Scientific Notation
Scientific notation (also called standard index form) is a way of writing very large or very small numbers compactly. Scientists and engineers use it to avoid writing long strings of zeros.
The Format
A number in scientific notation is written as:
a × 10ⁿ where 1 ≤ a < 10 and n is an integer
The part a is called the coefficient or significand; 10ⁿ is the power of 10.
Converting Large Numbers
- Move the decimal point left until only one non-zero digit is to the left of it.
- Count the number of places moved – this becomes the positive exponent.
6,400,000 = 6.4 × 10⁶ (moved 6 places left)
93,000,000 = 9.3 × 10⁷
Converting Small Numbers
- Move the decimal point right until one non-zero digit is to its left.
- Count the places moved – this becomes the negative exponent.
0.0047 = 4.7 × 10⁻³ (moved 3 places right)
0.000000085 = 8.5 × 10⁻⁸
Converting Back to Standard Form
| Scientific Notation | Standard Form |
|---|---|
| 3.2 × 10⁴ | 32,000 |
| 7.05 × 10⁶ | 7,050,000 |
| 1.6 × 10⁻³ | 0.0016 |
| 9.9 × 10⁻¹ | 0.99 |
Real-World Uses
| Quantity | Scientific Notation |
|---|---|
| Distance from Earth to Sun | ≈ 1.5 × 10¹¹ m |
| Mass of an electron | ≈ 9.1 × 10⁻³¹ kg |
| World population | ≈ 8 × 10⁹ |
| Width of a human hair | ≈ 7 × 10⁻⁵ m |
Key Points
- Format: a × 10ⁿ where 1 ≤ a < 10.
- Large numbers have a positive exponent.
- Small (less than 1) numbers have a negative exponent.
- The exponent equals the number of places the decimal point moved.
Quick Practice
- Write 450,000 in scientific notation.
- Write 0.0000072 in scientific notation.
- Convert 3.8 × 10⁵ to standard form.
- Convert 6.02 × 10⁻⁴ to standard form.
- Which is larger: 2.5 × 10⁸ or 9.9 × 10⁷?
Summary
Scientific notation combines a coefficient between 1 and 10 with a power of 10 to express any number concisely. Moving the decimal point left gives a positive exponent (large numbers); moving it right gives a negative exponent (small numbers). It is an essential tool in science, engineering, computing and finance.