TAOCP 4.4 Exercise 11
We are asked to convert $(5772)_{10}$ to decimal.
Exercise 11. [16] Convert $(5772)_{10}$ to decimal.
Verified: yes
Solve time: 1m16s
Solution
We are asked to convert $(5772)_{10}$ to decimal. Since the source and target radix are both $10$, this is an identity conversion. Therefore, the decimal representation of $5772$ is the same as the original number. Explicitly, writing the digits from left to right in the usual positional notation, we have
$(5772)_{10} = 5 \cdot 10^3 + 7 \cdot 10^2 + 7 \cdot 10^1 + 2 \cdot 10^0.$
Evaluating each term gives
$$ 5 \cdot 10^3 = 5000, \qquad 7 \cdot 10^2 = 700, \qquad 7 \cdot 10^1 = 70, \qquad 2 \cdot 10^0 = 2. $$
Summing these contributions yields
$$ 5000 + 700 + 70 + 2 = 5772. $$
Thus, the decimal digits are
$$ U_3 = 5, \quad U_2 = 7, \quad U_1 = 7, \quad U_0 = 2. $$
Hence, the decimal representation of $5772$ is
$$ \boxed{5772}. $$
This completes the proof.
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