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Dealing With Fractional Parts

Understand how base conversion rules change for numbers with a decimal point (fractional parts).

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Decimal Fractions to Other Bases

Explain repeated multiplication by base.

When dealing with fractional parts (like .25 or .675), the rules for base conversion flip. Instead of dividing the integer part, we use Repeated Multiplication on the fractional part.

The Repeated Multiplication Method

Follow these steps to convert a decimal fraction to another base:

  • Step 1: Multiply the fractional part by the target base (e.g., 2, 8, or 16).
  • Step 2: Record the resulting integer part (this becomes a digit in your new base).
  • Step 3: Take the new fractional part and multiply it by the base again.
  • Step 4: Repeat until the fractional part is 0 (or until it starts repeating).
  • Step 5: Read the recorded integer parts from TOP to BOTTOM.
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Fractional Multiplication Process

Visual of 0.675 x 8 for octal conversion.

Polished process flowchart showing repeated multiplication of 0.675 by 8. Node 1 shows 0.675 x 8 = 5.400 with extracted integer 5. Node 2 shows 0.400 x 8 = 3.200 with extracted integer 3. Node 3 shows 0.200 x 8 = 1.600 with extracted integer 1. Node 4 shows 0.600 x 8 = 4.800 with extracted integer 4. Node 5 shows 0.800 x 8 = 6.400 with extracted integer 6. A large downward-pointing arrow beside the extracted integers 5, 3, 1, 4, 6 is labeled 'Top to Bottom'. Pastel gradient fills, elegant sans-serif typography, generous whitespace, subtle connecting arrows, light neutral background.
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Reading the extracted integers from top to bottom, the decimal 0.6750.675 is approximately:

(0.675)10≈(0.53146)8(0.675)_{10} \approx (0.53146)_8

If you continued the process, the next step would be 0.400×80.400 \times 8, which restarts the cycle from Node 2. This means the octal value is actually a repeating fraction:

(0.675)10=(0.53146‾)8(0.675)_{10} = (0.5\overline{3146})_8

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Other Base Fractions to Decimal

Explain negative positional weights.

When converting numbers with a decimal point back to base-10, the positional values flip from positive exponents to negative exponents.

The Rule for Fractional Weights

Just as integer positions increase to the left (0,1,2...0, 1, 2...), fractional positions decrease to the right. The first digit immediately after the decimal point is position -1, the next is -2, and so on.

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Visualizing Negative Positional Weights

Diagram showing positional values for integers vs fractions.

Clean scientific diagram visualizing the binary number 101.101 spanning across a central decimal point. The integer digits 1, 0, 1 have position powers 2^0, 2^1, 2^2 branching upwards and reading right-to-left. The fractional digits 1, 0, 1 have position powers 2^-1, 2^-2, 2^-3 branching downwards and reading left-to-right. Pastel color palette, elegant typography, precise labeling, translucent layers for cross-sections, white background, soft shadows.
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Integer powers branch up and left; fractional powers branch down and right.

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Grouping Fractional Bits

Explain how grouping direction changes for fractions.

🔑 Key Takeaway: Grouping Direction Flips!
When converting binary to Octal or Hexadecimal by grouping bits, pay close attention to the decimal point:
→ Integer Part: Group from Right-to-Left (start at the decimal and move left).
→ Fractional Part: Group from Left-to-Right (start at the decimal and move right).
Note: If you run out of bits at the end of the fraction, pad with 0s on the right to complete the group.
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Fractional Hex Grouping

Convert fractional binary to hex.

An engineer needs to convert the binary number (10101100.010111)2(10101100.010111)_2 into hexadecimal to configure a color display setting. What is the correct hexadecimal representation?