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Learn why physics needs standard units, how to read SI units, derived units, and metric prefixes.

---

## Numbers Need a Shared Language

Imagine reading the measurement result $$12$$ without a unit. Is it $$12 \text{ cm}$$, $$12 \text{ m}$$, $$12 \text{ kg}$$, or $$12 \text{ s}$$? The number is the same, but the meaning can change completely.

Visible text: Imagine reading the measurement result without a unit. Is it , , , or ? The number is the same, but the meaning can change completely.

A unit is a reference size used to compare a quantity. In physics, units make measurements from different people, tools, and places readable in the same language.

```math
\text{measurement result} = \text{measured value} \times \text{unit}
```

The measured value answers *how much*. The unit answers *which reference it is compared with*.

## From Many Systems to SI

Before one standard became widely used, several unit systems existed side by side. Three names often appear in physics books:

| System | Example base units | How to read them |
| :----- | :----------------- | :--------------- |
| FPS | $$\text{ft},\ \text{lb},\ \text{s}$$ | foot, pound, and second |
| CGS | $$\text{cm},\ \text{g},\ \text{s}$$ | centimeter, gram, and second |
| MKS | $$\text{m},\ \text{kg},\ \text{s}$$ | meter, kilogram, and second |

Visible text: | System | Example base units | How to read them |
| :----- | :----------------- | :--------------- |
| FPS | | foot, pound, and second |
| CGS | | centimeter, gram, and second |
| MKS | | meter, kilogram, and second |

These differences make measurements hard to compare when the unit system is unclear. That is why modern science uses the **International System of Units** or **SI**. SI comes from the French name *Système international d'unités*. BIPM, the International Bureau of Weights and Measures, is the international organization that maintains the SI standard.

The official BIPM source for measurement units can be opened through [this source link](https://www.bipm.org/en/measurement-units).

> SI is not just a list of abbreviations. SI is an agreement that $$1 \text{ m}$$, $$1 \text{ kg}$$, and $$1 \text{ s}$$ mean the same thing in different schools, laboratories, factories, and countries.

Visible text: > SI is not just a list of abbreviations. SI is an agreement that , , and mean the same thing in different schools, laboratories, factories, and countries.

## The Seven SI Base Units

SI base units are the building blocks. From them, other units such as $$\text{m}^2$$, $$\text{m/s}$$, $$\text{N}$$, $$\text{J}$$, and $$\text{W}$$ are built.

Visible text: SI base units are the building blocks. From them, other units such as , , , , and are built.

| Base quantity | Common symbol | SI unit | Unit symbol | Dimension |
| :------------ | :------------ | :------ | :---------- | :-------- |
| Length | $$\ell,\ x,\ r$$ | meter | $$\text{m}$$ | $$[\mathrm{L}]$$ |
| Mass | $$m$$ | kilogram | $$\text{kg}$$ | $$[\mathrm{M}]$$ |
| Time | $$t$$ | second | $$\text{s}$$ | $$[\mathrm{T}]$$ |
| Electric current | $$I$$ | ampere | $$\text{A}$$ | $$[\mathrm{I}]$$ |
| Thermodynamic temperature | $$T$$ | kelvin | $$\text{K}$$ | $$[\Theta]$$ |
| Amount of substance | $$n$$ | mole | $$\text{mol}$$ | $$[\mathrm{N}]$$ |
| Luminous intensity | $$I_v$$ | candela | $$\text{cd}$$ | $$[\mathrm{J}]$$ |

Visible text: | Base quantity | Common symbol | SI unit | Unit symbol | Dimension |
| :------------ | :------------ | :------ | :---------- | :-------- |
| Length | | meter | | |
| Mass | | kilogram | | |
| Time | | second | | |
| Electric current | | ampere | | |
| Thermodynamic temperature | | kelvin | | |
| Amount of substance | | mole | | |
| Luminous intensity | | candela | | |

One detail often causes confusion: the SI base unit for mass is $$\text{kg}$$, not $$\text{g}$$. So when mass is written in grams, convert it to kilograms if you want the full SI base unit form.

Visible text: One detail often causes confusion: the SI base unit for mass is , not . So when mass is written in grams, convert it to kilograms if you want the full SI base unit form.

```math
\begin{aligned}
1 \text{ g} &= 10^{-3} \text{ kg} \\
250 \text{ g} &= 250 \times 10^{-3} \text{ kg} = 0.25 \text{ kg}
\end{aligned}
```

## Derived Units Come from Operations

A derived unit appears when a derived quantity is formed from base quantities. If the formula multiplies or divides quantities, the units are multiplied or divided too.

| Derived quantity | Quantity formula | SI unit | Dimension |
| :--------------- | :--------------- | :------ | :-------- |
| Area | $$A = l \times w$$ | $$\text{m}^2$$ | $$[\mathrm{L}]^2$$ |
| Volume | $$V = l \times w \times h$$ | $$\text{m}^3$$ | $$[\mathrm{L}]^3$$ |
| Density | $$\rho = \frac{m}{V}$$ | $$\text{kg/m}^3$$ | $$[\mathrm{M}][\mathrm{L}]^{-3}$$ |
| Speed | $$v = \frac{\Delta s}{\Delta t}$$ | $$\text{m/s}$$ | $$[\mathrm{L}][\mathrm{T}]^{-1}$$ |
| Acceleration | $$a = \frac{\Delta v}{\Delta t}$$ | $$\text{m/s}^2$$ | $$[\mathrm{L}][\mathrm{T}]^{-2}$$ |
| Force | $$F = m \times a$$ | $$\text{N}$$ | $$[\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2}$$ |
| Work | $$W = F \times \Delta s$$ | $$\text{J}$$ | $$[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2}$$ |
| Power | $$P = \frac{W}{t}$$ | $$\text{W}$$ | $$[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}$$ |

