Scientific Notation Calculator

Convert numbers to and from scientific notation

Enter a standard decimal number to convert it into scientific notation.

Other Titles
Understanding Scientific Notation: A Comprehensive Guide
Learn how scientific notation is used to express very large or very small numbers concisely and perform calculations with them.

What is Scientific Notation?

Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal form. It simplifies arithmetic and makes comparing the magnitude of numbers much easier. It is widely used by scientists, mathematicians, and engineers.
The format is always:
a × 10ᵇ
Where:

Conversion Examples

  • **Large Number:** The speed of light is about 299,792,458 m/s. In scientific notation, this is 2.99792458 × 10⁸ m/s.
  • **Small Number:** The mass of an electron is about 0.000000000000000000000000000910938356 kg. In scientific notation, this is 9.10938356 × 10⁻³¹ kg.

Step-by-Step Guide to Using the Calculator

This calculator handles conversions in both directions.
To Convert a Number TO Scientific Notation:
To Convert a Number FROM Scientific Notation:

Manual Conversion Rules

  • **Decimal to Scientific:** Move the decimal point until you have a number between 1 and 10. The number of places you moved the decimal is the exponent `b`. If you moved the decimal to the left, the exponent is positive. If you moved it to the right, the exponent is negative. (e.g., 47,000 -> 4.7, moved 4 places left -> 4.7 x 10⁴).
  • **Scientific to Decimal:** If the exponent `b` is positive, move the decimal point `b` places to the right. If it's negative, move the decimal point `b` places to the left. (e.g., 2.1 x 10⁻³ -> move 3 left -> 0.0021).

Real-World Applications of Scientific Notation

Scientific notation is indispensable in many fields.
Astronomy:
Chemistry and Physics:
Computing:

Practical Examples

  • A biologist measuring the size of a bacterium (0.000002 m) would write it as 2 × 10⁻⁶ m.
  • An economist discussing the national debt ($30 trillion) might write it as $3 × 10¹³ for use in formulas.

Common Misconceptions and E Notation

Misconception: The Exponent is the Number of Zeros
A common error is to think that the exponent b in 10ᵇ is simply the number of zeros. While this is sometimes true (e.g., 10² = 100), it's not a general rule. For example, 5.4 x 10³ is 5400, which has two zeros, not three. The exponent represents how many places the decimal point moves, not the count of zeros.
Understanding E Notation
On calculators and in programming, scientific notation is often displayed as E notation. The 'E' (or 'e') stands for 'exponent' and replaces the '× 10^' part.

Key Takeaways

  • Scientific notation is for very large or very small numbers.
  • The format is a number between 1 and 10, times a power of 10.
  • A positive exponent means a large number; a negative exponent means a small number (less than 1).

Mathematical Derivation and Examples

Converting to scientific notation is an exercise in factoring out powers of 10.
Derivation for 352,000
Derivation for 0.0078

Operations in Scientific Notation

  • **Multiplication:** (a × 10ᵇ) * (c × 10ᵈ) = (a * c) × 10⁽ᵇ⁺ᵈ⁾
  • **Division:** (a × 10ᵇ) / (c × 10ᵈ) = (a / c) × 10⁽ᵇ⁻ᵈ⁾