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181 lines
6.7 KiB
HTML
181 lines
6.7 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/numbers/basic-operations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:45:28 GMT -->
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<!-- #BeginEditable "doctitle" -->
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<title>Basic Operations</title>
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<meta name="keywords" content="math, maths, mathematics, school, homework, education">
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</head>
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<body id="bodybg">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu">
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<h2><br></h2></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Basic Operations</h1>
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<p>Ah, the familiar four basic operations:</p>
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<p class="center"><a href="addition.html"><img src="../images/style/plus.svg" alt="add" style="vertical-align:middle;" height="44" width="44"></a>
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<a href="subtraction.html"><img src="../images/style/minus.svg" alt="Subtract" style="vertical-align:middle;" height="44" width="44"></a>
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<a href="../tables.html"><img src="../images/style/multiply.svg" alt="Multiply" style="vertical-align:middle;" height="44" width="44"></a>
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<a href="division.html"><img src="../images/style/divide.svg" alt="Divide" style="vertical-align:middle;" height="44" width="44"></a></p>
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<p>But there is more to the story!</p>
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<h2>Addition and Subtraction</h2>
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<p>Addition says how many steps to take.</p>
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<p>Subtraction is actually addition, we just change the sign of the second number before we add.</p>
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<div class="example">Example: 7 − 5 = 7 + (−5) = 2</div>
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<p>This idea is very useful!</p>
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<div class="example">
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<h3>Example: Computer programs (or computer chips)</h3>
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<p>To handle numbers we first find a good way to "add"</p>
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<p>And then to subtract we simply change the sign of the second number, then add.</p>
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</div>
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<p>Play with it here (sliders):</p>
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<div class="script" style="height: 270px;">
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images/number-line-add.js?sub=n
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</div>
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<h2>Multiplication and Division</h2>
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<p>Multiplication says how many adds to do.</p>
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<p>Division is actually multiplication, we just do the <a href="../reciprocal.html">reciprocal</a> (<span class="intbl"><em>1</em><strong>value</strong></span>) of the second number before we multiply.</p>
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<div class="example">
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Example: 12 ÷ 4 = 12 × <span class="intbl"><em>1</em><strong>4</strong></span> = 3
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</div>
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<p>This idea is very useful!</p>
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<div class="example">
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It helps explain fractions: <span class="intbl"><em>3</em><strong>4</strong></span> is multiplication of 3 and <span class="intbl"><em>1</em><strong>4</strong></span>
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</div>
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<div class="example">
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In some areas of mathematics (such as Matrix Algebra) we cannot divide, but we can do an inverse, so we multiply by the inverse and the job is done.
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</div>
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<h2>Wait, What?</h2>
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<p>So the four basic operators are just two?</p>
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<p class="center"><a href="addition.html"><img src="../images/style/plus.svg" alt="add" style="vertical-align:middle;" height="44" width="44"></a>
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<a href="../tables.html"><img src="../images/style/multiply.svg" alt="Multiply" style="vertical-align:middle;" height="44" width="44"></a></p>
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<p>Much simpler don't you think? </p>
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<p>We just need to be happy with the concept of <a href="inverse.html">inverse</a>:</p>
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<ul>
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<li>Additive inverse (change sign)</li>
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<li>Multiplicative inverse (<span class="intbl"><em>1</em><strong>value</strong></span>)</li></ul>
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<p>They are also both better behaved:</p>
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<p>Addition is <a href="../associative-commutative-distributive.html">commutative</a>: 3 + 5 = 5 + 3. But subtraction is not: 3 − 5 <b>≠</b> 5 − 3</p>
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<p>Multiplication is commutative: 3 × 5 = 5 × 3. But division is not: 3/5 <b>≠</b> 5/3</p>
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<p>Subtraction and division are both still important ... we just see them now from a higher level.</p>
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<h2>Exponents and Logarithms</h2>
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<p>We can go one step further:</p>
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<p><a href="../exponent.html">Exponents</a> say how many multiplies to do.</p>
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<p>The inverse of exponent is <a href="../algebra/logarithms.html">logarithm</a>.</p>
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<h2>Order of Operations</h2>
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<p>Now <a href="../operation-order-pemdas.html">order of operations</a> becomes simpler. PEMDAS becomes <b>PEMA</b>:</p>
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<ul>
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<li>Parentheses (overrides the usual order)</li>
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<li>Exponents (how many multiplies)</li>
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<li>Multiply (how many adds)</li>
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<li>Add (how many steps)</li></ul>
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<h2></h2>
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<h2>Summary</h2>
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<p>Knowing the concept of inverses lets us simplify to this:</p>
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<ul>
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<li><b>Addition</b> says how many steps</li>
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<ul>
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<li>Inverse is changing sign/direction</li></ul>
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<li><b>Multiplication</b> says how many adds</li>
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<ul>
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<li>Inverse is the reciprocal (<span class="intbl"><em>1</em><strong>value</strong></span>)</li></ul>
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<li><b>Exponents</b> says how many multiplies</li>
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<ul>
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<li>Inverse is logarithm</li></ul></ul>
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<p><br></p>
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<p><br></p>
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<div class="related">
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<a href="evolution-of-numbers.html">Evolution of Numbers</a>
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<a href="../algebra/index.html">Algebra Index</a>
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<a href="index.html">Numbers Index</a>
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</div>
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</article>
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