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Cleanup narrative, fix broken lineno equations

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John Winans 2018-05-18 23:04:42 -05:00
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book/binary/chapter.tex
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@ -147,14 +147,16 @@ In a line of code the above might read like this: \verb@output = A ^ B@
A binary integer is constructed with only 1s and 0s in the same
manner as decimal numbers are constructed with values from 0 to 9.
Counting in binary is the same as in decimal. For example, when
adding 1 to 9, the carry is added to the next place value. When
subtracting 1 from 0, a borrow is required and so on.
Counting in binary (base-2) uses the same basic rules as decimal (base-10).
The difference comes in when we consider that there are ten decimal digits and
only two binary digits. Therefore, in base-10, we must carry when adding one to
nine (because there is no digit representing a ten) and, in base-2, we must
carry when adding one to one (because there is no digit representing a two.)
Figure~\autoref{Figure:integers} shows an abridged table of the
decimal, binary and hexadecimal values from 0 to 129.
\autoref{Figure:integers} shows an abridged table of the decimal, binary and hexadecimal
values ranging from $0_{10}$ to $129_{10}$.
\begin{figure}[ht]
\begin{figure}[t]
\begin{center}
\begin{tabular}{|c|c|c||c|c|c|c|c|c|c|c||c|c|}
\hline
@ -196,17 +198,27 @@ $10^2$ & $10^1$ & $10^0$ & $2^7$ & $2^6$ & $2^5$ & $2^4$ & $2^3$ & $2^2$ & $2^1$
\label{Figure:integers}
\end{figure}
One way to look at this table is on a per-row basis where each place
value is represented by the base raised to the power of the place value
position (shown in the column headings.) This is useful when
converting arbitrary values between bases. For example to interpret
the decimal value on the fourth row:
\[ 0 \times 10^2 + 0 \times 10^1 + 3 \times 10^0 = 3_{10} \]
And to interpret binary value on the same row by converting it to decimal:
\[ 0 \times 2^7 + 0 \times 2^6 +0 \times 2^5 +0 \times 2^4 +0 \times 2^3 +0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 3_{10} \]
And the same for the hexadecimal value:
\[ 0 \times 16^1 + 3 \times 16^0 = 3_{10} \]
\begin{equation}
0 \times 10^2 + 0 \times 10^1 + 3 \times 10^0 = 3_{10}
\end{equation}
Interpreting the binary value on the fourth row by converting it to decimal:
\begin{equation}
0 \times 2^7 + 0 \times 2^6 +0 \times 2^5 +0 \times 2^4 +0 \times 2^3 +0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 3_{10}
\end{equation}
Interpreting the hexadecimal value on the fourth row by converting it to decimal:
\begin{equation}
0 \times 16^1 + 3 \times 16^0 = 3_{10}
\end{equation}
Another way to look at this table is on a per-column basis. When