Summary of SI Units and Scientific Calculations

## Introduction Understanding units is essential in science. Units tell you what a number means and ensure measurements from different experiments can be compared. This guide explains S.I. units, common prefixes, how to convert between units, and how to check units in equations. ## S.I. Units Are Used Worldwide Scientists use a standard set of units called the S.I. (Système International) units so measurements are comparable. > **Definition:** S.I. units are the internationally agreed base units for physical quantities. ### Common S.I. Base Units | Quantity | S.I. Base Unit | | --- | --- | | mass | kilogram, kg | | length | metre, m | | time | second, s | | amount of substance | mole, mol | | temperature | kelvin, K | ## Scaling Prefixes for Large and Small Quantities Quantities range from enormous to tiny. Prefixes attach to base units to make numbers easier to use. > **Definition:** A prefix multiplies the base unit by a power of ten. | Prefix | Symbol | Factor | | --- | --- | ---: | | tera | T | $10^{12}$ | | giga | G | $10^{9}$ | | mega | M | $10^{6}$ | | kilo | k | $10^{3}$ | | centi | c | $10^{-2}$ | | milli | m | $10^{-3}$ | | micro | $\mu$ | $10^{-6}$ | | nano | n | $10^{-9}$ | - One kilometre is $10^{3}$ metres. One millimetre is $10^{-3}$ metres. Did you know that using prefixes makes scientific notation easier and reduces the chance of mistakes when comparing very different sizes? ### Using Conversion Factors - To convert from a larger unit to a smaller unit, multiply by the conversion factor. - To convert from a smaller unit to a larger unit, divide by the conversion factor. Examples of common conversions: - Mass: $1\ \text{kg} = 1000\ \text{g}$ - Energy: $1\ \text{kJ} = 1000\ \text{J}$ - Volume: $1\ \text{dm}^{3} = 1\ \text{L} = 1000\ \text{cm}^{3}$ - Length: $1\ \text{m} = 100\ \text{cm} = 1000\ \text{mm}$ Practical example: Convert $250\ \text{cm}$ to metres. $$250\ \text{cm} \div 100 = 2.50\ \text{m}$$ Fun fact: Prefixes like kilo and milli are the same across different quantities, so $1\ \text{kg} = 1000\ \text{g}$ and $1\ \text{km} = 1000\ \text{m}$ both use the factor $10^{3}$. ## Checking Units in Equations Always ensure the units you use in formulas match the units the formula expects. > **Definition:** A conversion factor is the number you multiply or divide by to change from one unit to another. 1. Write the formula and identify the units required. 2. Convert your measured values to those units before substituting. 3. Track units through each step to confirm the result has the correct unit. Example: Wave speed formula. Wave speed equals frequency times wavelength: $$\text{wave speed} = \text{frequency} \times \text{wavelength}$$ If you need frequency: $$\text{frequency} = \frac{\text{wave speed}}{\text{wavelength}}$$ Make sure wave speed is in metres per second and wavelength in metres so frequency comes out in hertz ($\text{Hz}$), where $1\ \text{Hz} = 1\ \text{s}^{-1}$. Common mistake corrected: When rearranging, divide both sides by the same quantity; do not add. For example, rearranging $v = f\lambda$ gives $f = v / \lambda$, not $v + \lambda$. Practical example: A wave travels at $340\ \text{m/s}$ with wavelength $0.5\ \text{m}$. Find frequency. $$f = \frac{v}{\lambda} = \frac{340\ \text{m/s}}{0.5\ \text{m}} = 680\ \text{s}^{-1} = 680\ \text{Hz}$$ ## Tips to Avoid Unit Errors - Always write units on each line of a calculation. - Convert all measurements to the units used by the formula before calculating. - Use prefixes rather than long strings of zeros. Did you know that tracking units step-by-step is called dimensional analysis and can reveal mistakes in algebraic rearrangements? ## Real-world Applications - Engineering: designing bridges requires consistent length and force units to calculate stress. - Medicine: drug dosages use mass per volume, e.g. $\text{mg/mL}$, so conversions between mg and g must be correct. - Environmental science: reporting pollutant concentrations o