Understanding Powers of 10
Powers of 10 are the foundation of our decimal number system. Each power of 10 represents a position in a number: ones (100), tens (101), hundreds (102), and so on. Understanding powers of 10 is essential for scientific notation, metric conversions, and order-of-magnitude reasoning.
Key Concepts
Positive Exponents
10^n means 1 followed by n zeros. Each increase by 1 multiplies by 10.
Negative Exponents
10^(-n) = 1/10^n. Moves the decimal point n places left.
Zero Exponent
Any non-zero number raised to the power 0 equals 1.
Scientific Notation
Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of 10. For example, 6,500,000 = 6.5 x 106. This notation makes it easy to work with very large or very small numbers in science, engineering, and everyday calculations.
Metric (SI) Prefixes
The metric system uses powers of 10 to define prefixes for units of measurement:
- Tera (T): 1012 = 1,000,000,000,000
- Giga (G): 109 = 1,000,000,000
- Mega (M): 106 = 1,000,000
- Kilo (k): 103 = 1,000
- Milli (m): 10-3 = 0.001
- Micro (u): 10-6 = 0.000001
- Nano (n): 10-9 = 0.000000001
- Pico (p): 10-12 = 0.000000000001
Real-World Scale
- 10-15 meters: Size of a proton (femtometer)
- 10-9 meters: Size of a molecule (nanometer)
- 100 meters: Human height
- 107 meters: Diameter of Earth
- 1011 meters: Distance from Earth to Sun
- 1026 meters: Observable universe diameter
Properties of Powers of 10
- 10^a x 10^b = 10^(a+b)
- 10^a / 10^b = 10^(a-b)
- (10^a)^b = 10^(a x b)
- log10(10^n) = n