Scientific Notation Converter
Convert numbers to and from scientific notation.
Understanding Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It's written as a number between 1 and 10 multiplied by a power of 10. This notation is essential in science, engineering, and mathematics when dealing with extreme values.
Format of Scientific Notation
Scientific notation has the form: a × 10n
- a (coefficient): A number between 1 and 10 (can be negative)
- n (exponent): An integer (positive or negative)
Examples:
- 300,000 = 3 × 105
- 0.00456 = 4.56 × 10-3
- 6,020,000,000,000 = 6.02 × 1012
Converting to Scientific Notation
For Large Numbers (Greater than 10)
- Move the decimal point to the left until you have a number between 1 and 10
- Count how many places you moved - this is your positive exponent
- Write in the form a × 10n
Example: 45,000
Move decimal 4 places left: 4.5 × 104
For Small Numbers (Between 0 and 1)
- Move the decimal point to the right until you have a number between 1 and 10
- Count how many places you moved - this is your negative exponent
- Write in the form a × 10-n
Example: 0.00034
Move decimal 4 places right: 3.4 × 10-4
E-Notation (Computer/Calculator Format)
Computers and calculators often display scientific notation using "e" or "E":
- 1.5e8 means 1.5 × 108 = 150,000,000
- 3.2e-5 means 3.2 × 10-5 = 0.000032
- 6.02e23 means 6.02 × 1023 (Avogadro's number)
Real-World Applications
Astronomy and Space
| Quantity | Value | Scientific Notation |
|---|---|---|
| Distance to Sun | 93,000,000 miles | 9.3 × 107 miles |
| Speed of light | 299,792,458 m/s | ≈3.0 × 108 m/s |
| Mass of Earth | 5,972,000,000,000,000,000,000,000 kg | 5.972 × 1024 kg |
| Light year | 9,460,730,000,000,000 meters | 9.46 × 1015 m |
Microscopic and Atomic Scale
| Quantity | Value | Scientific Notation |
|---|---|---|
| Width of DNA strand | 0.0000002 cm | 2 × 10-7 cm |
| Size of atom | 0.0000000001 m | 1 × 10-10 m |
| Mass of electron | 0.00000000000000000000000000000091 kg | 9.1 × 10-31 kg |
| Planck constant | 0.000000000000000000000000000000000662607 J·s | 6.626 × 10-34 J·s |
Chemistry
This is the number of atoms or molecules in one mole of a substance - a fundamental constant in chemistry.
Operations with Scientific Notation
Multiplication
Multiply the coefficients and add the exponents:
(2 × 103) × (3 × 104) = (2 × 3) × 103+4 = 6 × 107
Division
Divide the coefficients and subtract the exponents:
(8 × 106) ÷ (2 × 103) = (8 ÷ 2) × 106-3 = 4 × 103
Addition and Subtraction
Convert to the same exponent, then add/subtract coefficients:
Example: (3 × 104) + (5 × 103)
Convert: (3 × 104) + (0.5 × 104)
Result: 3.5 × 104
Significant Figures in Scientific Notation
Scientific notation makes it easy to indicate precision:
- 2.5 × 103 has 2 significant figures
- 2.50 × 103 has 3 significant figures
- 2.500 × 103 has 4 significant figures
Common Scientific Constants
| Constant | Symbol | Value (Scientific Notation) |
|---|---|---|
| Speed of light | c | 2.998 × 108 m/s |
| Gravitational constant | G | 6.674 × 10-11 N·m²/kg² |
| Planck constant | h | 6.626 × 10-34 J·s |
| Elementary charge | e | 1.602 × 10-19 C |
| Avogadro's number | NA | 6.022 × 1023 mol-1 |
Why Use Scientific Notation?
- Compactness: Easier to write and read very large or small numbers
- Clarity: Shows the order of magnitude immediately
- Precision: Clearly indicates significant figures
- Calculations: Simplifies multiplication and division
- Standardization: Universal format in science and engineering
Quick Reference
Format: a × 10n
- a: between 1 and 10
- n: positive (large) or negative (small)
- E-notation: 1.5e8 = 1.5 × 108
Powers of 10
- 103 = 1,000 (thousand)
- 106 = 1,000,000 (million)
- 109 = billion
- 1012 = trillion
- 10-3 = 0.001 (milli)
- 10-6 = 0.000001 (micro)
- 10-9 = nano