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updated preliminaries xml
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‎mat101-book.xml

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@@ -38,10 +38,11 @@ along with MathBook XML. If not, see <http://www.gnu.org/licenses/>.
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<!-- such as a chapter or section -->
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<mathbook xmlns:xi="http://www.w3.org/2001/XInclude" xml:lang="en-US">
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<!-- <mathbook xmlns:xi="http://www.w3.org/2001/XInclude" xml:lang="fr-FR"> -->
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<docinfo>
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<latex-image-preamble>
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\usepackage{tikz}
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\usepackage{pgfplots}
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</latex-image-preamble>
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</docinfo>
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<book xml:id="mat101-book">

‎preliminaries.xml

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@@ -27,7 +27,7 @@ along with MathBook XML. If not, see <http://www.gnu.org/licenses/>.
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<!-- % Chapters now begin with Chapter 1 -->
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<title>Set 00-Preliminaries: Signed Numbers and Operations of Numbers</title>
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<introduction>
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<introduction>
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<p>Objectives:
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<ul>
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<li>Add, subtract, multiply and divide positive and negative
@@ -42,112 +42,109 @@ along with MathBook XML. If not, see <http://www.gnu.org/licenses/>.
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<title>Integer Addition and Subtraction</title>
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<p>A small picture of the integers is provided on the number line below:</p>
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<figure xml:id="figure-numberline">
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<image xml:id="numberline">
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<image xml:id="numberline" width="70%">
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<latex-image-code><![CDATA[
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\begin{tikzpicture} [scale = 0.4]
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\draw[<->] (-5.5,0) -- (5.5,0) node [below] {$\mathbb{R}$};
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\begin{tikzpicture} [scale = 0.5, transform shape]
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\draw[<->] (-5.5,0) -- (5.5,0);
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {\x};
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {$\x$};
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\end{tikzpicture}]]>
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</latex-image-code>
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</image>
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<caption> The Real Number Line </caption>
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</figure>
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<p>The number line actually goes on forever in both directions. The <b>positive integers</b> are the whole numbers to the right of (or greater than) <m>0</m>: <m>1, 2, 3, \dots</m>. The <b>negative integers</b> are the whole numbers to the left of (or less than) <m>0</m>: <m>-1, -2, -3, \dots</m>. Some interesting facts to note are:
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<p>The number line actually goes on forever in both directions. The <term>positive integers</term> are the whole numbers to the right of (or greater than) <m>0</m>: <m>1, 2, 3, \dots</m>. The <term>negative integers</term> are the whole numbers to the left of (or less than) <m>0</m>: <m>-1, -2, -3, \dots</m>. Some interesting facts to note are:
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<ul>
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<li>The integer <m>0</m> is neither positive nor negative.</li>
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<li>As we move to the right on the number line, the integers increase in value. <m>3</m> is greater than <m>2</m>, written <m>3\gt 2</m>. But <m>-2</m> is greater than <m>-3</m>, written <m>-2\gt -3</m>.</li>
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</ul>
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</p>
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<p>Recall that adding a <m>\color{green}{\text{positive}}</m> integer equates to moving <m>\color{green}{\text{right}}</m> on the number line.</p>
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<example xml:id="example-numberline-ex-1plus3">
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<example xml:id="example-numberline-ex-1plus3">
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<title>Add Positive <m>3</m> to <m>1</m></title>
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<!-- Our own-->
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<p><m>1+3</m> Start at <m>1</m> on the number line. </p>
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<figure xml:id="figure-numberline-ex-1plus3">
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<image xml:id="numberline-ex-1plus3">
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<p><md>
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<mrow>1+3\amp\amp\amp\text{Start at 1 on the number line.}</mrow></md>
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Move three positions right:</p>
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<figure xml:id="figure-numberline-ex-1plus3">
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<image xml:id="numberline-ex-1plus3" width="70%">
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<latex-image-code><![CDATA[
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\begin{tikzpicture} [scale = 0.4]
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\draw[<->] (-5.5,0) -- (5.5,0) node [below] {$\mathbb{R}$};
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\begin{tikzpicture} [scale = 0.5, transform shape]
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\draw[<->] (-5.5,0) -- (5.5,0);
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {\x};
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {$\x$};
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\draw[color=green, ->] (0:1) arc (180:15:.5);
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\draw[color=green, ->] (0:2) arc (180:15:.5);
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\draw[color=green, ->] (0:3) arc (180:15:.5);
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\filldraw[color=blue] (1,0) circle (.1);
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\filldraw[color=green] (4,0) circle (.1);
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\end{tikzpicture}]]>
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</latex-image-code>
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</image>
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<caption> <m>1+3</m> </caption>
