Convert numbers to scientific notation, E-notation, and engineering notation. Perform calculations with very large or very small numbers using professional scientific notation tools.
Scientific notation is a mathematical method for expressing very large or very small numbers in a compact, standardized format. It's extensively used in science, engineering, and mathematics to simplify calculations and make numbers more manageable. The format consists of a coefficient (between 1 and 10) multiplied by 10 raised to an integer power.
This notation system is essential for fields dealing with extreme values like astronomy (distances to stars), chemistry (molecular masses), physics (atomic scales), and engineering (precision measurements). Understanding scientific notation enables you to work efficiently with numbers ranging from subatomic particles to cosmic distances.
Where: 1 ≤ |a| < 10 and n is an integer
| Standard Form | Scientific Notation | E-notation | Engineering Notation |
|---|---|---|---|
| 5 | 5.0 × 10⁰ | 5.0E0 | 5.0 × 10⁰ |
| 700 | 7.0 × 10² | 7.0E2 | 700 × 10⁰ |
| 1,000,000 | 1.0 × 10⁶ | 1.0E6 | 1.0 × 10⁶ |
| 0.0004212 | 4.212 × 10⁻⁴ | 4.212E-4 | 421.2 × 10⁻⁶ |
| -5,000,000,000 | -5.0 × 10⁹ | -5.0E9 | -5.0 × 10⁹ |
| 0.000000123 | 1.23 × 10⁻⁷ | 1.23E-7 | 123 × 10⁻⁹ |
| 6,020,000,000,000 | 6.02 × 10¹² | 6.02E12 | 6.02 × 10¹² |
Example: 1.23E-4 = 1.23 × 10⁻⁴
Exponents are multiples of 3 (0, ±3, ±6, ±9, ...)
Large Numbers (≥ 10): Move the decimal point left until you have one non-zero digit before it. Count the moves - this becomes your positive exponent.
Small Numbers (< 1): Move the decimal point right until you have one non-zero digit before it. Count the moves - this becomes your negative exponent.
Numbers between 1 and 10: Already in proper form with exponent 0.
Astronomy: Distance to Proxima Centauri = 4.24 × 10¹³ km. Without scientific notation, this would be 42,400,000,000,000 km.
Chemistry: Avogadro's number = 6.022 × 10²³ particles/mol. Essential for calculating molecular quantities.
Physics: Planck's constant = 6.626 × 10⁻³⁴ J·s. Critical for quantum mechanics calculations.
Biology: Size of a virus = 1.0 × 10⁻⁷ m. Much clearer than 0.0000001 meters.
Technology: Computer processing speeds measured in nanoseconds (10⁻⁹ seconds) or clock frequencies in gigahertz (10⁹ Hz).
Precision: Scientific notation clearly shows significant figures. In 2.30 × 10⁵, there are three significant figures, including the trailing zero.
Rounding: When performing calculations, maintain appropriate precision based on the least precise measurement in your data.
Leading Zeros: Never count as significant. In 0.00456, only 4, 5, and 6 are significant (written as 4.56 × 10⁻³).
Coefficient Range: The coefficient must be between 1 and 10. Writing 12.3 × 10⁴ is incorrect; it should be 1.23 × 10⁵.
Exponent Signs: Remember that positive exponents mean large numbers, negative exponents mean small numbers (less than 1).
Order of Operations: In calculations, follow proper order: exponents first, then multiplication/division, finally addition/subtraction.