754 Floating-Point Bit Tool

IEEE 754 Binary Converter

Convert decimal numbers to IEEE 754 binary or decode IEEE 754 bit patterns back to decimal. Analyze the sign bit, biased exponent, fraction field, unbiased exponent, hexadecimal representation, number classification and precision format.

32-bit Single Precision 64-bit Double Precision Decimal to IEEE 754 IEEE 754 to Decimal Sign / Exponent / Fraction Hex Representation

IEEE 754 Encoder & Decoder 32-bit + 64-bit

Decimal to IEEE 754 Binary

Encode a decimal value into its actual IEEE 754 stored bit pattern.

Decimal Value You may also enter Infinity, -Infinity or NaN.

IEEE 754 Format

IEEE 754 Encoded Result

–

Sign –

Exponent –

Fraction / Mantissa Field –

Stored Exponent –

Unbiased Exponent –

Exponent Bias –

Classification –

Hexadecimal –

Precision –

Total Width –

IEEE 754 Binary to Decimal

Decode a complete 32-bit or 64-bit IEEE 754 bit pattern.

IEEE 754 Binary Bits Enter exactly 32 or 64 binary digits. Spaces are allowed and removed automatically.

IEEE 754 Decoded Result

–

Sign –

Exponent –

Fraction / Mantissa Field –

Stored Exponent –

Unbiased Exponent –

Exponent Bias –

Classification –

Hexadecimal –

Precision –

Total Width –

What Is IEEE 754 Floating-Point?

IEEE 754 is a widely used standard for representing floating-point numbers in computers. Instead of storing a binary number as a simple integer and fraction, IEEE 754 divides the bit pattern into a sign field, exponent field and fraction field.

The exponent is stored using a bias rather than as an ordinary signed integer. Normal finite numbers also use an implicit leading binary digit, allowing one extra bit of effective significand precision without physically storing that bit.

The standard also reserves special bit patterns for positive and negative zero, subnormal numbers, positive and negative infinity, and NaN values.

IEEE 754 Single vs Double Precision

Property	32-bit Single Precision	64-bit Double Precision
Total bits	32	64
Sign bits	1	1
Exponent bits	8	11
Fraction bits	23	52
Exponent bias	127	1023
Effective significand precision	24 binary digits for normal values	53 binary digits for normal values
Approximate decimal precision	About 7 significant digits	About 15 to 17 significant digits

How IEEE 754 Bit Fields Work

Sign Bit

A sign bit of 0 represents a positive value, while a sign bit of 1 represents a negative value.

0 = positive

1 = negative

Exponent Field

The exponent field stores a biased exponent. For normal single-precision values the true exponent is the stored exponent minus 127.

Single: E = stored exponent – 127

Double: E = stored exponent – 1023

Fraction Field

For normal numbers, IEEE 754 assumes an implicit leading 1 before the binary fraction field.

Fraction bits: 101…

Significand: 1.101…

Number Classification

Exponent and fraction combinations determine whether the value is normal, subnormal, zero, infinity or NaN.

IEEE 754 Normal Number Formula

For a normal finite IEEE 754 binary floating-point value:

Value = (-1)^Sign x (1 + Fraction) x 2^(Stored Exponent – Bias)

The fraction represents the sum of the fraction bits using binary place values such as 2^-1, 2^-2, 2^-3 and so on.

IEEE 754 Special Value Patterns

Exponent Field	Fraction Field	Classification	Meaning
All zeros	All zeros	Zero	Positive or negative zero depending on sign
All zeros	Nonzero	Subnormal	Very small finite number without implicit leading 1
Neither all zero nor all one	Any	Normal	Ordinary finite floating-point value
All ones	All zeros	Infinity	Positive or negative infinity
All ones	Nonzero	NaN	Not-a-Number

Worked Example 1: Decimal 13.25 to IEEE 754 Single Precision

Decimal 13.25 can be represented exactly in binary.

Decimal value: 13.25

Binary value: 1101.01

Normalized: 1.10101 x 2^3

Sign bit: 0

True exponent: 3

Exponent bias: 127

Stored exponent: 3 + 127 = 130

130 binary: 10000010

Fraction field: 10101000000000000000000

IEEE 754:

0 10000010 10101000000000000000000

Full bits: 01000001010101000000000000000000

Hex: 0x41540000

Worked Example 2: Decode 32-bit IEEE 754

Bits: 01000001010101000000000000000000

Sign: 0

Exponent: 10000010

Fraction: 10101000000000000000000

Stored exponent: 130

True exponent: 130 – 127 = 3

Significand: 1.10101 binary

1.10101 binary = 1.65625 decimal

1.65625 x 2^3 = 13.25

Answer: 13.25

Worked Example 3: Negative IEEE 754 Number

For decimal -7.5:

7.5 binary = 111.1

Normalized = 1.111 x 2^2

Sign bit = 1

True exponent = 2

Stored single exponent = 2 + 127 = 129

129 binary = 10000001

Fraction = 11100000000000000000000

IEEE 754 = 1 10000001 11100000000000000000000

Worked Example 4: Decimal 0.1 and Rounding

Decimal 0.1 does not have a finite binary expansion. Its binary representation repeats, so IEEE 754 must store a rounded approximation.

