openfreebooks-bookgenerator/content/math/written-calculation/_index.md

+++
title = "Written calculation"
description = "Add, subtract, multiply, and divide whole numbers using vertical written form: align columns, regroup, and read the result."
template = "chapter.html"
weight = 2
[extra]
subject = "math"
chapter_id = "written-calculation"
strand = "Number & Algebra"
+++

<div class="book-prose">
  <p>
    You can read number sentences from <strong>basic math notation</strong> and use the
    <strong>properties of operations</strong> to see why reordering or splitting sums can still give
    the same answer. This chapter teaches
    <strong>written calculation</strong> — how to add, subtract, multiply, and divide on paper by
    stacking digits in <strong>vertical form</strong>: one number above another, digit under digit.
    The ideas work for any size number; the visualizer below works best with about eight digits or
    fewer per line.
  </p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Vertical form</p>
    <p>
      Line up the same places in columns, work from right to left, and write the answer under a horizontal
      line. Carry and borrow keep each column within a single digit.
    </p>
  </div>

  <h2 id="why-columns" class="book-prose__heading">Why columns?</h2>
  <p>
    In <strong>counting and numbers</strong> you learned <strong>place value</strong>: the ones column,
    tens column, hundreds column. In vertical form, digits that share a place must sit in the same
    column. If you shift a digit left or right, you are changing its value.
  </p>
  <p>
    Always line up the <strong>ones</strong> under the ones, then tens under tens, and so on. Draw a
    horizontal line under the stack before you write the answer.
  </p>

  <h2 id="carry-borrow" class="book-prose__heading">Carry and borrow</h2>
  <p>
    In the four operations, a column sometimes gives a digit from <strong>0 to 9</strong> and sometimes
    gives <strong>10 or more</strong> (or too small to subtract). Then you adjust the next column to the
    <strong>left</strong>.
  </p>
  <ul>
    <li>
      <strong>Carry</strong> — in addition, when a column sums to 10 or more, you write the ones
      digit of that sum in the column and pass the extra ten to the next column on the left.
    </li>
    <li>
      <strong>Borrow</strong> — in subtraction, when the top digit is smaller than the bottom
      digit, you take one ten from the column on the left, add 10 to the current column, then subtract.
    </li>
  </ul>
  <p>Work from the <strong>right</strong> (ones) toward the <strong>left</strong>, one column at a time.</p>

  <h3 id="carry-example" class="book-prose__heading">Carry: 27 + 59 = 86</h3>
  <pre class="va-static" aria-label="27 plus 59 equals 86 with carry"><span class="va-static__carry">¹</span>
  27
+ 59
────
  86</pre>
  <p>
    In the ones column: \(7 + 9 = 16\). Write <strong>6</strong> in the ones place. Carry
    <strong>1</strong> to the tens column (written small above the tens). In the tens column:
    \(2 + 5 + 1 = 8\).
  </p>

  <h3 id="borrow-example" class="book-prose__heading">Borrow: 47 − 19 = 28</h3>
  <pre class="va-static" aria-label="47 minus 19 equals 28 with borrow"><span class="va-static__borrow">−1</span>
  47
− 19
────
  28</pre>
  <p>
    In the ones column: \(7 - 9\) does not work. Borrow 10 from the tens: the <strong>4</strong> becomes
    <strong>3</strong>, and the ones digit becomes <strong>17</strong>. Then \(17 - 9 = 8\). In the tens
    column: \(3 - 1 = 2\).
  </p>

  <h2 id="addition" class="book-prose__heading">Addition</h2>
  <p>
    Write the numbers one above the other, plus sign to the left of the bottom number, line underneath,
    sum below. Add each column from right to left; carry when needed.
  </p>

  <h2 id="subtraction" class="book-prose__heading">Subtraction</h2>
  <p>
    Put the <strong>larger</strong> number on top. Subtract column by column. When the top digit is too
    small, borrow from the left (decrease that digit by 1, add 10 to the current column). Example with a
    zero in the middle: \(503 - 278 = 225\) needs borrowing across the tens column.
  </p>

  <h2 id="multiplication" class="book-prose__heading">Multiplication</h2>
  <p>
    Write the larger number on top (often). Multiply the top number by each digit of the bottom number,
    one <strong>partial product</strong> per digit, shifted so ones line up with ones. Add the partial
    products for the final answer.
  </p>
  <pre class="va-static" aria-label="23 times 14 equals 322">
   23
 × 14
 ────
   92
  230
 ────
  322</pre>
  <p>Here \(23 \times 4 = 92\) and \(23 \times 10 = 230\); \(92 + 230 = 322\).</p>

  <h2 id="division" class="book-prose__heading">Division</h2>
  <p>
    <strong>Long division</strong> writes the divisor outside a bracket, the dividend inside, and the
    quotient on top. At each step: how many times does the divisor fit into the current part of the
    dividend? Write that digit in the quotient, multiply, subtract, then bring down the next digit. If
    the division is not exact, the amount left is the <strong>remainder</strong>.
  </p>
  <p>
    To calculate and check a remainder, always use
    \( \text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder} \), with
    \(0 \le \text{remainder} < \text{divisor}\). For example:
  </p>
  <p>
    \[
    \begin{aligned}
    157 \div 12 &= 13 \text{ remainder } 1 \\
    157 &= 12 \times 13 + 1
    \end{aligned}
    \]
  </p>
  <p>
    The remainder can also be written as a fraction of the divisor:
    \[
    \begin{aligned}
    157 \div 12 &= 13 + \frac{1}{12}
    \end{aligned}
    \]
  </p>

