openfreebooks-bookgenerator/content/math/integer-foundations/core.html

<div class="book-prose">
  <p>
    Integers extend whole numbers so we can model values above, below, and exactly at a reference
    point. They appear in temperature, sea level, game scores, electricity, and money balances.
    Without integers, many real situations would be awkward to calculate.
  </p>

  <h2 id="history" class="book-prose__heading">A short history of the number line idea</h2>
  <p>
    The modern number line grew over time. Early printed works arranged numbers in order, and later
    mathematicians described arithmetic as movement along a line. By the late 1600s, this “walk
    forward / backward” idea was used to explain addition and subtraction in a geometric way.
  </p>
  <p>
    The key shift was not just drawing numbers in order, but treating position on a line as a model
    for operations. That model is now foundational in school mathematics.
  </p>

  <h2 id="number-line" class="book-prose__heading">Drawing and reading a number line</h2>
  <p>
    A number line is a straight line with equally spaced marks. Numbers increase to the right and
    decrease to the left. Zero is the center reference point for positive and negative values.
  </p>
  <ul>
    <li><strong>Right of 0:</strong> positive integers (\(+1, +2, +3, \dots\)).</li>
    <li><strong>Left of 0:</strong> negative integers (\(-1, -2, -3, \dots\)).</li>
    <li><strong>At 0:</strong> neither positive nor negative.</li>
  </ul>
  <p>
    In coordinate graphs, both axes are number lines: the horizontal \(x\)-axis and vertical
    \(y\)-axis.
  </p>
  <div class="book-callout book-callout--formula" role="note">
    <p class="book-callout__label">Compare quickly</p>
    <p class="book-formula">\(a &gt; b\) means \(a\) is to the right of \(b\) on the number line.</p>
  </div>

  <h2 id="compare" class="book-prose__heading">Comparing integers with position</h2>
  <p>
    Position gives order immediately:
  </p>
  <ul>
    <li>If \(a\) is to the right of \(b\), then \(a &gt; b\).</li>
    <li>If \(a\) is to the left of \(b\), then \(a &lt; b\).</li>
  </ul>
  <p>
    Example: \(-2 &gt; -5\) because \(-2\) is to the right of \(-5\), even though both are negative.
  </p>
  <div class="math-widget-mount" data-widget="integer-compare-demo" data-pagefind-ignore></div>

  <h2 id="absolute-value" class="book-prose__heading">Absolute value and opposites</h2>
  <p>
    Absolute value means distance from zero, so it is never negative: \(|-7|=7\), \(|7|=7\).
    Opposite numbers are equally far from zero in opposite directions, such as \(+5\) and \(-5\).
  </p>
  <p>
    Distance between two integers can be read from the line:
    \[
      \text{distance between } a \text{ and } b = |a-b|.
    \]
    For example, the distance from \(-4\) to \(3\) is \(|-4-3|=7\).
  </p>

  <h2 id="operations" class="book-prose__heading">Operations as movement and grouping</h2>
  <h3 class="book-prose__heading">Addition and subtraction</h3>
  <p>
    Think of arithmetic as moves on the line:
  </p>
  <ul>
    <li>\(a+b\): start at \(a\), move \(b\) units (right if \(b&gt;0\), left if \(b&lt;0\)).</li>
    <li>\(a-b = a+(-b)\): subtracting means adding the opposite.</li>
  </ul>
  <p>
    Examples:
    \[
      -3+8=5,\qquad 4-9=-5,\qquad 6-(-4)=10.
    \]
  </p>
  <h3 class="book-prose__heading">Multiplication</h3>
  <p>
    Multiplication can be seen as repeated equal jumps. For instance, \(5\times3\) means three jumps
    of length 5 to the right, ending at 15.
  </p>
  <p>
    Sign rules come from direction:
  </p>
  <ul>
    <li>\((+)\times(+)=+\)</li>
    <li>\((+)\times(-)=-\)</li>
    <li>\((-)\times(+)=-\)</li>
    <li>\((-)\times(-)=+\)</li>
  </ul>
  <h3 class="book-prose__heading">Division</h3>
  <p>
    Division asks how many equal lengths fit into another length. For example, \(6\div2=3\) because
    three jumps of 2 reach 6. Integer sign rules mirror multiplication:
    \[
      (+)\div(+)=+,\quad (+)\div(-)=-,\quad (-)\div(+)=-,\quad (-)\div(-)=+.
    \]
  </p>

  <h2 id="zero-principle" class="book-prose__heading">Zero principle with tiles</h2>
  <p>
    A positive tile and a negative tile form a zero pair. Add or remove zero pairs to simplify
    without changing value.
  </p>

  <details class="book-question">
    <summary class="book-question__prompt">Checkpoint: evaluate \(-3 + 8\)</summary>
    <div class="book-question__solution">
      <p>\(-3 + 8 = 5\). Start at \(-3\), move 8 units right.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Checkpoint: evaluate \(4 - 9\)</summary>
    <div class="book-question__solution">
      <p>\(4 - 9 = -5\). Equivalent to \(4 + (-9)\).</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Checkpoint: explain why \(2-(-2)=4\)</summary>
    <div class="book-question__solution">
      <p>
        Subtracting \(-2\) means adding its opposite \(+2\), so \(2-(-2)=2+2=4\).
      </p>
    </div>
  </details>

  <h2 id="why-zero-negative" class="book-prose__heading">Why zero and negatives matter</h2>
  <p>
    Zero and negative numbers make arithmetic coherent. Before negatives were accepted, people often
    restricted results to non-negative values, which forced case-by-case rules and extra wording.
  </p>
  <p>
    With integers, one set of laws works smoothly:
  </p>
  <ul>
    <li>Subtraction is always possible: \(a-b=a+(-b)\).</li>
    <li>Opposites are explicit: every \(a\) has \(-a\), and \(a+(-a)=0\).</li>
    <li>Distributive structure stays consistent across signs:
      \(k(a+b)=ka+kb\), even when values are negative.</li>
  </ul>
  <p>
    This consistency is why integer arithmetic is not just “extra symbols”; it is the foundation for
    algebra, equations, and later topics like vectors and coordinate geometry.
  </p>
</div>