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<title>3.3 Number Encoding * - Hello Algo</title>
<title>3.3 Number encoding * - Hello Algo</title>
@@ -153,7 +153,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
</div>
@@ -201,7 +201,13 @@
<li class="md-select__item">
<a href="/" hreflang="zh" class="md-select__link">
中文
简体中文
</a>
</li>
<li class="md-select__item">
<a href="/zh-hant/" hreflang="zh-Hant" class="md-select__link">
繁體中文
</a>
</li>
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<span class="md-ellipsis">
0.1 About This Book
0.1 About this book
</span>
@@ -414,7 +420,7 @@
<span class="md-ellipsis">
0.2 How to Read
0.2 How to read
</span>
@@ -491,7 +497,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
<span class="md-ellipsis">
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</span>
@@ -507,7 +513,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
Chapter 1. Introduction to Algorithms
Chapter 1. Introduction to algorithms
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -524,7 +530,7 @@
<span class="md-ellipsis">
1.1 Algorithms are Everywhere
1.1 Algorithms are everywhere
</span>
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<span class="md-ellipsis">
1.2 What is an Algorithm
1.2 What is an algorithm
</span>
@@ -626,7 +632,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
<span class="md-ellipsis">
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</span>
@@ -642,7 +648,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity Analysis
Chapter 2. Complexity analysis
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
2.1 Algorithm Efficiency Assessment
2.1 Algorithm efficiency assessment
</span>
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<span class="md-ellipsis">
2.2 Iteration and Recursion
2.2 Iteration and recursion
</span>
@@ -701,7 +707,7 @@
<span class="md-ellipsis">
2.3 Time Complexity
2.3 Time complexity
</span>
@@ -722,7 +728,7 @@
<span class="md-ellipsis">
2.4 Space Complexity
2.4 Space complexity
</span>
@@ -805,7 +811,7 @@
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
<span class="md-ellipsis">
Chapter 3. Data Structures
Chapter 3. Data structures
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
Chapter 3. Data Structures
Chapter 3. Data structures
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -838,7 +844,7 @@
<span class="md-ellipsis">
3.1 Classification of Data Structures
3.1 Classification of data structures
</span>
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<span class="md-ellipsis">
3.2 Fundamental Data Types
3.2 Fundamental data types
</span>
@@ -889,7 +895,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
@@ -900,7 +906,7 @@
<span class="md-ellipsis">
3.3 Number Encoding *
3.3 Number encoding *
</span>
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<li class="md-nav__item">
<a href="#331-integer-encoding" class="md-nav__link">
<span class="md-ellipsis">
3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
</span>
</a>
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<li class="md-nav__item">
<a href="#332-floating-point-number-encoding" class="md-nav__link">
<span class="md-ellipsis">
3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
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<span class="md-ellipsis">
3.4 Character Encoding *
3.4 Character encoding *
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Array and linked list
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1094,7 +1100,7 @@
<span class="md-ellipsis">
4.2 Linked List
4.2 Linked list
</span>
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<span class="md-ellipsis">
4.4 Memory and Cache
4.4 Memory and cache
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</span>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stack and queue
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1290,7 +1296,7 @@
<span class="md-ellipsis">
5.3 Double-ended Queue
5.3 Double-ended queue
</span>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
<span class="md-ellipsis">
Chapter 6. Hash Table
Chapter 6. Hash table
</span>
@@ -1385,7 +1391,7 @@
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash Table
Chapter 6. Hash table
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1402,7 +1408,7 @@
<span class="md-ellipsis">
6.1 Hash Table
6.1 Hash table
</span>
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<span class="md-ellipsis">
6.2 Hash Collision
6.2 Hash collision
</span>
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<span class="md-ellipsis">
6.3 Hash Algorithm
6.3 Hash algorithm
</span>
@@ -2019,7 +2025,7 @@
<li class="md-nav__item">
<a href="#331-integer-encoding" class="md-nav__link">
<span class="md-ellipsis">
3.3.1 &nbsp; Integer Encoding
3.3.1 &nbsp; Integer encoding
</span>
</a>
@@ -2028,7 +2034,7 @@
<li class="md-nav__item">
<a href="#332-floating-point-number-encoding" class="md-nav__link">
<span class="md-ellipsis">
3.3.2 &nbsp; Floating-Point Number Encoding
3.3.2 &nbsp; Floating-point number encoding
</span>
</a>
@@ -2070,12 +2076,12 @@
<!-- Page content -->
<h1 id="33-number-encoding">3.3 &nbsp; Number Encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<h1 id="33-number-encoding">3.3 &nbsp; Number encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>
</div>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer Encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<h2 id="331-integer-encoding">3.3.1 &nbsp; Integer encoding<a class="headerlink" href="#331-integer-encoding" title="Permanent link">&para;</a></h2>
