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UPSC 2016 Maths Optional Paper 2 Q5d — Step-by-Step Solution

10 marks · Section B

Number systems · Numerical Analysis · asked 7× in 13 yrs · Read the full method →

Question

Convert the following decimal numbers to equivalent binary and hexadecimal numbers:

• (i) $4096$
• (ii) $0.4375$
• (iii) $2048.0625$

Technique

Recognise the powers of two ($4096=2^{12}$, $2048=2^{11}$, $0.0625=2^{-4}$) for instant binary; repeated $\times2$ for the $0.4375$ fraction; group binary digits in nibbles of four to read hex directly.

Solution

Method. Integer part: repeated division by $2$ (binary) / by $16$ (hex), reading remainders bottom-up. Fractional part: repeated multiplication by $2$ / by $16$, reading the integer carries top-down. Each hex digit groups $4$ binary digits.

(i) $4096$

$4096=2^{12}$, so in binary it is a $1$ followed by twelve $0$s:

Grouping in fours from the right: $0001\ 0000\ 0000\ 0000_2 = (1)(0)(0)(0)_{16}$, i.e. hex $1000$:

(Check: $16^3=4096$, so $1000_{16}=4096$. ✓)

(ii) $0.4375$

Multiply the fraction by $2$ repeatedly, recording carries:

Reading carries top-down: $0.0111_2$. Indeed $\tfrac14+\tfrac18+\tfrac1{16}=0.4375$. For hex, $0.4375\times16=7.0$, a single digit $7$:

(Consistency: group the binary fraction in fours after the point — $0111_2=7_{16}$. ✓)

(iii) $2048.0625$

Integer part $2048=2^{11}$: a $1$ followed by eleven $0$s,

Fractional part $0.0625=\tfrac{1}{16}=2^{-4}$:

So $2048.0625=(100\,0000\,0000.0001)_2$. Hex. $2048=16^2\cdot 8=8\cdot256$, so integer part $=800_{16}$; fraction $0.0625\times16=1.0\Rightarrow 0.1_{16}$:

Answer