In binary, decimal fractions can be represented in a manner similar to how they are in base 10. The position of each digit after the binary point represents a power of 2, just as in base 10 the digits represent powers of 10.

Representation of a Binary Decimal

Let’s consider the binary decimal 0.101₂ as an example.

Each position after the binary point represents a negative power of 2:

• The first digit after the point represents $2^{-1}$,
• The second digit represents $2^{-2}$,
• The third digit represents $2^{-3}$, and so on.

Now, let's break down 0.101₂:

$0.101₂ = 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3}$

Calculating each term:

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

Now sum these values:

$0.5 + 0 + 0.125 = 0.625$

So, $0.101₂$ is equal to $0.625_{10}$ in base 10.

Converting a Base 10 Decimal into a Binary Decimal

To convert a base 10 decimal into a binary decimal, the approach is to repeatedly multiply the decimal part by 2 and record the integer part (either 0 or 1) until you reach a repeating pattern or desired accuracy.

Example:

Convert $0.625_{10}$ to binary.

1. Multiply $0.625 \times 2 = 1.25$. The integer part is 1.
2. Take the fractional part $0.25$ and multiply by 2: $0.25 \times 2 = 0.5$. The integer part is 0.
3. Take the fractional part $0.5$ and multiply by 2: $0.5 \times 2 = 1.0$. The integer part is 1, and there’s no remaining fraction.

So, $0.625_{10} = 0.101₂$.

Summary

• To represent a binary decimal: Use powers of 2, similar to how powers of 10 work in decimal notation.
• To convert from binary to decimal: Multiply each digit by decreasing powers of 2, starting from $2^{-1}$.
• To convert from decimal to binary: Multiply the decimal part by 2, taking the integer part at each step, and continue with the fractional part.

Would you like more details on binary conversions, or do you have any questions?

Related Questions:

1. How would you convert a repeating decimal in binary to base 10?
2. What are some methods to handle rounding errors in binary decimals?
3. How does the concept of precision impact floating-point binary numbers?
4. Can binary decimals represent all real numbers accurately?
5. What role do binary decimals play in computer arithmetic?

Tip: When converting decimal to binary, be cautious of repeating patterns. They often signal an infinite repeating binary fraction, just like $\frac{1}{3}$ in decimal becomes $0.333...$.