Place Value & Ordering
<p>Understanding place value is like knowing the address of every digit in a number. Each position tells you how much that digit is worth, and mastering this skill is the foundation for almost everything else in maths.</p>
<h2>How Place Value Works</h2>
<p>In our number system, each column is worth ten times more than the column to its right. Look at the number 47,582:</p>
<ul>
<li><strong>4</strong> is in the ten-thousands column, so it is worth 40,000</li>
<li><strong>7</strong> is in the thousands column, so it is worth 7,000</li>
<li><strong>5</strong> is in the hundreds column, so it is worth 500</li>
<li><strong>8</strong> is in the tens column, so it is worth 80</li>
<li><strong>2</strong> is in the ones column, so it is worth 2</li>
</ul>
<p>We can write this as: 47,582 = 40,000 + 7,000 + 500 + 80 + 2. This is called <strong>partitioning</strong>.</p>
<h2>Decimals and Place Value</h2>
<p>Place value continues after the decimal point, but now each column is ten times <em>smaller</em>. In the number 3.64:</p>
<ul>
<li><strong>3</strong> is worth 3 ones</li>
<li><strong>6</strong> is worth 6 tenths (6/10 or 0.6)</li>
<li><strong>4</strong> is worth 4 hundredths (4/100 or 0.04)</li>
</ul>
<h2>Ordering Numbers</h2>
<p>To put numbers in order, compare them column by column starting from the left. This works because the leftmost column is always the most valuable.</p>
<p><strong>Worked Example:</strong> Put these in order from smallest to largest: 3,847 3,874 3,478</p>
<p>Step 1: The thousands digit is 3 in all three numbers, so move to the hundreds.<br>
Step 2: The hundreds digits are 8, 8, and 4. Since 4 is smallest, 3,478 comes first.<br>
Step 3: For the two numbers with 8 in the hundreds, compare the tens: 4 and 7. Since 4 < 7, we get 3,847 before 3,874.<br>
Answer: 3,478 < 3,847 < 3,874</p>
<h2>Ordering Decimals</h2>
<p>The trick with decimals is to make all numbers have the same number of decimal places by adding zeros. This does not change the value.</p>
<p><strong>Worked Example:</strong> Order these: 0.7 0.65 0.605</p>
<p>Rewrite them all with three decimal places: 0.700, 0.650, 0.605. Now compare: 605 < 650 < 700, so the order is 0.605, 0.65, 0.7.</p>
<h2>Rounding</h2>
<p>Rounding means replacing a number with a simpler nearby value. The rule is: look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.</p>
<p><strong>Worked Example:</strong> Round 4,672 to the nearest hundred.</p>
<p>The hundreds digit is 6. Look at the tens digit: it is 7 (which is 5 or more), so round up. Answer: 4,700.</p>
<h2>Negative Numbers</h2>
<p>On a number line, numbers to the left of zero are negative. A common mistake is thinking that -8 is bigger than -3. Think of temperature: -8°C is colder (and therefore smaller) than -3°C. The further from zero on the negative side, the smaller the number.</p>
<h2>Practice Tips</h2>
<ul>
<li>When ordering, always line up the columns carefully — write the numbers one below the other</li>
<li>With decimals, pad with zeros so every number has the same length</li>
<li>For negative numbers, draw a quick number line if you get confused</li>
<li>In the exam, double-check whether the question asks for smallest-to-largest or largest-to-smallest</li>
</ul>