Fixed and Floating Point Representations

💡 Data Representation – Fixed and Floating Point Representations

Computers deal with numbers all the time — from counting your files to calculating distances in a video game.
But here’s the tricky part: computers can’t use “decimal points” like we do on paper. They only understand 0s and 1s — bits!

So, how does a computer store numbers like 25, -3.5, or 0.0000042 using just 0s and 1s?
That’s where fixed-point and floating-point representations come in.

🧮 1. Fixed-Point Representation

Let’s start simple.

Imagine you’re writing numbers on paper but always keeping the decimal point in one fixed position.
That’s basically what a fixed-point representation does — the decimal (or binary point, in a computer) never moves.

🔹 How It Works

In fixed-point form:

Some bits are used for the whole number part (integer part).
Some bits are used for the fractional part (after the point).

The position of the point is fixed — it doesn’t change for different numbers.

🧠 Example

Suppose we have 8 bits, and we decide:

5 bits for the integer part
3 bits for the fractional part

So, a number like this:

10110.011

means:

Integer part: 10110 = 22
Fractional part: .011 = 0.375 (in decimal)
→ So, total = 22.375

Notice how the “binary point” is fixed after 5 bits.

⚙️ Key Point

Because the point is fixed, it’s great for simple calculations where precision and range are limited — like when you’re working with money or small measurements.

But there’s a catch — you can only represent numbers within a certain range.
If a number gets too big or too small, the fixed-point format can’t handle it.

🎯 Analogy

Think of fixed-point like a ruler that only measures between 0 and 30 cm with millimeter marks.
You can measure short distances very accurately, but if someone asks for 10 meters, you’re stuck!
You’d need a new ruler — or a new system — to handle larger values.

🌊 2. Floating-Point Representation

Now, let’s talk about the more flexible (and more magical) one — floating-point representation.

Here, the decimal (or binary) point can “float” — meaning it can move left or right depending on the number.
This allows computers to handle very large and very small numbers easily.

🔹 How It Works

In floating-point form, a number is split into three parts:

Part	What it does	Analogy
Sign bit	Tells whether the number is + or –	Like a “+” or “–” sign
Exponent	Decides where the point “floats”	Like shifting the decimal point
Mantissa (or Significand)	Holds the actual digits of the number	Like the main part of a number

So, the general formula is:

Value = (−1)^Sign × Mantissa × 2^(Exponent)

🧠 Example

Let’s say the floating-point number is stored like this (in binary):

0  10000010  10100000000000000000000

Breaking it down:

Sign bit (0) → Positive
Exponent (10000010) → 130 in decimal → (130 – 127 = 3, since we use a bias of 127)
Mantissa (1.101) → 1.625 in decimal

Now plug into the formula:

Value = +1.625 × 2^3 = 13.0

So, this binary pattern actually represents the number 13.0!

⚙️ Why Use Floating Point?

Floating-point is awesome because:

It can represent huge numbers (like distances between planets)
And tiny numbers (like size of an atom)
All using the same number of bits!

That’s why scientific calculators, physics simulations, and 3D games all depend on floating-point math.

🎯 Analogy

Imagine the floating-point system like scientific notation in math.

For example:

3,000 = 3 × 10³
0.0045 = 4.5 × 10⁻³

In both, the decimal “floats” — moving left or right to fit the number neatly.
Computers do the same thing, but in binary instead of decimal.

🧩 Diagram: Fixed vs Floating Point Representation

         +---------------------------------------------+
         |           Data Representation               |
         +----------------+-----------------------------+
                          |
          +---------------+----------------+
          |                                |
          v                                v
 +-----------------------+       +-----------------------------+
 |  Fixed Point Format   |       |  Floating Point Format       |
 +-----------------------+       +-----------------------------+
 | Sign | Integer | Fraction |  | Sign | Exponent | Mantissa  |
 | bit  |  bits   |  bits    |  | bit  |  bits    | bits      |
 +-----------------------+       +-----------------------------+
 | Decimal point is FIXED |      | Decimal point can FLOAT     |
 | Range is limited       |      | Large range of numbers      |
 | Example: 10110.011     |      | Example: 1.101 × 2^3        |
 +-----------------------+       +-----------------------------+

⚖️ Comparison Between Fixed and Floating Point

Feature	Fixed Point	Floating Point
Decimal/Binary Point	Fixed	Moves (floats)
Range	Small	Very large
Precision	High (for small numbers)	Varies (can handle big/small)
Speed	Faster and simpler	Slightly slower, more complex
Used in	Simple devices, embedded systems	Scientific and engineering apps

🌟 In Simple Words

Fixed Point: The point doesn’t move. Best for steady, small-range numbers.
→ Like having a ruler with fixed marks.
Floating Point: The point can move. Best for a huge range of values.
→ Like scientific notation that adapts to the number’s size.

💬 Real-Life Example

Imagine two friends measuring things:

Fixed-Point Fred uses a small ruler — great for measuring a pen, bad for a mountain.
Floating-Point Fiona uses a telescope and a microscope — she can measure stars or sand grains with equal ease.

That’s exactly how computers switch between fixed and floating-point systems depending on what they’re doing!