# Units

Nomenclature and precision are important for understanding the internals of Phoenix Legacy. These quantities and conversions enable a robust way to understand how numerical data flows through the matching engine.

### Atoms and Units

On Solana, all token quantities can be interpreted as a decimal value.

<table><thead><tr><th width="129">Token</th><th width="109">Decimals</th><th width="193">Amount (atoms)</th><th>Decimal interpretation (units)</th></tr></thead><tbody><tr><td>SOL</td><td>9</td><td>1,000,000,000</td><td>1.0 SOL</td></tr><tr><td>USDC</td><td>6</td><td>1,000,000</td><td>1.0 USDC</td></tr></tbody></table>

In the context of Phoenix Legacy, the smallest possible quantity of a token is referred to as an **atom**. The associated decimal value for a number of **atoms** is measured in **units**.

To convert from an amount of **atoms** to **units**, we apply the following formula:

$$
\begin{align\*}\
1 \text{ unit} &= 10^{\text{decimals}} \text{ atoms}
\ A \text{ atoms} &= \frac{K}{10^{\text{decimals}}} \text{ units}
\end{align\*}
$$

### Base and Quote

Every Phoenix Legacy market represents a venue for exchange between 2 assets (e.g. SOL/USDC). The asset being traded (in our example, SOL) is referred to as the **base** token and the asset that prices the base token is referred to as the **quote** token (in our example, USDC).

### Lots

A **lot** is the smallest discrete unit of a token that the Phoenix Legacy matching engine can process. Every quantity that is passed into Phoenix Legacy must first be converted into the an integer number of **lots**. Note that lot sizes are set on a per-market level.

The smallest amount of base token (in **atoms**) the matching engine can work with is referred to as the **base lot size**, which is a quantity measured in **base atoms per base lot.**

The smallest amount of quote token (in **atoms**) the matching engine can work with is referred to as the **quote lot size**, which is a quantity measured in **quote atoms per quote lot.**

#### Converting Between Atoms and Lots

<table><thead><tr><th width="130.33333333333331">Token</th><th width="119">Type</th><th width="188">Lot size (units)</th><th>Lot size (atoms per lot)</th></tr></thead><tbody><tr><td>SOL</td><td>Base</td><td>0.001</td><td>1,000,000</td></tr><tr><td>USDC</td><td>Quote</td><td>0.00001</td><td>10</td></tr></tbody></table>

We know that the number of **base atoms per base** **unit** (for SOL) is $$10^9$$. So if the base lot size in **base units per base lot** is 0.001 SOL, we can apply the following conversion to get the base lot size in base atoms per base lot:

$$
\begin{equation\*} 0.001 \frac{\text{base units}}{\text{base lot}} \times 10^9 \frac{\text{base atoms}}{\text{base units}} = 10^6 \frac{\text{base atoms}}{\text{base lot}} \end{equation\*}
$$

We can perform a similar conversion for the quote token.

$$
0.00001 \frac{\text{quote units}}{\text{quote lot}} \times 10^6 \frac{\text{quote atoms}}{\text{quote unit}} = 10 \frac{\text{quote atoms}}{\text{quote lot}}
$$

### Ticks

A **tick** is the smallest discrete unit of price by which 2 orders on the book can differ; all valid prices on a market are an integer number of ticks. The size of a **tick** is measured in an integer number of **quote lots per base unit**.

{% hint style="info" %}
Prices on the book are measured in an integer number of ticks.
{% endhint %}

Suppose the market has $$Q$$ **quote lots per quote unit** and a tick size of $$T$$ **quote lots per base unit per tick**. The following formula converts a price of $$P$$ **ticks** to the human-readable **quote units per base unit**:

$$
P \text{ ticks} \times T \frac{\text{quote lots}}{\text{base unit} \times \text{tick}} \times \frac{1}{Q} \frac{\text{quote units}}{\text{quote lot}} = \frac{PT}{Q} \frac{\text{quote units}}{\text{base unit}}
$$

Observe some examples in the table below:

<table><thead><tr><th width="138">Price (P)</th><th width="102.33333333333331">Tick size (T)</th><th width="225">Quote Lots/Quote Unit</th><th>Price (quote units/base unit)</th></tr></thead><tbody><tr><td>2000</td><td>1000</td><td>10000</td><td>20.00</td></tr><tr><td>2000</td><td>100</td><td>1000</td><td>20.00</td></tr><tr><td>400</td><td>5000</td><td>10000</td><td>20.00</td></tr></tbody></table>

### Order Sizes

Every order on the book is represented with a size $$S$$ in **base lots** at some price $$P$$ in ticks.

To compute the number of **quote lots** $$Q$$ this order represents (amount posted for bids and amount required to fill the order for offers), use the following formula. Assume a tick size of $$T$$ **quote lots per base unit per tick** and $$B$$ **base lots per base unit**.

$$
P \text{ ticks} \times T \frac{\text{quote lots}}{\text{base unit} \times \text{tick}} \times S \text{ base lots} \times \frac{1}{B} \frac{\text{base units}}{\text{base lot}} = \frac{PTS}{B} \text{ quote lots}
$$

### Invariants

One important guarantee is that the book will enable trades of an integer number of **quote lots** for a single **base lot** at any price. What this means is that at a price of 1 **tick,** selling a 1 **base lot**, will return $$K$$**quote lots** where $$K$$is a nonnegative integer. We can rearrange the formula from the previous section to get the following equivalence:

$$K\* b \ \frac{quote\ lots}{base\ unit\ \*\ tick} = T\ \frac{quote\ lots}{base\ unit\ \*\ tick}$$

{% hint style="info" %}
This indicates that the **tick size** must be an integer multiple of the number of **base lots** found in a **base unit**
{% endhint %}


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