IEEE 754 Floating Point Representation — Explained Simply

Introduction

IEEE 754 is the standard that defines how computers store decimal (floating-point) numbers in binary. If you’ve ever wondered why 0.1 + 0.2 doesn’t exactly equal 0.3 in programming, the answer lies in IEEE 754 representation. This topic is essential for computer science courses, GATE preparation, and understanding numerical computing.

This guide explains the format simply with worked examples so you can decode any floating-point bit pattern.

Why IEEE 754?

Computers work in binary, but we need to store numbers like 3.14, −0.001, and 1000000.5. Fixed-point representation wastes bits for very large or very small numbers. IEEE 754 uses a scientific notation approach:

Value = (−1)^sign × 1.mantissa × 2^(exponent − bias)

This allows a wide range of values with reasonable precision.

Format Structure

Single Precision (32-bit / float)

Field	Bits	Purpose
Sign	1 bit	0 = positive, 1 = negative
Exponent	8 bits	Biased exponent (bias = 127)
Mantissa	23 bits	Fractional part (implicit leading 1)

Total: 1 + 8 + 23 = 32 bits

Double Precision (64-bit / double)

Field	Bits	Purpose
Sign	1 bit	0 = positive, 1 = negative
Exponent	11 bits	Biased exponent (bias = 1023)
Mantissa	52 bits	Fractional part (implicit leading 1)

Total: 1 + 11 + 52 = 64 bits

Step-by-Step Conversion: Decimal to IEEE 754

Let’s convert −6.75 to 32-bit IEEE 754.

Step 1: Determine the Sign Bit

Number is negative → Sign = 1

Step 2: Convert Absolute Value to Binary

6.75 in binary:

• Integer part: 6 = 110
• Fractional part: 0.75 = 0.11 (0.75 × 2 = 1.5 → 1; 0.5 × 2 = 1.0 → 1)
• Combined: 110.11

Step 3: Normalize (Scientific Notation in Binary)

110.11 = 1.1011 × 2²

The implicit leading 1 is not stored. Mantissa = 1011 (padded with zeros to 23 bits).

Step 4: Calculate Biased Exponent

Actual exponent = 2 Biased exponent = 2 + 127 = 129 = 10000001 in binary

Step 5: Assemble

Sign	Exponent	Mantissa
1	10000001	10110000000000000000000

Result: 1 10000001 10110000000000000000000

Hex: 0xC0D80000

Step-by-Step: IEEE 754 to Decimal

Given: 0 10000010 10100000000000000000000

Step 1: Sign

Bit 0 = 0 → Positive

Step 2: Exponent

10000010 = 130 decimal Actual exponent = 130 − 127 = 3

Step 3: Mantissa

Stored: 10100000… Full significand: 1.101 (add implicit 1)

Step 4: Calculate Value

Value = 1.101 × 2³ = 1101.0 binary = 13.0 decimal

Special Values

Value	Sign	Exponent	Mantissa
+0	0	00000000	00…0
−0	1	00000000	00…0
+∞	0	11111111	00…0
−∞	1	11111111	00…0
NaN	0/1	11111111	non-zero

• Zero exponent + zero mantissa = Zero (positive or negative)
• All-ones exponent + zero mantissa = Infinity
• All-ones exponent + non-zero mantissa = NaN (Not a Number)

Why 0.1 + 0.2 ≠ 0.3

The decimal 0.1 cannot be represented exactly in binary (it becomes an infinite repeating fraction like 0.000110011…). When stored in finite bits, a tiny rounding error occurs. Adding two such imprecise values produces a result slightly off from the expected 0.3.

This is not a bug — it’s a fundamental property of binary floating-point representation.

Precision Limits

Format	Significant digits	Range
Single (32-bit)	~7 decimal digits	±3.4 × 10³⁸
Double (64-bit)	~15 decimal digits	±1.8 × 10³⁰⁸

Try It Online

Use the free IEEE 754 Converter on Numverto to convert any decimal number to its 32-bit and 64-bit representations. The tool shows sign, exponent, and mantissa fields separately with hex output. Verify your exam answers instantly.

Frequently Asked Questions

What is the bias in IEEE 754 single precision?

The bias is 127. The stored exponent = actual exponent + 127. This allows representing both positive and negative exponents without a separate sign bit for the exponent.

Why is there an implicit leading 1?

In normalized numbers, the binary scientific notation always starts with 1.xxx. Since this leading 1 is always present, storing it would waste a bit. By leaving it implicit, we gain one extra bit of precision for free.

What is a denormalized (subnormal) number?

When the exponent field is all zeros but mantissa is non-zero, the number is denormalized. These represent very small values close to zero, with implicit leading 0 instead of 1, allowing gradual underflow.

Can IEEE 754 represent all decimal numbers exactly?

No. Only numbers that can be expressed as a finite sum of negative powers of 2 are exact. For example, 0.5 (= 2⁻¹) is exact, but 0.1 (= infinite binary fraction) is not. This causes the famous floating-point rounding errors.

What languages use IEEE 754?

Virtually all modern languages: C/C++ (float/double), Java (float/double), Python (float), JavaScript (all numbers are 64-bit doubles), and Rust (f32/f64) all follow IEEE 754.