Number & Operations | Free ACT Math Course

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<title>ACT Math: Numbers & Operations | The School of Mathematics</title>

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<h1>ACT Math: Numbers & Operations</h1>

Numbers & Operations is one of the foundational topic categories in ACT Math, and it covers the number-sense skills that show up early in the test and underpin everything that follows. This free, complete lesson covers classifying numbers (natural, whole, integer, rational, and irrational), the order of operations (PEMDAS) and absolute value, exponent rules, and scientific notation. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-numbers-operations-quiz-1">Practice Numbers & Operations Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#classification">Number Types and Classification</a></li>

<li><a href="#order-of-operations">Order of Operations and Absolute Value</a></li>

<li><a href="#exponents">Exponent Rules</a></li>

<li><a href="#scientific-notation">Scientific Notation</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="classification">1. Number Types and Classification</h2>

<p>Numbers are grouped into nested categories, each one a subset of the next:</p>

<tr><th>Type</th><th>Description</th><th>Examples</th></tr>

<tr><td>Natural numbers</td><td>Counting numbers</td><td>1, 2, 3, 4, ...</td></tr>

<tr><td>Whole numbers</td><td>Natural numbers plus zero</td><td>0, 1, 2, 3, ...</td></tr>

<tr><td>Integers</td><td>Whole numbers and their negatives</td><td>..., −2, −1, 0, 1, 2, ...</td></tr>

<tr><td>Rational numbers</td><td>Can be written as a fraction of two integers</td><td>−3, 0.75, <span class="frac"><span class="num">2</span><span class="den">3</span></span></td></tr>

<tr><td>Irrational numbers</td><td>Cannot be written as a fraction; decimal never repeats or terminates</td><td>π, √2</td></tr>

</table>

<p><strong>Worked Example:</strong> Classify the number −7.</p>

<p>−7 is an integer, and every integer is also rational and real. It is not a whole number or a natural number, since both of those exclude negative values.</p>

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<p class="prompt">1. Which of the following best classifies the number 0.75?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-1',true)">A) Rational</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">B) Irrational</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">C) Integer</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">D) Natural number</button>

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<p><span class="label">Explanation:</span> 0.75 = <span class="frac"><span class="num">3</span><span class="den">4</span></span>, a fraction of two integers, so it's rational. It's not an integer or natural number since it isn't a whole count.</p>

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<p class="prompt">2. Which of the following numbers is irrational?</p>

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<p><span class="label">Explanation:</span> √2 cannot be written as a fraction of two integers, its decimal expansion never terminates or repeats, so it's irrational. The other three choices are all rational.</p>

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<p class="prompt">3. Which set of numbers includes all natural numbers, whole numbers, and integers?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-3',true)">A) Rational numbers</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">B) Irrational numbers</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">C) Prime numbers</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">D) Even numbers</button>

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<p><span class="label">Explanation:</span> Every natural number, whole number, and integer can be written as a fraction over 1, so all of them are rational numbers.</p>

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<p class="prompt">4. Is −5 a whole number?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-4',true)">A) No, whole numbers cannot be negative</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">B) Yes, all integers are whole numbers</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">C) Yes, since −5 is a real number</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">D) No, because −5 is irrational</button>

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<p><span class="label">Explanation:</span> Whole numbers are 0, 1, 2, 3, ..., never negative. −5 is an integer, but not a whole number.</p>

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<p class="prompt">5. For which of the following values of n is √n a rational number?</p>

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<p><span class="label">Explanation:</span> The square root of an integer is rational only when that integer is a perfect square. 36 = 6², so √36 = 6, a rational number. 20, 40, and 50 are not perfect squares, so their square roots are irrational.</p>

</div>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-6',true)">A) Negative</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-6',false)">B) Positive</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-6',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> Raising a negative number to an odd power keeps it negative, so a³ < 0. Raising a negative number to an even power makes it positive, so b² > 0. A negative times a positive is negative.</p>

