# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/physics/measurement/quantity
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/measurement/quantity/en.mdx

Learn what a physical quantity is, how base and derived quantities differ, and how quantities connect to SI units and dimensions.

---

## Reading Quantities from Measurements

When we write a measurement result such as $$0.60 \text{ m}$$, $$2$$ parts are working together.

Visible text: When we write a measurement result such as , parts are working together.

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

The value $$0.60$$ tells us *how much*. The unit $$\text{m}$$ tells us *which standard it is compared with*. The quantity is the property of an object or event being measured, such as length, mass, time, temperature, or speed.

Visible text: The value tells us *how much*. The unit tells us *which standard it is compared with*. The quantity is the property of an object or event being measured, such as length, mass, time, temperature, or speed.

> Without a quantity, a number has no direction. Without a unit, a number cannot be compared.

## Base and Derived Quantities

In SI, the **International System of Units**, $$7$$ base quantities are used as foundations. Other quantities can be built from combinations of those base quantities.

Visible text: In SI, the **International System of Units**, base quantities are used as foundations. Other quantities can be built from combinations of those base quantities.

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

Visible text: | Base quantity | Typical symbol | SI unit | Dimension |
| :------------ | :------------- | :------ | :-------- |
| Length | | | |
| Mass | | | |
| Time | | | |
| Electric current | | | |
| Thermodynamic temperature | | | |
| Amount of substance | | | |
| Luminous intensity | | | |

A derived quantity appears when base quantities are combined. For example, area comes from length multiplied by width.

```math
\begin{aligned}
A &= l \times w \\
\text{unit of } A &= \text{m} \times \text{m} = \text{m}^2 \\
\dim A &= \mathrm{L} \times \mathrm{L} = \mathrm{L}^2
\end{aligned}
```

Speed is also a derived quantity because it comes from distance divided by time.

```math
\begin{aligned}
v &= \frac{s}{t} \\
\text{unit of } v &= \frac{\text{m}}{\text{s}} = \text{m s}^{-1} \\
\dim v &= \frac{\mathrm{L}}{\mathrm{T}} = \mathrm{L}\mathrm{T}^{-1}
\end{aligned}
```

## Dimension as a Code

Dimension is a compact way to see which base quantities build a quantity. It does not replace units, but it helps us check whether a formula makes sense.

Component: Mermaid
Props:
- title: How to check a quantity
- description: Start from the measured quantity, then connect its base quantities to the SI unit and dimension.
```mermaid
flowchart TD
  A["Choose quantity"] --> B["Find base quantities"]
  B --> C["Write the SI unit"]
  B --> D["Write the dimension"]
  C --> E["Check result"]
  D --> E
```

For example, the area of a desk and the area of a sheet of paper can have different values, but they are the same kind of quantity. Both are area, both use the SI unit $$\text{m}^2$$, and both have dimension $$\mathrm{L}^2$$.

Visible text: For example, the area of a desk and the area of a sheet of paper can have different values, but they are the same kind of quantity. Both are area, both use the SI unit , and both have dimension .

On the other hand, length, width, height, diameter, radius, distance, and displacement can all use length units. They all have dimension $$\mathrm{L}$$.

Visible text: On the other hand, length, width, height, diameter, radius, distance, and displacement can all use length units. They all have dimension .

## One Area Calculation

A sheet of paper has length $$0.30 \text{ m}$$ and width $$0.20 \text{ m}$$. Its area is:

Visible text: A sheet of paper has length and width . Its area is:

```math
\begin{aligned}
A &= l \times w \\
&= 0.30 \text{ m} \times 0.20 \text{ m} \\
&= 0.060 \text{ m}^2
\end{aligned}
```

The value $$0.060 \text{ m}^2$$ is the measurement result for area. Since area is built from $$2$$ length quantities, area is a derived quantity.

Visible text: The value is the measurement result for area. Since area is built from length quantities, area is a derived quantity.

Notice the pattern: once a unit is built from operations between other units, the quantity is no longer base. This way of reading values, units, and dimensions helps us check whether units and formulas agree.