0.1 decimal = repeating binary fraction

Single precision hex = 0x3DCCCCCD

Single precision stored value is approximately 0.10000000149

Double precision stores many more fraction bits and therefore provides a closer approximation.

Why IEEE 754 Uses an Exponent Bias

IEEE 754 exponent fields are unsigned binary fields. A bias allows both positive and negative mathematical exponents to be represented without a separate exponent sign bit.

Format	Exponent Bits	Bias	Example Stored Exponent	True Exponent
Single	8	127	130	3
Single	8	127	125	-2
Double	11	1023	1026	3
Double	11	1023	1021	-2

Normal vs Subnormal IEEE 754 Numbers

Normal Numbers

Normal numbers have a nonzero, non-all-ones exponent field and use an implicit leading binary 1 in the significand.

Subnormal Numbers

Subnormal numbers use an all-zero exponent field and a nonzero fraction. They do not use the normal implicit leading 1, allowing representation of values closer to zero.

Single vs Double Precision Accuracy

Increasing the fraction field from 23 stored bits in single precision to 52 stored bits in double precision dramatically improves numerical precision.

Feature	Single Precision	Double Precision
Storage	32 bits	64 bits
Fraction field	23 bits	52 bits
Effective normal significand	24 bits	53 bits
Approximate decimal digits	About 7	About 15 to 17
Memory use	Lower	Higher
Typical precision	Moderate	High

Common IEEE 754 Single-Precision Examples

Decimal	IEEE 754 Hex	Classification
0	0x00000000	Positive zero
-0	0x80000000	Negative zero
1	0x3F800000	Normal
-1	0xBF800000	Normal
2	0x40000000	Normal
0.5	0x3F000000	Normal
13.25	0x41540000	Normal
Positive Infinity	0x7F800000	Infinity
Negative Infinity	0xFF800000	Infinity

IEEE 754 vs General Binary Floating-Point

Feature	IEEE 754	General Binary Floating-Point Notation
Purpose	Computer storage standard	Mathematical representation
Sign	Dedicated sign bit	Usually written with + or –
Exponent	Biased stored field	Ordinary signed exponent
Significand	Fraction field plus implicit leading bit for normals	Written explicitly
Special values	Zero, subnormal, infinity and NaN	Not inherently defined

Important IEEE 754 Considerations

Important: many decimal fractions cannot be represented exactly in binary floating-point. The stored IEEE 754 value may therefore differ slightly from the decimal value entered.

This is expected behavior and is one reason calculations such as decimal 0.1 plus 0.2 can produce a value extremely close to, but not mathematically identical to, 0.3 in binary floating-point systems.

Also remember that 32-bit single precision and 64-bit double precision can produce different stored approximations because they use different numbers of exponent and fraction bits.

Related BinaryCon Tools

IEEE 754 Binary Converter FAQs

What is IEEE 754?

IEEE 754 is a standard for floating-point arithmetic and binary floating-point representations used by computers and programming languages.

What are the three main parts of an IEEE 754 value?

The main fields are the sign bit, exponent field and fraction field. For normal values, the fraction contributes to a significand with an implicit leading 1.

How many bits are in IEEE 754 single precision?

Single precision uses 32 total bits: 1 sign bit, 8 exponent bits and 23 fraction bits.

How many bits are in IEEE 754 double precision?

Double precision uses 64 total bits: 1 sign bit, 11 exponent bits and 52 fraction bits.

What is the exponent bias in IEEE 754?

Single precision uses an exponent bias of 127, while double precision uses a bias of 1023.

What is the hidden or implicit leading bit?

Normal IEEE 754 values assume a leading binary 1 before the stored fraction field. Because that 1 is implied, it does not need to consume a stored fraction bit.

What is a subnormal IEEE 754 number?

A subnormal value has an all-zero exponent field and a nonzero fraction field. It represents very small values close to zero and does not use the usual implicit leading 1.

How is infinity represented in IEEE 754?

Infinity uses an exponent field containing all ones and a fraction field containing all zeros. The sign bit determines positive or negative infinity.

How is NaN represented?

NaN uses an all-ones exponent field and a nonzero fraction field. Different fraction patterns can represent different NaN encodings.

Why cannot IEEE 754 represent 0.1 exactly?

Decimal 0.1 has a repeating binary fractional expansion. Because an IEEE 754 value contains only a finite number of fraction bits, the repeating value must be rounded.

What is the difference between single and double precision?

Double precision uses more exponent and fraction bits, providing a much wider range and substantially greater numerical precision than single precision.

Is IEEE 754 the same as binary scientific notation?

No. Binary scientific notation describes a mathematical value using a significand and power-of-two exponent. IEEE 754 specifies exactly how floating-point values are encoded into stored bit fields.

Can this converter decode both 32-bit and 64-bit values?

Yes. A 32-bit input is interpreted as IEEE 754 single precision, while a 64-bit input is interpreted as IEEE 754 double precision.