</div>

<h2 id="order-of-operations">2. Order of Operations and Absolute Value</h2>

<p>The <strong>order of operations</strong>, often remembered as <strong>PEMDAS</strong>, tells you the sequence to follow: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). <strong>Absolute value</strong>, written |x|, gives the distance a number is from 0 and is always non-negative.</p>

<p><strong>Worked Example:</strong> Evaluate 6 + 4 × (10 − 7)² ÷ 2.</p>

<p class="step-math">6 + 4 × (3)² ÷ 2   (parentheses)</p>

<p class="step-math">6 + 4 × 9 ÷ 2   (exponent)</p>

<p class="step-math">6 + 36 ÷ 2   (multiply)</p>

<p class="step-math">6 + 18   (divide)</p>

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<p class="prompt">1. Evaluate: 8 − 2 × 3 + 5</p>

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<p><span class="label">Explanation:</span> Multiply first: 2 × 3 = 6. Then: 8 − 6 + 5 = 7.</p>

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<p class="prompt">2. Evaluate: 3 + 5² − 4</p>

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<p><span class="label">Explanation:</span> Apply the exponent first: 5² = 25. Then: 3 + 25 − 4 = 24.</p>

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<p class="prompt">3. Evaluate: |−8| + |3 − 10|</p>

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<p><span class="label">Explanation:</span> |−8| = 8. Inside the second bars: 3 − 10 = −7, so |3 − 10| = 7. Then 8 + 7 = 15.</p>

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<p class="prompt">4. Evaluate: 2 × (6 + 3)² − 20 ÷ 4</p>

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<p><span class="label">Explanation:</span> Parentheses: 6 + 3 = 9. Exponent: 9² = 81. Multiply: 2 × 81 = 162. Divide: 20 ÷ 4 = 5. Subtract: 162 − 5 = 157.</p>

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<p class="prompt">5. Evaluate: |5 − 12| − |−3|</p>

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<p><span class="label">Explanation:</span> |5 − 12| = |−7| = 7. |−3| = 3. Then 7 − 3 = 4.</p>

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<h2 id="exponents">3. Exponent Rules</h2>

<tr><th>Rule</th><th>Formula</th></tr>

<tr><td>Product of powers</td><td>a<sup>m</sup> × a<sup>n</sup> = a<sup>m+n</sup></td></tr>

<tr><td>Quotient of powers</td><td>a<sup>m</sup> ÷ a<sup>n</sup> = a<sup>m−n</sup></td></tr>

<tr><td>Power of a power</td><td>(a<sup>m</sup>)<sup>n</sup> = a<sup>mn</sup></td></tr>

<tr><td>Zero exponent</td><td>a<sup>0</sup> = 1   (for a ≠ 0)</td></tr>

<tr><td>Negative exponent</td><td>a<sup>−n</sup> = <span class="frac"><span class="num">1</span><span class="den">a<sup>n</sup></span></span></td></tr>

</table>

<p><strong>Worked Example:</strong> Simplify (x³)(x⁵) ÷ x².</p>

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<p class="prompt">1. Simplify: 3⁴ × 3²</p>

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<p><span class="label">Explanation:</span> Add the exponents: 3<sup>4+2</sup> = 3⁶ = 729.</p>

</div>

<p class="prompt">2. Simplify: (2³)⁴</p>

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<p><span class="label">Explanation:</span> Multiply the exponents: 2<sup>3×4</sup> = 2²² (that is, 2 to the 12th power) = 4,096.</p>

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<p class="prompt">3. Simplify: x⁷ ÷ x³</p>

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<p><span class="label">Explanation:</span> Subtract the exponents: x<sup>7−3</sup> = x⁴.</p>

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<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) Undefined</button>

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<p><span class="label">Explanation:</span> Any nonzero number raised to the power of 0 equals 1.</p>

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<p class="prompt">5. Simplify: 2<sup>−3</sup></p>

