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Bug fixes and improvements (#1348)

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* Add "reference" for EN version. Bug fixes.

* Unify the figure reference as "the figure below" and "the figure above".
Bug fixes.

* Format the EN markdown files.

* Replace "" with <u></u> for EN version and bug fixes

* Fix biary_tree_dfs.png

* Fix biary_tree_dfs.png

* Fix zh-hant/biary_tree_dfs.png

* Fix heap_sort_step1.png

* Sync zh and zh-hant versions.

* Bug fixes

* Fix EN figures

* Bug fixes

* Fix the figure labels for EN version
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Yudong Jin
2024-05-06 14:44:48 +08:00
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parent 8e60d12151
commit c4a7966882
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10
en/docs/chapter_data_structure/number_encoding.md
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@@ -14,11 +14,11 @@ Firstly, it's important to note that **numbers are stored in computers using the
- **One's complement**: The one's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by inverting all bits except the sign bit.
- **Two's complement**: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding $1$ to their one's complement.
The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:
The figure below illustrates the conversions among sign-magnitude, one's complement, and two's complement:
![Conversions between sign-magnitude, one's complement, and two's complement](number_encoding.assets/1s_2s_complement.png)
Although sign-magnitude is the most intuitive, it has limitations. For one, **negative numbers in sign-magnitude cannot be directly used in calculations**. For example, in sign-magnitude, calculating $1 + (-2)$ results in $-3$, which is incorrect.
Although <u>sign-magnitude</u> is the most intuitive, it has limitations. For one, **negative numbers in sign-magnitude cannot be directly used in calculations**. For example, in sign-magnitude, calculating $1 + (-2)$ results in $-3$, which is incorrect.
$$
\begin{aligned}
@@ -29,7 +29,7 @@ $$
\end{aligned}
$$
To address this, computers introduced the **one's complement**. If we convert to one's complement and calculate $1 + (-2)$, then convert the result back to sign-magnitude, we get the correct result of $-1$.
To address this, computers introduced the <u>one's complement</u>. If we convert to one's complement and calculate $1 + (-2)$, then convert the result back to sign-magnitude, we get the correct result of $-1$.
$$
\begin{aligned}
@@ -51,7 +51,7 @@ $$
\end{aligned}
$$
Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the **two's complement**. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:
Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the <u>two's complement</u>. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:
$$
\begin{aligned}
@@ -125,7 +125,7 @@ $$
![Example calculation of a float in IEEE 754 standard](number_encoding.assets/ieee_754_float.png)
Observing the diagram, given an example data $\mathrm{S} = 0$, $\mathrm{E} = 124$, $\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$, we have:
Observing the figure above, given an example data $\mathrm{S} = 0$, $\mathrm{E} = 124$, $\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$, we have:
$$
\text{val} = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875