What is the hinary equivalent of the decimal number 85.125?

The correct answer is (1010101.001).

To convert a decimal number with a fractional part to binary, we handle the integer part and the fractional part separately, then combine them with a binary point.

Step 1: Convert the Integer Part (85) to Binary

We repeatedly divide 85 by 2 and record the remainders (read from bottom to top):

85 ÷ 2 = 42    remainder   1
42 ÷ 2 = 21    remainder   0
21 ÷ 2 = 10    remainder   1
10 ÷ 2 = 5     remainder   0
5 ÷ 2 = 2      remainder   1
2 ÷ 2 = 1      remainder   0
1 ÷ 2 = 0      remainder   1

Reading the remainders from bottom to top:

85 in binary = 1010101

Verification:

$2^6+2^4+2^2+2^0=64+16+4+1=85✓$

Step 2: Convert the Fractional Part (0.125) to Binary

We repeatedly multiply the fractional part by 2 and record the integer part of each result (read from top to bottom):

0.125 × 2 = 0.250    → integer part = 0
0.250 × 2 = 0.500    → integer part = 0
0.500 × 2 = 1.000    → integer part = 1

Reading the integer parts from top to bottom:

0.125 in binary = .001

Verification:

$2^{-3}=\frac{1}{8}=0.125✓$

Step 3: Combine Both Parts

Join the integer binary and fractional binary parts with a binary point:

Integer part:    1010101
Fractional part: .001

Combined: 1010101.001

Final Answer:

The binary equivalent of the decimal number 85.125 is:

(1010101.001)