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</figure>
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<p>Move three positions right.
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<m>4</m> Our Solution: We knew that!</p>
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</figure>
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<p><md>
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<mrow>4\amp\amp\amp\text{Our Solution: We knew that!}</mrow>
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</md></p>
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</example>
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<p>Similarly, adding a \(\color{red}{\text{negative}}\) integer equates to moving \(\color{red}{\text{left}}\) on the number line.</p>
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<example xml:id="example-numberline-ex-1minus3">
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<title>Add Negative <m>3</m> to <m>1</m></title>
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<!-- Our own-->
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<p><m>1+(-3)</m> Start at <m>1</m> on the number line. </p>
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<p><md><mrow>1+(-3)\amp\amp\amp\text{Start at }1\text{ on the number line.}</mrow></md></p>
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<p>Move three positions left.</p>
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<figure xml:id="figure-numberline-ex-1minus3">
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<image xml:id="numberline-ex-1minus3">
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<image xml:id="numberline-ex-1minus3" width="70%">
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<latex-image-code><![CDATA[
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\begin{tikzpicture} [scale = 0.4]
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\draw[<->] (-5.5,0) -- (5.5,0) node [below] {$\mathbb{R}$};
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {\x};
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\begin{tikzpicture} [scale = 0.5, transform shape]
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\draw[<->] (-5.5,0) -- (5.5,0);
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {$\x$};
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\draw[color=red, ->] (0:1) arc (0:165:.5);
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\draw[color=red, ->] (0:0) arc (0:165:.5);
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\draw[color=red, ->] (0:-1) arc (0:165:.5);
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\filldraw[color=blue] (1,0) circle (.1);
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\filldraw[color=red] (-2,0) circle (.1);
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\end{tikzpicture}]]>
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</latex-image-code>
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</image>
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<!--<caption> <m>1+(-3)</m></caption> -->
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</figure>
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<p>Move three positions left. <m>-2</m> Our Solution</p>
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<p><md><mrow>-2\amp\amp\amp\text{Our Solution}</mrow></md></p>
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</example>
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<example xml:id="example-numberline-ex-neg1minus2">
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<title>Add Negative <m>2</m> to <m>-1</m></title>
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<!-- Our own-->
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<p><m>-1+(-2)</m> Start at <m>-1</m> on the number line. </p>
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<figure xml:id="figure-numberline-ex-neg1minus2">
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<image xml:id="numberline-ex-neg1minus2">
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<image xml:id="numberline-ex-neg1minus2" width="70%">
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<latex-image-code><![CDATA[
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\begin{tikzpicture} [scale = 0.4]
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\draw[<->] (-5.5,0) -- (5.5,0) node [below] {$\mathbb{R}$};
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {\x};
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\begin{tikzpicture} [scale = 0.5, transform shape]
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\draw[<->] (-5.5,0) -- (5.5,0);
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\foreach \x in {-5,...,5}
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\draw (\x, 0.1) -- (\x, -0.1) node [below] {$\x$};
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\draw[color=red, ->] (0:-1) arc (0:165:.5);
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\draw[color=red, ->] (0:-2) arc (0:165:.5);
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\filldraw[color=blue] (-1,0) circle (.1);
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\filldraw[color=red] (-3,0) circle (.1);
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\end{tikzpicture}]]>
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</latex-image-code>
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</image>
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<caption> <m>-1+(-2)</m></caption>
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</figure>
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<p>Move two positions left. <m>-3</m> Our Solution</p>
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</example>
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<p>Notice in Example<nbsp /><xref ref="example-numberline-ex-neg1minus2" /> that adding two negative numbers is equivalent to adding the two corresponding positive numbers, but the final result is negative: <m>1+2=3</m> and <m>-1+(-2) = -3</m>. However in Example<nbsp /><xref ref="example-numberline-ex-1minus3" /> we see that adding a negative number to a positive number is actually related to subtraction: <m>1+(-3)=-2</m> can be computed by first subtracting the corresponding positive integers, but the final result is negative: <m>3-1=2</m> and <m>1+(-3)=-2</m>.</p>
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<example xml:id="example-add-two-neg">
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<title>Add Two Negative Integers</title>
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<p><md><mrow>-4+(-11)\amp\amp\amp\text{Adding two negatives: add positives}</mrow>
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<mrow>4+11=15\amp\amp\amp\text{The result is negative}</mrow>
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<mrow>-15\amp\amp\amp\text{Our Solution}</mrow>
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</md></p>
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</example>
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</section>
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<section xml:id="sec-int-mult-div">
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<section xml:id="sec-work-with-vars">
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<title>Working with Variables</title>
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</section>
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