The octal equivalent of the hexadecimal number (A.B)16 is:

Concept: hexadecimal and octal are both powers of two, so binary is the natural bridge between them — one hexadecimal digit is exactly 4 bits and one octal digit is exactly 3 bits. To convert hexadecimal to octal, expand every hexadecimal digit into its 4-bit group and then regroup that same bit string into 3-bit clusters, counting outward from the radix point: leftward through the integer part and rightward through the fractional part. Zero padding is added only at the two far ends, never next to the radix point, because a bit inserted there would shift the place value of every other bit.

Application to (A.B)16:

1. Expand each hexadecimal digit into 4 bits: A16 = 10102 and B16 = 10112, so (A.B)16 = (1010.1011)2.

2. Regroup the integer bits 1010 into 3-bit clusters counting leftward from the radix point: the cluster nearest the point is 010 and a single 1 is left over, so pad at the far left to get 001 010.

3. Read those clusters as octal digits: 0012 = 1 and 0102 = 2, so the integer part is 128.

4. Regroup the fractional bits 1011 into 3-bit clusters counting rightward from the radix point: the cluster nearest the point is 101 and a single 1 is left over, so pad at the far right to get 101 100.

5. Read those clusters as octal digits: 1012 = 5 and 1002 = 4, so the fractional part is .548.

6. Combine the two parts: (A.B)16 = (12.54)8.

Cross-check by an independent decimal route: A16 = 10 and the digit B after the point contributes 11/16, so (A.B)16 = 10.6875 in decimal. Dividing the integer 10 by 8 gives quotient 1 and remainder 2, i.e. 128. Multiplying the fraction repeatedly by 8 gives 0.6875 x 8 = 5.5, yielding the digit 5, and then 0.5 x 8 = 4.0, yielding the digit 4, i.e. .548. The process terminates exactly, so both methods agree: (A.B)16 = (12.54)8, with no rounding involved.

Common trap: padding the fractional bits on the wrong side. The fractional group must be completed with zeros at the far right, away from the radix point; padding it on the left instead shifts every fractional bit by one place and produces a different octal fraction.