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One-Dimensional (Linear) Unit Conversion
One-dimensional unit conversion deals with linear measurements - quantities that have only one dimension, such as length, time, mass, temperature, and electric current.
Metric Prefixes Table
The International System of Units (SI) uses prefixes to indicate multiples and submultiples of base units:
| Prefix | Symbol | Factor | Decimal | Scientific Notation | Example |
|---|---|---|---|---|---|
| tera | T | 1 000 000 000 000 | 1 trillion | 10¹² | 1 Tm = 10¹² m |
| giga | G | 1 000 000 000 | 1 billion | 10⁹ | 1 GHz = 10⁹ Hz |
| mega | M | 1 000 000 | 1 million | 10⁶ | 1 MHz = 10⁶ Hz |
| kilo | k | 1 000 | 1 thousand | 10³ | 1 km = 10³ m |
| hecto | h | 100 | 1 hundred | 10² | 1 hm = 10² m |
| deca | da | 10 | ten | 10¹ | 1 dam = 10¹ m |
| base unit | — | 1 | one | 10⁰ | 1 m, 1 g, 1 s |
| deci | d | 0.1 | one tenth | 10⁻¹ | 1 dm = 10⁻¹ m |
| centi | c | 0.01 | one hundredth | 10⁻² | 1 cm = 10⁻² m |
| milli | m | 0.001 | one thousandth | 10⁻³ | 1 mm = 10⁻³ m |
| micro | μ | 0.000001 | one millionth | 10⁻⁶ | 1 μm = 10⁻⁶ m |
| nano | n | 0.000000001 | one billionth | 10⁻⁹ | 1 nm = 10⁻⁹ m |
| pico | p | 0.000000000001 | one trillionth | 10⁻¹² | 1 pm = 10⁻¹² m |
Conversion Factors
Step-by-Step Process
- Find equality: You can use more than one equality to relate different units
- Put the equality as fraction equal to 1: Place units that you want to change in the opposite position
- Calculate and check units: Perform the arithmetic and ensure all units have been converted
Understanding Conversion Factors
A conversion factor is a fraction that equals 1, created from any equality between units. These fractions allow us to change the units of a measurement without changing its value.
Key principle: Every equality gives us two conversion factors. For example, from the equality 1 km = 1 000 m, we can create:\[ \frac{1 \text{ km}}{1 000 \text{ m}} = 1 \quad \text{or} \quad \frac{1 000 \text{ m}}{1 \text{ km}} = 1\]
How to choose the right factor: Use the conversion factor where the unwanted unit cancels out, leaving only the desired unit.
Conversion Examples Table
| Desired Conversion | Equivalence Relationship | Formula and Calculation |
|---|---|---|
| Length: km → m 2.5 km to meters | 1 km = 1 000 m | \(2.5 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 2.5 \times 1000 \text{ m} = 2500 \text{ m}\) |
| Length: mm → m 750 mm to meters | 1 000 mm = 1 m | \(750 \text{ mm} \times \frac{1 \text{ m}}{1000 \text{ mm}} = \frac{750}{1000} \text{ m} = 0.75 \text{ m}\) |
| Length: cm → mm 4.2 cm to millimeters | 1 cm = 10 mm | \(4.2 \text{ cm} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 4.2 \times 10 \text{ mm} = 42 \text{ mm}\) |
| Time: h → min 3.5 hours to minutes | 1 h = 60 min | \(3.5 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} = 3.5 \times 60 \text{ min} = 210 \text{ min}\) |
| Time: min → s 45 minutes to seconds | 1 min = 60 s | \(45 \text{ min} \times \frac{60 \text{ s}}{1 \text{ min}} = 45 \times 60 \text{ s} = 2700 \text{ s}\) |
