UPSC 2023 Maths Optional Paper 2 Q5c-i — Step-by-Step Solution

5 marks · Section B

Algebra of Binary Numbers · Numerical Analysis · asked 2× in 13 yrs · Read the full method →

Question

Evaluate, using the binary arithmetic, the following numbers in their given system: (i) $(634.235)_8-(132.223)_8$

3-bit-per-octal-digit conversion; binary subtraction with borrow propagation; regroup.

The procedure:

Step 1 — Convert to binary.

(634.235)_8=110\,011\,100\,.\,010\,011\,101_2,\qquad(132.223)_8=001\,011\,010\,.\,010\,010\,011_2.

Step 2 — Binary subtraction (rightmost to leftmost; borrow when needed).

Position	$2^{-9}$	$2^{-8}$	$2^{-7}$	$2^{-6}$	$2^{-5}$	$2^{-2}$
minuend	1	0	1	1	1	1
subtrahend	1	1	0	0	1	1
difference	0	1 (borrow)	0	1	0	0

Position	$2^1$	$2^2$	$2^3$	$2^4$	$2^6$	$2^7$	$2^8$
minuend	0	1	1	1	0	1	1
subtrahend	1	0	1	1	1	0	0
difference	1 (borrow)	0	0	0	1 (borrow)	0	1

Reading bits from $2^8$ down to $2^{-9}$ : $\boxed{101\,000\,010\,.\,000\,001\,010_2}$ .

Step 3 — Regroup in threes back to octal.

101\;000\;010\,.\,000\;001\;010_2=(5\,0\,2\,.\,0\,1\,2)_8.

\boxed{\;(634.235)_8-(132.223)_8=(502.012)_8.\;}