<p>In the table from the previous section, we observed that all integer types can represent one more negative number than positive numbers, such as the <code>byte</code> range of <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.</p>
<p>Firstly, it's important to note that <strong>numbers are stored in computers using the two's complement form</strong>. Before analyzing why this is the case, let's define these three encoding methods:</p>
<ul>
@@ -2084,8 +2090,8 @@
<li><strong>Two's complement</strong>: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding <span class="arithmatex">\(1\)</span> to their one's complement.</li>
</ul>
<p>The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:</p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between Sign-Magnitude, One's Complement, and Two's Complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p>
<p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Conversions between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
<p align="center"> Figure 3-4 &nbsp; Conversions between sign-magnitude, one's complement, and two's complement </p>
<p>Although sign-magnitude is the most intuitive, it has limitations. For one, <strong>negative numbers in sign-magnitude cannot be directly used in calculations</strong>. For example, in sign-magnitude, calculating <span class="arithmatex">\(1 + (-2)\)</span> results in <span class="arithmatex">\(-3\)</span>, which is incorrect.</p>
<div class="arithmatex">\[
@@ -2139,7 +2145,7 @@
<p>It's important to note that this doesn't mean computers can only perform addition. <strong>By combining addition with basic logical operations, computers can execute a variety of other mathematical operations</strong>. For example, the subtraction <span class="arithmatex">\(a - b\)</span> can be translated into <span class="arithmatex">\(a + (-b)\)</span>; multiplication and division can be translated into multiple additions or subtractions.</p>
<p>We can now summarize the reason for using two's complement in computers: with two's complement representation, computers can use the same circuits and operations to handle both positive and negative number addition, eliminating the need for special hardware circuits for subtraction and avoiding the ambiguity of positive and negative zero. This greatly simplifies hardware design and enhances computational efficiency.</p>
<p>The design of two's complement is quite ingenious, and due to space constraints, we'll stop here. Interested readers are encouraged to explore further.</p>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-Point Number Encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-point number encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
<p>You might have noticed something intriguing: despite having the same length of 4 bytes, why does a <code>float</code> have a much larger range of values compared to an <code>int</code>? This seems counterintuitive, as one would expect the range to shrink for <code>float</code> since it needs to represent fractions.</p>
<p>In fact, <strong>this is due to the different representation method used by floating-point numbers (<code>float</code>)</strong>. Let's consider a 32-bit binary number as:</p>
<div class="arithmatex">\[
@@ -2166,8 +2172,8 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
(1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} \times 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]</div>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example Calculation of a float in IEEE 754 Standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example Calculation of a float in IEEE 754 Standard </p>
<p><a class="glightbox" href="../number_encoding.assets/ieee_754_float.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Example calculation of a float in IEEE 754 standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></a></p>
<p align="center"> Figure 3-5 &nbsp; Example calculation of a float in IEEE 754 standard </p>
<p>Observing the diagram, given an example data <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, we have:</p>
<div class="arithmatex">\[
@@ -2176,7 +2182,7 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
<p>Now we can answer the initial question: <strong>The representation of <code>float</code> includes an exponent bit, leading to a much larger range than <code>int</code></strong>. Based on the above calculation, the maximum positive number representable by <code>float</code> is approximately <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>, and the minimum negative number is obtained by switching the sign bit.</p>
<p><strong>However, the trade-off for <code>float</code>'s expanded range is a sacrifice in precision</strong>. The integer type <code>int</code> uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a <code>float</code>, the greater the difference between adjacent numbers.</p>
<p>As shown in the Table 3-2 , exponent bits <span class="arithmatex">\(E = 0\)</span> and <span class="arithmatex">\(E = 255\)</span> have special meanings, <strong>used to represent zero, infinity, <span class="arithmatex">\(\mathrm{NaN}\)</span>, etc.</strong></p>
<p align="center"> Table 3-2 &nbsp; Meaning of Exponent Bits </p>
<p align="center"> Table 3-2 &nbsp; Meaning of exponent bits </p>
<div class="center-table">
<table>
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3.2 Fundamental Data Types
3.2 Fundamental data types
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3.4 Character encoding *
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