# Floating Point Numbers (Floats)

A **Float** (Floating Point Number) is a data type used in computer programming to represent real numbers that have a fractional part (decimals).

## 1. The Core Concept

Computers store numbers in binary (0s and 1s).
*   **Integers:** Store whole numbers exactly (e.g., `5`, `100`, `-42`).
*   **Floats:** Store numbers using a formula similar to scientific notation ($m \times 2^e$). This allows them to represent a massive range of values, from extremely small ($1.5 \times 10^{-45}$) to extremely large ($3.4 \times 10^{38}$), by "floating" the decimal point.

## 2. IEEE 754 Standard

Most modern computers follow the **IEEE 754** standard for representing floating-point numbers. A float is stored in three parts:
1.  **Sign Bit:** Positive or Negative.
2.  **Exponent:** Determines the magnitude (where the decimal point sits).
3.  **Mantissa (Significand):** The actual digits of the number.

### Types of Precision
*   **Single Precision (Float):** Uses 32 bits. Accurate to about 7 decimal digits.
*   **Double Precision (Double):** Uses 64 bits. Accurate to about 15 decimal digits. (This is the default number type in JavaScript and Python).

## 3. The Precision Problem

Because computers use base-2 (binary) and humans use base-10 (decimal), some numbers cannot be represented exactly. Just like $1/3$ in decimal is $0.3333...$ (repeating), $0.1$ in binary is an infinitely repeating fraction.

This leads to the infamous rounding error:

```javascript
// JavaScript / Python
console.log(0.1 + 0.2); 
// Output: 0.30000000000000004
```

**Rule of Thumb:** Never use floats for financial calculations (money). Use Integers (cents) or a `Decimal` type instead.

## 4. Comparing Floats

Because of precision issues, you should **never** compare floats using `==`.

*   **Bad:** `if (a == b)`
*   **Good:** Check if the difference is smaller than a tiny number (Epsilon).

```python
a = 0.1 + 0.2
b = 0.3
epsilon = 0.00001

if abs(a - b) < epsilon:
    print("They are equal")
```

## 5. Special Values

Floats can represent special non-number states:
*   **Infinity (`inf`):** Result of dividing a positive number by zero.
*   **NaN (Not a Number):** Result of invalid operations like `0 / 0` or `sqrt(-1)`.

[[programming/common-syntax]]
[[programming/bit-manipulation]]