What is Scientific Notation?
Scientific notation is a standardized way of writing numbers that are very large or very small, making them easier to read, understand, and use in calculations. It is expressed as the product of a number between 1 and 10 (the coefficient) and a power of 10 (the exponent). The format is `a × 10ᵇ`, where `1 ≤ |a| < 10` and `b` is an integer. For example, the number 5,972,000,000,000,000,000,000,000 kg (the mass of the Earth) is much simpler to write as `5.972 × 10²⁴` kg. Similarly, a very small number like 0.0000000001 meters becomes `1 × 10⁻¹⁰` meters.
Arithmetic in Scientific Notation
- Addition and Subtraction: To add or subtract numbers in scientific notation, their exponents must be the same. If they aren't, you must adjust one of the numbers. For example, to add `(2 × 10³)` and `(3 × 10²)`, you would rewrite the second number as `(0.3 × 10³)`. Then, you can add the coefficients: `(2 + 0.3) × 10³ = 2.3 × 10³`.
- Multiplication: Multiply the coefficients and add the exponents. For example, `(2 × 10⁵) × (3 × 10⁴) = (2 × 3) × 10⁵⁺⁴ = 6 × 10⁹`.
- Division: Divide the coefficients and subtract the exponents. For example, `(8 × 10⁷) / (2 × 10³) = (8 / 2) × 10⁷⁻³ = 4 × 10⁴`.
E-Notation and Engineering Notation
Besides standard scientific notation, there are two other common forms:
- E-Notation: This is a format used by calculators and programming languages to represent scientific notation in plain text. The 'e' stands for "exponent." For example, `5.972 × 10²⁴` is written as `5.972e+24` or `5.972E24`.
- Engineering Notation: This is a variation where the exponent is always a multiple of 3 (e.g., 3, 6, -3, -9). This aligns with common metric prefixes like kilo (10³), mega (10⁶), milli (10⁻³), and micro (10⁻⁶). The coefficient in this case will be between 1 and 1000.
Why is Scientific Notation So Important?
This notation is not just for convenience; it's a fundamental tool in many fields:
- Science and Engineering: From the vast distances in astronomy to the minuscule sizes in quantum physics, scientific notation is essential for expressing measurements clearly. It's the language of scale.
- Mathematics: It simplifies arithmetic with very large or small numbers. Multiplying `(2 × 10⁵)` by `(3 × 10⁴)` is much easier than multiplying 200,000 by 30,000. It's an application of the rules you can practice with our Exponent Calculator.
- Computing: Computers often use a form of scientific notation called floating-point representation to store and calculate with a wide range of numbers.
For more information and practice problems, the Khan Academy has excellent tutorials on scientific notation.