💡 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:
10110.011means:
- 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 10100000000000000000000Breaking 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 = 13.0So, 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!
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