Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 4 - Number Representation and Calculation - 4.2 Number Bases in Positional Systems - Exercise Set 4.2: 36

Answer

$3013_{\text{four}}$

Work Step by Step

The place values of base four are: $..., 4^4, 4^3, 4^2, 4^1, 1$. The place values that are less than or equal to $199$ are $4^3$, $4^2$, $4^1$, and $1$ Divide $199$ by $4^3$ or $64$ to obtain: $199 \div 64 = 3$ remainder $7$ Divide the remainder $7$ by $4^2$ or $16$ to obtain: $7 \div 16 = 0$ remainder $7$ Divide the remainder $7$ by $4^1$ or $4$ to obtain: $7 \div 4 = 1$ remainder $3$ Divide the remainder $3$ by $1$ to obtain: $3 \div 1 = 3$ remainder $0$ Thus, $199=(3 \times 4^3) + (0 \times 4^2) + (1 \times 4^1) + (3 \times 1) \\199=3013_{\text{four}}$
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