| Time: h → s (multi-step) 2.5 hours to seconds | 1 h = 60 min 1 min = 60 s | \(2.5 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{60 \text{ s}}{1 \text{ min}} = 2.5 \times 60 \times 60 \text{ s} = 9000 \text{ s}\) |
| Mass: kg → g 3.2 kg to grams | 1 kg = 1 000 g | \(3.2 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} = 3.2 \times 1000 \text{ g} = 3200 \text{ g}\) |
| Mass: mg → g 850 mg to grams | 1 000 mg = 1 g | \(850 \text{ mg} \times \frac{1 \text{ g}}{1000 \text{ mg}} = \frac{850}{1000} \text{ g} = 0.85 \text{ g}\) |
| Mass: tonnes → kg 0.75 tonnes to kg | 1 tonne = 1 000 kg | \(0.75 \text{ tonnes} \times \frac{1000 \text{ kg}}{1 \text{ tonne}} = 0.75 \times 1000 \text{ kg} = 750 \text{ kg}\) |
| Frequency: MHz → Hz 95.5 MHz to Hz | 1 MHz = 10⁶ Hz | \(95.5 \text{ MHz} \times \frac{10^6 \text{ Hz}}{1 \text{ MHz}} = 95.5 \times 10^6 \text{ Hz} = 9.55 \times 10^7 \text{ Hz}\) |
| Frequency: GHz → MHz 2.4 GHz to MHz | 1 GHz = 1 000 MHz | \(2.4 \text{ GHz} \times \frac{1000 \text{ MHz}}{1 \text{ GHz}} = 2.4 \times 1000 \text{ MHz} = 2400 \text{ MHz}\) |
| Volume: L → mL 1.5 L to milliliters | 1 L = 1 000 mL | \(1.5 \text{ L} \times \frac{1000 \text{ mL}}{1 \text{ L}} = 1.5 \times 1000 \text{ mL} = 1500 \text{ mL}\) |
Solved Examples
Example 1: Convert 2.5 × 10⁶ μm to km
Method 1: Direct conversion
1 μm = 10⁻⁶ m and 1 km = 10³ m
So 1 μm = 10⁻⁶ m ÷ 10³ m/km = 10⁻⁹ km
\[2.5 \times 10^6 \text{ μm} \times 10^{-9} \frac{ \text{km}}{ \text{μm}} = 2.5 \times 10^{-3} \text{ km} = 0.0025 \text{ km}\]
Method 2: Step-by-step
\[2.5 \times 10^6 \text{ μm} \times \frac{10^{-6} \text{ m}}{1 \text{ μm}} \times \frac{1 \text{ km}}{10^3 \text{ m}} = 0.0025 \text{ km}\]
Result: 2.5 × 10⁶ μm = 0.0025 km = 2.5 mmExample 2: A movie lasts 2 hours and 15 minutes. Express this in seconds.
Step 1: Convert to total minutes
\[2 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} + 15 \text{ min} = 120 \text{ min} + 15 \text{ min} = 135 \text{ min}\]
Step 2: Convert minutes to seconds
\[135 \text{ min} \times \frac{60 \text{ s}}{1 \text{ min}} = 8 100 \text{ s}\]
Result: 2 h 15 min = 8 100 sExample 3: Convert 3.2 × 10⁻⁴ kg to mg
Step 1: Convert kg to g
\[3.2 \times 10^{-4} \text{ kg} \times \frac{1 000 \text{ g}}{1 \text{ kg}} = 3.2 \times 10^{-4} \times 10^3 \text{ g} = 3.2 \times 10^{-1} \text{ g}\]
Step 2: Convert g to mg
\[3.2 \times 10^{-1} \text{ g} \times \frac{1 000 \text{ mg}}{1 \text{ g}} = 3.2 \times 10^{-1} \times 10^3 \text{ mg} = 3.2 \times 10^2 \text{ mg} = 320 \text{ mg}\]
Result: 3.2 × 10⁻⁴ kg = 320 mgExample 4: A radio wave has a frequency of 95.5 MHz. Convert this to Hz and kHz.
Step 1: Convert MHz to Hz
1 MHz = 10⁶ Hz
\[95.5 \text{ MHz} \times \frac{10^6 \text{ Hz}}{1 \text{ MHz}} = 95.5 \times 10^6 \text{ Hz} = 9.55 \times 10^7 \text{ Hz}\]
Step 2: Convert Hz to kHz
1 kHz = 10³ Hz, so 1 Hz = 10⁻³ kHz
\[9.55 \times 10^7 \text{ Hz} \times \frac{1 \text{ kHz}}{10^3 \text{ Hz}} = 9.55 \times 10^4 \text{ kHz} = 95 500 \text{ kHz}\]
Result: 95.5 MHz = 9.55 × 10⁷ Hz = 95 500 kHz