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88 lines
3.3 KiB
HTML
88 lines
3.3 KiB
HTML
<div class="book-prose">
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<p>
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Numbers on a line can lie to the <strong>right</strong> of zero (positive) or to the
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<strong>left</strong> (negative). Temperatures below freezing, bank overdrafts, and positions
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below sea level are everyday examples of <strong>directed numbers</strong>.
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</p>
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<h2 id="number-line" class="book-prose__heading">The number line</h2>
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<p>
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On a horizontal number line, moving right means the value <em>increases</em>; moving left means it
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<em>decreases</em>. The distance from zero is the <strong>absolute value</strong>
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\(|a|\). For example, \(|-5| = 5\) and \(|3| = 3\).
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</p>
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<div class="book-callout book-callout--formula" role="note">
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<p class="book-callout__label">Comparing</p>
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<p class="book-formula">
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\(a > b\) means \(a\) lies to the right of \(b\). Any negative number is less than any
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positive number.
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</p>
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</div>
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<h2 id="adding-subtracting" class="book-prose__heading">Adding and subtracting</h2>
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<p>
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Think in terms of movement on the line:
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</p>
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<ul>
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<li><strong>\(a + b\)</strong> — start at \(a\), move \(b\) units (right if \(b > 0\), left if \(b < 0\)).</li>
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<li><strong>\(a - b\)</strong> — start at \(a\), move \(-b\) units (equivalently, add the opposite of \(b\)).</li>
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</ul>
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<p>
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Subtracting a negative is adding a positive: \(5 - (-3) = 5 + 3 = 8\).
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</p>
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<h2 id="multiplying-signs" class="book-prose__heading">Multiplying and dividing</h2>
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<div class="book-table-wrap">
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<table class="book-table">
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<thead>
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<tr>
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<th scope="col">Signs</th>
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<th scope="col">Result</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>positive × positive, negative × negative</td>
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<td>positive</td>
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</tr>
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<tr>
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<td>positive × negative (either order)</td>
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<td>negative</td>
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</tr>
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</tbody>
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</table>
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</div>
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<h2 id="ratios" class="book-prose__heading">Ratios</h2>
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<p>
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A <strong>ratio</strong> compares quantities of the same kind. The ratio \(2:3\) means “2 parts
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to 3 parts” — five equal parts in total. To share \(40\) in the ratio \(2:3\), one part is
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\(40 \div 5 = 8\), so the shares are \(16\) and \(24\).
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</p>
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<div class="book-callout book-callout--formula" role="note">
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<p class="book-callout__label">Sharing in ratio \(m:n\)</p>
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<p class="book-formula">
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\[ \text{one part} = \frac{\text{total}}{m+n}, \quad
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\text{first share} = \frac{m}{m+n}\times\text{total}, \quad
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\text{second share} = \frac{n}{m+n}\times\text{total} \]
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</p>
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</div>
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<h2 id="rates" class="book-prose__heading">Rates</h2>
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<p>
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A <strong>rate</strong> compares different units. <strong>Speed</strong> is distance per time:
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\[ \text{speed} = \frac{\text{distance}}{\text{time}}. \]
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A <strong>unit rate</strong> tells “how much per one” — for example \(60\) km in \(1.5\) h gives
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\(60 \div 1.5 = 40\) km/h.
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</p>
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<h2 id="try-it-yourself" class="book-prose__heading">Try it yourself</h2>
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<p>
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Use the number-line explorer to pick two directed numbers and see their sum on the line. Then try
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the ratio sharer in the applications section below.
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</p>
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</div>
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<div class="math-widget-mount" data-widget="number-line" data-pagefind-ignore></div>
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