Overview
This lecture covers metric prefixes, how to convert between different metric units using powers of ten, and how these conversions change when dealing with units squared (area) and cubed (volume).
Metric Prefixes and Base Units
- Metric prefixes represent specific powers of ten and are combined with base units like meters (m), grams (g), and liters (L).
- Each prefix has a symbol (e.g., kilo–k, mega–M, nano–n), representing a number expressed as a power of ten.
- Base units measure quantities (length, mass, volume, etc.) and can be combined with any metric prefix.
Converting Between Metric Prefixes
- Use dimensional analysis: Write the quantity and unit, multiply by a fraction with the original unit on the bottom and the desired unit on top.
- Place 1 with the larger unit; assign the smaller unit a power of ten found by subtracting prefix exponents.
- Example: 2.6 kg to decigrams: kilo (3) – deci (–1) = 4 → 2.6 kg × 10⁴ dg/1 kg = 26,000 dg.
- When converting from a smaller to a larger unit, process is similar: subtract exponents, larger unit gets 1.
Applying Prefixes to Various Units
- Prefixes can be attached to any metric unit (liters, seconds, joules, pascals, watts, etc.).
- Examples: milliliter (10⁻³ L), microsecond (10⁻⁶ s), kilopascal (10³ Pa), gigawatt (10⁹ W).
Conversions with Squared and Cubed Units
- For area (squared units), square the numerical relationship: (e.g., 10³ m in 1 km becomes 10⁶ m² in 1 km²).
- For volume (cubed units), cube the numerical relationship: (e.g., 10¹ cm in 1 dm becomes 10³ cm³ in 1 dm³).
- Always adjust the power of ten by squaring or cubing for area or volume conversions.
Key Terms & Definitions
- Metric Prefix — A symbol and name representing a specific power of ten used with base metric units.
- Base Unit — The fundamental unit for a type of measurement (meter, gram, liter, etc.).
- Dimensional Analysis — A method for converting between units using conversion factors.
- Exponent — The power of ten associated with each prefix (e.g., kilo = 10³).
- Area — The measure of a surface, using squared units.
- Volume — The measure of space, using cubed units.
Action Items / Next Steps
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Memorize metric prefixes and their corresponding exponents.
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Practice dimensional analysis for unit conversions.
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Be prepared to perform conversions for squared and cubed units without a reference chart.
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Converting to a smaller unit: Move the decimal to the right (the number gets bigger).
- Example: km to m (kilometers to meters) — meters are smaller, so multiply by 1000 → decimal moves right 3 places.
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Converting to a larger unit: Move the decimal to the left (the number gets smaller).
- Example: cm to m (centimeters to meters) — meters are larger, so divide by 100 → decimal moves left 2 places.
How to decide:
- Look at the exponents of the prefixes (powers of ten).
- Subtract: (exponent of desired unit) – (exponent of given unit).
- If the result is positive, move the decimal to the left that many places (dividing).
- If the result is negative, move the decimal to the right that many places (multiplying).
Example:
Convert 3.5 km to m:
- kilo (k) = 10³
- meter (m) = 10⁰
- Exponent difference = 0 – 3 = –3 (negative)
- Move decimal 3 places to the right:
[ 3.5 \rightarrow 3500 ]