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.
To convert a number to scientific notation: 1) Move the decimal point until you have a coefficient between 1 and 10, 2) Count how many places you moved (this becomes your exponent), 3) If you moved left, exponent is positive; if right, it's negative. Example: 45,000 → move decimal 4 places left → 4.5 × 10⁴. For 0.00067 → move decimal 4 places right → 6.7 × 10⁻⁴. Our scientific notation calculator handles this conversion instantly. For related calculations, use our exponent calculator and decimal converter.
E notation and scientific notation express the same concept differently. Scientific notation uses superscripts (4.5 × 10³), while E notation uses 'E' for "times ten to the power of" (4.5E3 or 4.5e3). Both equal 4,500. E notation is preferred in: programming (Python, JavaScript), calculators, spreadsheets (Excel), databases. Scientific notation is used in: academic papers, textbooks, formal documentation. The Math is Fun guide explains both formats comprehensively. Our calculator displays both formats for convenience. Convert between formats with our number format converter.
To multiply in scientific notation: 1) Multiply the coefficients, 2) Add the exponents, 3) Adjust if coefficient exceeds 10. Example: (3 × 10⁴) × (2 × 10⁵) = (3 × 2) × 10⁽⁴⁺⁵⁾ = 6 × 10⁹. If result is (4 × 10³) × (5 × 10²) = 20 × 10⁵, adjust to 2.0 × 10⁶ (move decimal one place, add 1 to exponent). Division works similarly but subtract exponents: (6 × 10⁸) ÷ (3 × 10⁵) = 2 × 10³. Our calculator performs these operations automatically with step-by-step explanations. Practice exponent rules with our algebra calculator.
Use scientific notation for: very large numbers (astronomy: 1.496 × 10⁸ km for Earth-Sun distance), very small numbers (chemistry: 1.67 × 10⁻²⁷ kg for proton mass), simplifying calculations (easier to multiply 10-power numbers), maintaining significant figures precision, scientific/engineering documentation. Use standard notation for: everyday numbers ($1,234.56), numbers between 0.01 and 10,000, general communication, financial statements, legal documents. According to NIST measurement guidelines, scientific notation ensures clarity in technical contexts. Convert as needed with our standard form calculator.
Adding/subtracting in scientific notation requires matching exponents first: 1) Convert numbers to same exponent, 2) Add/subtract coefficients, 3) Keep the common exponent. Example: (3 × 10⁴) + (5 × 10³) → convert to same power: (3 × 10⁴) + (0.5 × 10⁴) = 3.5 × 10⁴ = 35,000. Alternative: convert both to standard form, calculate, then convert back. Subtraction: (8 × 10⁶) - (2 × 10⁵) → (8 × 10⁶) - (0.2 × 10⁶) = 7.8 × 10⁶. Unlike multiplication/division, you cannot simply add/subtract exponents. Our calculator handles exponent matching automatically. For complex operations, use our general calculator and advanced math calculator.
In scientific notation, all digits in the coefficient are significant figures. 3.45 × 10⁶ has 3 significant figures; 3.450 × 10⁶ has 4. Rules: 1) All non-zero digits are significant, 2) Zeros between non-zero digits are significant (3.05 × 10² has 3 sig figs), 3) Trailing zeros in coefficient are significant (3.00 × 10⁴ has 3 sig figs). When multiplying/dividing, result should have same sig figs as least precise number: (2.3 × 10³) × (4.567 × 10²) = 1.0 × 10⁶ (2 sig figs). For addition/subtraction, align decimal places in standard form first. The Khan Academy sig figs guide provides detailed rules. Calculate with precision using our rounding calculator.
To convert scientific notation to decimal: 1) Identify the exponent, 2) Move decimal point right for positive exponents, left for negative, 3) Add zeros as needed. Examples: 4.5 × 10³ → move decimal 3 places right → 4,500. 6.7 × 10⁻⁴ → move decimal 4 places left → 0.00067. Large positive exponents (10⁶, 10⁹) make big numbers; negative exponents (10⁻³, 10⁻⁶) make small decimals. Our calculator instantly converts both directions. For very large conversions, remember: 10³=thousand, 10⁶=million, 10⁹=billion, 10¹²=trillion. Practice with our decimal calculator and big number calculator for extreme values.
Scientific notation is essential because it: 1) Handles extreme scales efficiently (Planck length 1.6 × 10⁻³⁵ m to observable universe 8.8 × 10²⁶ m), 2) Preserves significant figures accurately during calculations, 3) Simplifies complex computations (multiplying powers of 10 is easier than long strings of zeros), 4) Prevents calculation errors in very large/small numbers, 5) Standardizes scientific communication globally. In chemistry, Avogadro's number (6.022 × 10²³) would be unwieldy in standard form. In physics, speed of light (3 × 10⁸ m/s) is clearer than 300,000,000 m/s. NASA, research labs, and engineering firms use scientific notation universally. Learn more through NASA educational resources. Apply in practical scenarios with our physics calculator and chemistry calculator.