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<p><span class="label">Explanation:</span> A negative exponent flips the base into a fraction: 2<sup>−3</sup> = <span class="frac"><span class="num">1</span><span class="den">2³</span></span> = <span class="frac"><span class="num">1</span><span class="den">8</span></span>.</p>

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<h2 id="scientific-notation">4. Scientific Notation</h2>

<p><strong>Scientific notation</strong> writes a number as a × 10<sup>n</sup>, where 1 ≤ |a| < 10 and n is an integer. It's a compact way to write very large or very small numbers.</p>

<p><strong>Worked Example:</strong> Write 47,000,000 in scientific notation.</p>

<p>Move the decimal point until only one nonzero digit remains to its left: 4.7, moved 7 places.</p>

</div>

<p class="prompt">1. Write 0.00058 in scientific notation.</p>

</div>

<p><span class="label">Explanation:</span> Moving the decimal point 4 places to the right of 0.00058 gives 5.8, so 0.00058 = 5.8 × 10<sup>−4</sup>. (Choices C and D aren't in proper scientific notation, since their leading factor isn't between 1 and 10.)</p>

</div>

<p class="prompt">2. Write 3.2 × 10⁵ in standard form.</p>

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<p><span class="label">Explanation:</span> Move the decimal point 5 places to the right: 3.2 × 10⁵ = 320,000.</p>

</div>

<p class="prompt">3. Simplify and write in scientific notation: (2 × 10³)(3 × 10⁵)</p>

</div>

<p><span class="label">Explanation:</span> Multiply the leading factors: 2 × 3 = 6. Add the exponents: 3 + 5 = 8. Result: 6 × 10⁸.</p>

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<p class="prompt">4. Simplify and write in scientific notation: (8 × 10⁶) ÷ (2 × 10²)</p>

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<p><span class="label">Explanation:</span> Divide the leading factors: 8 ÷ 2 = 4. Subtract the exponents: 6 − 2 = 4. Result: 4 × 10⁴.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Mixing up whole numbers and natural numbers.</strong> Whole numbers include 0; natural numbers start at 1. A question that hinges on whether 0 counts is testing exactly this distinction.</li>

<li><strong>Doing addition or subtraction before multiplication or division.</strong> PEMDAS groups multiplication and division together (worked left to right), and addition and subtraction together (also left to right). Neither pair has one step that always comes first within the pair.</li>

<li><strong>Treating absolute value bars like parentheses that don't affect sign.</strong> Simplify what's inside the absolute value bars first, then apply the absolute value, and remember the result is always non-negative.</li>

<li><strong>Adding exponents when the operation calls for multiplying them, or vice versa.</strong> Multiplying same-base powers adds exponents; raising a power to another power multiplies them. Confusing the two is one of the most common exponent errors.</li>

<li><strong>Writing scientific notation with a leading factor outside the range 1 to 10.</strong> 58 × 10<sup>−5</sup> is mathematically equal to 5.8 × 10<sup>−4</sup>, but only the second is in proper scientific notation, and ACT answer choices test this distinction directly.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>Does the ACT allow a calculator on every math question?</h3>

<p>Yes, a permitted calculator is allowed throughout the entire ACT Math section, so number-sense questions like these are about knowing the concept and doing the arithmetic accurately and quickly, not about being restricted from calculator use.</p>

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<h3>How many answer choices does each ACT Math question have?</h3>

<p>Under the current Enhanced ACT format, each Math question has 4 answer choices.</p>

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<h3>Why does classifying numbers matter if I can just compute the answer?</h3>

<p>Several ACT questions ask directly about number classification, whether a value is rational, whether a set is closed under an operation, or which classification a computed result falls into, so recognizing the categories quickly saves time and avoids second-guessing.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-numbers-operations-quiz-1">Numbers & Operations quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-numbers-operations-quiz-1">Practice Numbers & Operations Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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