Why must (5A.B)16 be converted to decimal first?

XS-3 is defined for decimal digits

2078 Kartik · 2 marks

Convert (5A.B)₁₆ to Excess-3

Short answer

First convert (5A.B)₁₆ to decimal (90.6875)₁₀. Then add 3 to each decimal digit’s 4-bit BCD to obtain excess-3: 1100 0011.1001 1011 1010 1000.

Convert (5A.B)₁₆ = (?)Excess-3.

Excess-3 code is defined for decimal digits, so first convert the hexadecimal number to decimal, then add 3 (0011) to the BCD code of each decimal digit.

Step 1: Hex to decimal

\begin{aligned} (5A.B)_{16} &= 5 \times 16^1 + 10 \times 16^0 + 11 \times 16^{-1} \\ &= 80 + 10 + 0.6875 \\ &= (90.6875)_{10} \end{aligned}

Step 2: Each digit + 3, written in 4 bits

Digit	9	0	.	6	8	7	5
Digit + 3	12	3	.	9	11	10	8
XS-3	1100	0011	.	1001	1011	1010	1000

Answer: $(5A.B)_{16} = (90.6875)_{10} = (1100\ 0011.1001\ 1011\ 1010\ 1000)_{XS\text{-}3}$

✓ Answer checked by Claude Opus 5.5 (high effort)

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