a power of 10. The number 93,000,000 (the approximate distance from Earth to the Sun in miles) becomes 9.3 × 10⁷, and 0.000000001 (one nanometer in meters) becomes 1 × 10⁻⁹. This notation serves three critical purposes: it makes extreme numbers readable, it clearly indicates the precision (significant figures) of a measurement, and it simplifies arithmetic with very large or small numbers. Scientists, engineers, programmers, and mathematicians use scientific notation daily across disciplines from astronomy (distances measured in light-years) to molecular biology (sizes measured in angstroms) to computing (data measured in terabytes and beyond). For related number format conversions, see our Decimal Fraction Converter and Binary Calculator.
To convert a standard number to scientific notation, move the decimal point until you have a number between 1 and 10, then count the positions moved. Moving the decimal left (for large numbers) gives a positive exponent; moving right (for small numbers) gives a negative exponent. For example, 4,560,000 → move decimal 6 places left → 4.56 × 10⁶. For 0.00032 → move decimal 4 places right → 3.2 × 10⁻⁴. To convert back, reverse the process: move the decimal point right for positive exponents and left for negative exponents, filling with zeros as needed. Common errors include forgetting the negative sign on the exponent for small numbers and miscounting decimal positions — off-by-one errors change the value by a factor of 10.
| Operation | Rule | Example |
|---|---|---|
| Multiplication | Multiply coefficients, add exponents | (3 × 10⁴)(2 × 10³) = 6 × 10⁷ |
| Division | Divide coefficients, subtract exponents | (8 × 10⁶)/(4 × 10²) = 2 × 10⁴ |
| Addition | Match exponents first, then add | 3 × 10⁴ + 5 × 10³ = 3.5 × 10⁴ |
| Subtraction | Match exponents first, then subtract | 7 × 10⁵ − 2 × 10⁴ = 6.8 × 10⁵ |
| Exponentiation | Raise coefficient to power, multiply exponents | (2 × 10³)² = 4 × 10⁶ |
Scientific notation inherently communicates measurement precision through significant figures. The number 5.00 × 10³ indicates three significant figures (measurement precise to the ones place: 5,000 ± 0.5), while 5 × 10³ indicates only one significant figure (5,000 ± 500). This distinction is impossible to convey in standard notation — is "5000" measured to one, two, three, or four significant figures? The ambiguity disappears in scientific notation. In laboratory and engineering contexts, reporting the correct number of significant figures prevents overstating measurement precision, which can lead to false confidence in results.
Rules for significant figures in calculations: for multiplication and division, the result should have the same number of significant figures as the input with the fewest significant figures. For addition and subtraction, the result should have the same number of decimal places as the input with the fewest decimal places. These rules exist because calculations cannot create precision that wasn't in the original measurements. A measurement of 3.2 × 10² meters (two significant figures) multiplied by a measurement of 4.178 × 10¹ meters (four significant figures) should be reported as 1.3 × 10⁴ m² (two significant figures), not 1.337 × 10⁴ m². Understanding precision is crucial in scientific work — calculate measurement errors with our Percent Error Calculator.
Engineering notation is a variant of scientific notation where the exponent is always a multiple of 3, aligning with metric prefixes. Instead of 4.7 × 10⁴, engineering notation uses 47 × 10³ (47 kilo-). This system maps directly to the metric system: 10³ = kilo (k), 10⁶ = mega (M), 10⁹ = giga (G), 10¹² = tera (T), 10⁻³ = milli (m), 10⁻⁶ = micro (μ), 10⁻⁹ = nano (n), 10⁻¹² = pico (p). Engineers prefer this notation because real-world measurements are typically expressed in these standard prefixes — a 47 kΩ resistor, a 3.3 GHz processor, a 500 mA current draw. Converting between metric prefixes is straightforward: moving up one prefix (milli to base, base to kilo) divides by 1,000; moving down multiplies by 1,000.
In computing, a hybrid system exists where data storage uses binary prefixes (1 kilobyte = 1,024 bytes in the traditional convention, though the SI standard defines 1 kB = 1,000 bytes). This discrepancy means a "1 TB" hard drive contains approximately 931 GiB (gibibytes) of usable space — a 7% difference that increases with larger units. The IEC introduced unambiguous binary prefixes (kibi-, mebi-, gibi-, tebi-) in 1998, but adoption remains inconsistent. For data-related calculations, see our File Size Calculator and for unit conversions across measurement systems, try our Unit Converter.
Most programming languages represent scientific notation using "E notation" — 6.02e23 means 6.02 × 10²³, and 1.6e-19 means 1.6 × 10⁻¹⁹. This notation appears frequently in scientific computing, financial calculations, and any domain dealing with extreme values. Floating-point arithmetic (the standard way computers handle non-integer numbers) stores all numbers internally in a form similar to scientific notation: a significand (mantissa) multiplied by a power of 2. This representation enables an enormous range of values (roughly 10⁻³⁰⁸ to 10³⁰⁸ for 64-bit doubles) but introduces precision limitations — floating-point numbers have approximately 15–17 significant decimal digits of precision, meaning 0.1 + 0.2 ≠ 0.3 exactly in most programming languages. Understanding these limitations is essential for writing correct numerical code, particularly in financial and scientific applications. Explore number representation with our Roman Numeral Converter and Number Base Converter.
Several scientific constants and measurements are worth memorizing in scientific notation: the speed of light is approximately 3.0 × 10⁸ meters per second, Avogadro's number is 6.022 × 10²³, the charge of an electron is 1.602 × 10⁻¹⁹ coulombs, the mass of an electron is 9.109 × 10⁻³¹ kilograms, and the distance from Earth to the Sun averages about 1.496 × 10⁸ kilometers. Familiarity with these magnitudes helps quickly assess whether calculation results are reasonable — a physics answer yielding 10³⁰ meters for a tabletop experiment signals an error immediately.
Scientific notation becomes essential when numbers are extremely large or small. The distance from Earth to the nearest star (Proxima Centauri) is 39,900,000,000,000 km — much easier to write as 3.99 × 10¹³ km. A hydrogen atom's diameter is 0.000000000106 meters, or 1.06 × 10⁻¹⁰ m. Engineering notation is a variant that restricts exponents to multiples of 3, aligning with metric prefixes: kilo (10³), mega (10⁶), giga (10⁹), nano (10⁻⁹), pico (10⁻¹²). This makes conversions between notation and named units intuitive — 4.7 × 10⁻⁹ farads is simply 4.7 nanofarads. Use our Scientific Calculator for calculations in scientific notation.
→ Double-check with manual calculation. Use the calculator to verify your work, or work the problem by hand first and check against the calculator's result.
→ Pay attention to units and precision. Make sure your inputs use consistent units. The calculator preserves decimal precision — round the final answer to the appropriate number of significant figures for your context.
→ Use for learning, not just answers. Review the formula and step-by-step breakdown to understand the underlying math — this builds intuition for future problems.
→ Try edge cases. Test with very large, very small, or zero values to build intuition about how the formula behaves across different inputs.
See also: Proportion Calculator · Equation Solver · Number Sequence Calculator · Correlation Calculator · Fraction Calculator