Binary fraction converter

What is a binary fraction?

A binary fraction is a number expressed in base 2 that includes digits after the binary point, just as decimal numbers have digits after the decimal point. The binary number system uses only two digits — 0 and 1 — and represents all values using powers of two. When a binary number includes a fractional part, each digit after the binary point represents a negative power of two.

For example, the binary number 101.101 represents:

$$1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 + 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3}$$ $$= 4 + 0 + 1 + 0.5 + 0 + 0.125 = 5.625$$

Thus, 101.101₂ = 5.625₁₀.

How the binary fraction converter works

The binary fraction converter helps you convert any fractional number between binary and decimal systems automatically. You can also convert binary fractions to other number systems such as octal (base 8), hexadecimal (base 16), or any custom base between 2 and 36.

The process involves:

1. Interpreting the integer part by summing the powers of 2 for each ‘1’ digit.
2. Converting the fractional part by summing the corresponding negative powers of 2.
3. Combining both parts to get the full decimal value or converting backward to binary by repeatedly dividing or multiplying by 2.

This converter operates instantly — there’s no need to press “calculate”, as results adjust automatically when input values change.

Step-by-step example

Let’s convert $10.625_{10}$ into binary.

1. Convert the integer part (10):

Reading remainders from bottom to top:

$$10_{10} = 1010_2$$

2. Convert the fractional part (0.625):

Thus, $0.625_{10} = 0.101_2$.

3. Combine both parts:

$$10.625_{10} = 1010.101_2$$

Converting a binary fraction to decimal

Convert $110.011_2$ to decimal:

$$(1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) + (0 \times 2^{-1}) + (1 \times 2^{-2}) + (1 \times 2^{-3})$$ $$= 4 + 2 + 0 + 0 + 0.25 + 0.125 = 6.375_{10}$$

Hence, 110.011₂ = 6.375₁₀.

Converting binary fractions to other bases

To octal (base 8)

Group bits in sets of three from the binary point outward (integer part left, fractional part right). Fill with zeros if necessary.

Example: $1010.101_2$

$$1010.101_2 = (001\ 010.101)_2 = 12.5_8$$

To hexadecimal (base 16)

Group bits in sets of four:

$$1010.101_2 = (1010.1010)_2 = A.A_16$$

Hence $1010.101_2 = A.A_{16}$.

Notes on binary fractions

• Some decimal fractions cannot be represented exactly in binary (e.g., 0.1, 0.2, 0.3). They form repeating binary sequences, similar to how 1/3 = 0.333… in decimal notation.
• Computers internally handle real numbers in floating-point format, rigorously adhering to binary fraction representations, which is why small rounding errors occasionally occur in programming.
• The maximum precision depends on the chosen number of bits for the fractional part — the more bits, the higher the accuracy.

Historical insight

The binary number system traces back to the 17th century, formalized by Gottfried Wilhelm Leibniz, who recognized its connection to logic using only two symbols: 0 and 1. In modern computing, binary fractions became the foundation for digital signal encoding and numerical computation, allowing devices to perform arithmetic operations with incredible precision.

Frequently Asked Questions

How to convert 7.75 to binary step by step?

Integer part: $7_{10} = 111_2$. Fractional part: $0.75 \times 2 = 1.5$ → 1; $0.5 \times 2 = 1.0$ → 1. Combine both parts → $7.75_{10} = 111.11_2$.

Why can’t some decimal fractions be exactly converted to binary?

Because binary represents fractions as sums of reciprocals of powers of two, only numbers expressible as a sum of $1/2, 1/4, 1/8, ...$ can be exact. Fractions like 0.1 (which require $1/10$) don’t end within this series, leading to an infinite repeating sequence.

How to convert binary fraction 0.011 to decimal?

Evaluate using formula:

$$(0 \times 2^{-1}) + (1 \times 2^{-2}) + (1 \times 2^{-3}) = 0 + 0.25 + 0.125 = 0.375_{10}$$