Visual breakdown of how these numbers are stored in Float variable 0.1f, 0.2f, and 0.3f

float.md

Here are clear visual-style breakdowns of how 0.1f, 0.2f, and 0.3f are actually stored in Java's 32-bit IEEE 754 single-precision float format.

1. 0.1f – Visual Breakdown

Decimal: 0.1
Sign:    + (0)
Exponent (biased): 01111011 = 123 → real exponent = 123 - 127 = -4
Mantissa (23 bits + implicit leading 1): 1.10011001100110011001101

Full 32-bit pattern:
   S     Exponent      Mantissa (23 bits)
   ↓         ↓                ↓
  0    01111011    10011001100110011001101

Binary:  0011 1101 1100 1100 1100 1100 1100 1101
Hex:     3D       CC       CC       CD
         └───────┬────────┬────────┘
                 0.1f

Approximate real value: 0.100000001490116119384765625

2. 0.2f – Visual Breakdown

Decimal: 0.2
Sign:    + (0)
Exponent (biased): 01111100 = 124 → real exponent = 124 - 127 = -3
Mantissa (23 bits + implicit 1): 1.10011001100110011001101

Full 32-bit pattern:
   S     Exponent      Mantissa
   ↓         ↓              ↓
  0    01111100    10011001100110011001101

Binary:  0011 1110 0100 1100 1100 1100 1100 1101
Hex:     3E       4C       CC       CD
         └───────┬────────┬────────┘
                 0.2f

Approximate real value: 0.20000000298023223876953125

3. 0.3f – Visual Breakdown

Decimal: 0.3
Sign:    + (0)
Exponent (biased): 01111101 = 125 → real exponent = 125 - 127 = -2
Mantissa (23 bits + implicit 1): 1.00110011001100110011010

Full 32-bit pattern:
   S     Exponent      Mantissa
   ↓         ↓              ↓
  0    01111101    00110011001100110011010

Binary:  0011 1110 1001 1001 1001 1001 1001 1010
Hex:     3E       99       99       9A
         └───────┬────────┬────────┘
                 0.3f

Approximate real value: 0.29999999701976776123046875

Bonus: The magic moment – 0.1f + 0.2f result

When you add 0.1f + 0.2f in float precision:

• The exact sum before final rounding ≈ 0.30000001192092896
• After rounding back to 23 mantissa bits + proper exponent → it lands exactly on the same bit pattern as 0.3f!

Resulting bit pattern is identical to 0.3f:

0011 1110 1001 1001 1001 1001 1001 1010   →  3E99999A  (hex)

That's why in float:
0.1f + 0.2f == 0.3f → true
(while in double it remains false)

Hope these bit-level breakdowns make the floating-point mystery a bit more visual and understandable! 🧩