Visible text: | Derived quantity | Quantity formula | SI unit | Dimension |
| :--------------- | :--------------- | :------ | :-------- |
| Area | | | |
| Volume | | | |
| Density | | | |
| Speed | | | |
| Acceleration | | | |
| Force | | | |
| Work | | | |
| Power | | | |

The chain of a derived unit shows where it came from. For example, force uses the unit newton, but a newton is built from kilogram, meter, and second.

```math
\begin{aligned}
F &= m \times a \\
1 \text{ N} &= 1 \text{ kg} \cdot \text{m/s}^2 \\
&= 1 \text{ kg m s}^{-2}
\end{aligned}
```

Once force is clear, work and power can be read step by step.

```math
\begin{aligned}
1 \text{ J} &= 1 \text{ N} \cdot \text{m}
= 1 \text{ kg m}^2\text{ s}^{-2} \\
1 \text{ W} &= \frac{1 \text{ J}}{1 \text{ s}}
= 1 \text{ kg m}^2\text{ s}^{-3}
\end{aligned}
```

## Metric Prefixes Save Zeros

When a size is very large or very small, we do not need to write long rows of zeros. SI uses metric prefixes to express powers of $$10$$.

Visible text: When a size is very large or very small, we do not need to write long rows of zeros. SI uses metric prefixes to express powers of .

Component: Mermaid
Props:
- title: Prefix Shortens Very Large or Small Units
- description: See how metric prefixes replace long strings of zeros with shorter unit names.
```mermaid
flowchart LR
  A["Read value"] --> B["Split prefix"]
  B --> C["Use a power of 10"]
  C --> D["Multiply the value"]
  D --> E["Write the final unit"]
```

Common prefixes include:

| Factor | Prefix | Symbol | Factor | Prefix | Symbol |
| :----- | :----- | :----- | :----- | :----- | :----- |
| $$10^1$$ | deca | $$\text{da}$$ | $$10^{-1}$$ | deci | $$\text{d}$$ |
| $$10^2$$ | hecto | $$\text{h}$$ | $$10^{-2}$$ | centi | $$\text{c}$$ |
| $$10^3$$ | kilo | $$\text{k}$$ | $$10^{-3}$$ | milli | $$\text{m}$$ |
| $$10^6$$ | mega | $$\text{M}$$ | $$10^{-6}$$ | micro | $$\mu$$ |
| $$10^9$$ | giga | $$\text{G}$$ | $$10^{-9}$$ | nano | $$\text{n}$$ |
| $$10^{12}$$ | tera | $$\text{T}$$ | $$10^{-12}$$ | pico | $$\text{p}$$ |
| $$10^{15}$$ | peta | $$\text{P}$$ | $$10^{-15}$$ | femto | $$\text{f}$$ |
| $$10^{18}$$ | exa | $$\text{E}$$ | $$10^{-18}$$ | atto | $$\text{a}$$ |
| $$10^{21}$$ | zetta | $$\text{Z}$$ | $$10^{-21}$$ | zepto | $$\text{z}$$ |
| $$10^{24}$$ | yotta | $$\text{Y}$$ | $$10^{-24}$$ | yocto | $$\text{y}$$ |

Visible text: | Factor | Prefix | Symbol | Factor | Prefix | Symbol |
| :----- | :----- | :----- | :----- | :----- | :----- |
| | deca | | | deci | |
| | hecto | | | centi | |
| | kilo | | | milli | |
| | mega | | | micro | |
| | giga | | | nano | |
| | tera | | | pico | |
| | peta | | | femto | |
| | exa | | | atto | |
| | zetta | | | zepto | |
| | yotta | | | yocto | |

Scope note: the list above follows prefixes commonly used in grade $$10$$ material. BIPM also lists newer SI prefixes that go larger and smaller, such as $$10^{30}$$ and $$10^{-30}$$.

Visible text: Scope note: the list above follows prefixes commonly used in grade material. BIPM also lists newer SI prefixes that go larger and smaller, such as and .

The official BIPM source for SI prefixes can be opened through [this source link](https://www.bipm.org/en/measurement-units/si-prefixes).

## Reading Prefixes in Calculations

A prefix is attached to a base unit. For example, $$\text{km}$$ means kilo-meter, not a separate unit unrelated to the meter.

Visible text: A prefix is attached to a base unit. For example, means kilo-meter, not a separate unit unrelated to the meter.

```math
\begin{aligned}
7.5 \text{ km}
&= 7.5 \times 10^3 \text{ m} \\
&= 7500 \text{ m}
\end{aligned}
```

For small sizes, prefixes keep writing compact. A diameter of $$0.1 \ \mu\text{m}$$ is equal to:

Visible text: For small sizes, prefixes keep writing compact. A diameter of is equal to:

```math
\begin{aligned}
0.1 \ \mu\text{m}
&= 0.1 \times 10^{-6} \text{ m} \\
&= 1.0 \times 10^{-7} \text{ m}
\end{aligned}
```

The extremely small mass of an electron is also easier to read with scientific notation.

```math
9.11 \times 10^{-31} \text{ kg}
```

The point is not to memorize every prefix at once. When you see a unit such as $$\text{cm}$$, $$\text{mg}$$, or $$\mu\text{m}$$, first separate the prefix from the base unit. Then convert the prefix into a power of $$10$$ and continue the calculation.

Visible text: The point is not to memorize every prefix at once. When you see a unit such as , , or , first separate the prefix from the base unit. Then convert the prefix into a power of and continue the calculation.