Scientific Notation Converter
Convert numbers to and from scientific notation. Enter standard form or scientific notation for instant conversion with examples.
Scientific Notation Converter
Comprehensive Guide: Scientific Notation, Significant Figures & Order of Magnitude
Learn normalized scientific notation, E-notation, significant digit preservation, arithmetic operations, and astronomical orders of magnitude.
1. Standard Form & Scientific Notation Definition
Scientific notation (also called standard form) is a standardized method of writing real numbers that are either extraordinarily large or infinitesimally small. A number is written as:
The term a is the significand (or mantissa), containing all significant figures, and b is the integer exponent indicating order of magnitude.
2. Arithmetic Operations in Scientific Notation
- Multiplication: Multiply significands and add exponents:
(a × 10ᵐ) × (b × 10ⁿ) = (a · b) × 10ᵐ⁺ⁿ. - Division: Divide significands and subtract exponents:
(a × 10ᵐ) ÷ (b × 10ⁿ) = (a / b) × 10ᵐ⁻ⁿ. - Addition & Subtraction: You must first shift the decimal point so both numbers share an identical exponent power before adding or subtracting significands.
3. Graduated Step-by-Step Worked Problems
Problem 1: Multiplication with Scientific Notation
Multiply: (4.5 × 10⁶) × (6.0 × 10⁸)
Step 1: Multiply significands: 4.5 × 6.0 = 27.0.
Step 2: Add exponents: 10⁶⁺⁸ = 10¹⁴.
Step 3: Normalize: 27.0 is not between 1 and 10. Shift decimal left: 27.0 = 2.7 × 10¹.
Step 4: Combine: 2.7 × 10¹ × 10¹⁴ = 2.7 × 10¹⁵.
Result: 2.7 × 10¹⁵.
Problem 2: Addition Requiring Exponent Alignment
Add: (3.2 × 10⁵) + (4.5 × 10⁴)
• Rewrite 4.5 × 10⁴ with exponent 5: 0.45 × 10⁵.
• Add significands: (3.2 + 0.45) × 10⁵ = 3.65 × 10⁵.
Result: 3.65 × 10⁵ (or 365,000).
Problem 3: Astronomical Mass Ratio via Division
Compare the mass of the Sun (1.989 × 10³⁰ kg) to the mass of the Earth (5.972 × 10²⁴ kg):
• Quotient: (1.989 × 10³⁰) ÷ (5.972 × 10²⁴) = (1.989 / 5.972) × 10^(30 − 24).
• 0.333054 × 10⁶ = 3.33054 × 10⁵ ≈ 333,000.
Result: The Sun is approximately 3.33 × 10⁵ times (333,000 times) more massive than Earth.
4. Physical Constants in Physics & Engineering
Modern STEM disciplines rely on scientific notation to express universal physical constants:
- Speed of Light (c):
2.99792458 × 10⁸ m/s - Planck’s Constant (h):
6.62607015 × 10⁻³⁴ J·s - Avogadro’s Number (N_A):
6.02214076 × 10²³ mol⁻¹ - Gravitational Constant (G):
6.67430 × 10⁻¹¹ m³·kg⁻¹·s⁻²
5. Frequently Asked Questions (FAQ)
What does 'E' or 'e' mean on a scientific calculator display?
'E' stands for 'Exponent of 10'. For example, 3.45E+08 is shorthand for 3.45 × 10⁸. It is strictly a formatting display convention, not Euler's mathematical constant e.
What is the difference between scientific and engineering notation?
In scientific notation, the exponent can be any integer, and the significand is strictly 1 ≤ a < 10. In engineering notation, the exponent must be a multiple of 3 (matching metric prefixes like micro 10⁻⁶, milli 10⁻³, kilo 10³, mega 10⁶), and the significand can range from 1 to 999.99.
How do significant figures work in scientific notation?
Scientific notation clarifies ambiguous trailing zeros. The number 400 could have 1, 2, or 3 sig figs. Writing 4.00 × 10² unambiguously specifies exactly 3 significant figures.
How are negative numbers written in scientific notation?
The negative sign is placed in front of the significand: −0.00045 = −4.5 × 10⁻⁴. A negative sign on the exponent indicates a magnitude between 0 and 1, whereas a negative sign on the significand indicates the number is located to the left of zero on the real number line.