  <h2 id="terminology" class="book-prose__heading">Names of the parts</h2>
  <div class="book-table-wrap">
    <table class="book-table written-terms-table">
      <thead>
        <tr>
          <th scope="col">Operation</th>
          <th scope="col">Parts</th>
          <th scope="col">Answer</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>Addition</td>
          <td>addend + addend</td>
          <td>sum</td>
        </tr>
        <tr>
          <td>Subtraction</td>
          <td>minuend − subtrahend</td>
          <td>difference</td>
        </tr>
        <tr>
          <td>Multiplication</td>
          <td>factor × factor</td>
          <td>product</td>
        </tr>
        <tr>
          <td>Division</td>
          <td>dividend ÷ divisor</td>
          <td>quotient (and remainder if any)</td>
        </tr>
      </tbody>
    </table>
  </div>

  <h2 id="try-it-yourself" class="book-prose__heading">Try it yourself</h2>
  <p>
    Use the visualizer below: enter two or more whole numbers (one per line), choose an operation, and
    click <strong>Generate visualization</strong>. Step through with <strong>Play</strong> or the arrow
    buttons to see each carry, borrow, and column update.
  </p>
  <div class="math-widget-mount" data-widget="vertical-arithmetic-workbench" data-pagefind-ignore></div>
</div>

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    People calculated with tokens, rods, and abacuses long before everyone used pencil and paper. In many
    places, including China, India, and the Arab world, scribes lined up digits in columns so addition
    and subtraction could be done digit by digit.
  </p>
  <p>
    <strong>Mechanical calculators</strong> (1600s–1900s) had to solve the same puzzle: when one wheel
    moved from 9 to 0, it had to add 1 to the wheel for the next place. Early machines often did addition
    first, then ran a separate <strong>carry cycle</strong> to update higher digits. Later designs passed
    the carry instantly along the whole machine. That idea appears again inside modern chips: a
    <strong>carry flag</strong> records whether the last step overflowed a column.
  </p>

  <h2 id="derivation" class="book-prose__heading">Derivation</h2>
  <h3 id="place-value-add" class="book-prose__heading">Why column addition works</h3>
  <p>
    Split each number by place value. For example,
    \(27 + 59 = (20 + 7) + (50 + 9) = (20 + 50) + (7 + 9) = 70 + 16 = 86\). The ones column adds
    \(7 + 9\); the tens column adds \(2 + 5\) plus any carry. Each column only mixes the same place —
    that is why alignment matters.
  </p>
  <h3 id="inverse" class="book-prose__heading">Subtraction and addition</h3>
  <p>
    Subtraction is the inverse of addition: \(c = a - b\) means the same as \(c + b = a\). Borrowing
    moves one ten from the left so the subtraction in the current column matches that story.
  </p>
  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>When you add 7 and 9 in the ones column, what do you write in the ones place? What do you carry?</p>
  </div>
  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>In \(47 - 19\), why does the 4 in the tens column become 3 before you subtract the ones?</p>
  </div>
  <div class="book-callout" role="note">
    <p class="book-callout__label">Checkpoint</p>
    <p>In \(23 \times 14\), why are there two rows under the line before the final sum?</p>
  </div>

  <h2 id="applications" class="book-prose__heading">Real life</h2>
  <ul>
    <li><strong>Shopping</strong> — adding prices on a receipt column by column.</li>
    <li><strong>Scores</strong> — finding the difference between two team totals.</li>
    <li><strong>Sharing</strong> — dividing a total equally (e.g. \(157\) stickers among \(12\) children: quotient 13, remainder 1).</li>
  </ul>

  <h2 id="questions-easy" class="book-prose__heading">Question bank — Easy</h2>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(34 + 25\).</summary>
    <div class="book-question__solution">
      <pre class="va-static">  34
+ 25
────
  59</pre>
      <p><strong>Answer:</strong> 59</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(61 - 28\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> 33 (borrow from tens in the ones column).</p>
    </div>
  </details>

  <h2 id="questions-intermediate" class="book-prose__heading">Intermediate</h2>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(27 + 59\).</summary>
    <div class="book-question__solution">
      <pre class="va-static"><span class="va-static__carry">¹</span>
  27
+ 59
────
  86</pre>
      <p><strong>Answer:</strong> 86</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(503 - 278\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> 225</p>
      <p><strong>Steps:</strong> Borrow across the tens (0) from hundreds; ones become 13−8=5; tens 9−7=2; hundreds 4−2=2.</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(23 \times 14\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> 322</p>
    </div>
  </details>

  <h2 id="questions-difficult" class="book-prose__heading">Difficult</h2>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(156 \div 12\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> 13</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Work out \(157 \div 12\) and state the remainder.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> Quotient 13, remainder 1.</p>
    </div>
  </details>

  <h2 id="questions-hardcore" class="book-prose__heading">Hardcore</h2>
  <details class="book-question">
    <summary class="book-question__prompt">A book has 864 pages. You read 379 pages. How many pages are left?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(864 - 379 = 485\) pages.</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">A classroom orders 24 packs of 15 pencils. How many pencils is that?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(24 \times 15 = 360\) pencils.</p>
    </div>
  </details>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Wikipedia — <em>Carry (arithmetic)</em> and <em>Subtraction</em>.</li>
    <li>Cajori, Florian — <em>A History of Mathematical Notations</em>.</li>
    <li>Other school methods (Austrian/additions, partial differences) exist; this chapter uses column borrow as in most HK primary texts.</li>
  </ul>
</div>