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152 lines
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<h1 class="center">Real Number Properties</h1>
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<p><a href="../numbers/real-numbers.html">Real Numbers</a> have properties!</p>
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<div class="example">
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<h3>Example: Multiplying by zero</h3>
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<p>When we multiply a real number by zero we get zero:</p>
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<ul>
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<li>5 × 0 = 0 </li>
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<li>−7 × 0 = 0 </li>
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<li>0 × 0.0001 = 0 </li>
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<li>etc!</li>
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</ul>
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<p>It is called the "Zero Product Property", and is listed below.</p>
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</div>
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<h2>Properties</h2>
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<p>Here are the main properties of the Real Numbers</p>
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<p>Real Numbers are <a href="../associative-commutative-distributive.html">Commutative, Associative and Distributive</a>:</p>
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<p class="item"><span class="item-head">Commutative</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a + b = b + a</span><span class="item-ex">2 + 6 = 6 + 2</span></p>
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<p class="item"><span class="item-pr">ab = ba</span><span class="item-ex">4 × 2 = 2 × 4</span></p>
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<p class="item"><span class="item-head">Associative</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">(a + b) + c = a + ( b + c ) </span><span class="item-ex">(1 + 6) + 3 = 1 + (6 + 3) </span></p>
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<p class="item"><span class="item-pr">(ab)c = a(bc)</span><span class="item-ex">(4 × 2) × 5 = 4 × (2 × 5)</span></p>
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<p class="item"><span class="item-head">Distributive</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a × (b + c) = ab + ac</span><span class="item-ex">3 × (6+2) = 3 × 6 + 3 × 2</span></p>
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<p class="item"><span class="item-pr">(b+c) × a = ba + ca</span><span class="item-ex">(6+2) × 3 = 6 × 3 + 2 × 3</span> </p>
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<p> </p>
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<p>Real Numbers are <a href="closure.html">closed</a> (the result is also a real number) under addition and multiplication:</p>
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<p class="item"><span class="item-head">Closure</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a+b is real</span><span class="item-ex">2 + 3 = 5 is real</span></p>
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<p class="item"><span class="item-pr">a×b is real</span><span class="item-ex">6 × 2 = 12 is real</span></p>
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<p> </p>
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<p>Adding zero leaves the real number unchanged, likewise for multiplying by 1:</p>
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<p class="item"><span class="item-head">Identity</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a + 0 = a</span><span class="item-ex">6 + 0 = 6</span></p>
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<p class="item"><span class="item-pr">a × 1 = a</span><span class="item-ex">6 × 1 = 6</span></p>
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<p> </p>
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<p>For addition the inverse of a real number is its negative, and for multiplication the inverse is its <a href="../reciprocal.html">reciprocal</a>:</p>
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<p class="item"><span class="item-head">Additive Inverse</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a + (−a ) = 0</span><span class="item-ex">6 + (−6) = 0</span></p>
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<p class="item"><span class="item-head">Multiplicative Inverse</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">a × (1/a) = 1</span><span class="item-ex">6 × (1/6) = 1</span></p>
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<p class="item"><i>But not for 0 as 1/0 is <a href="../numbers/dividing-by-zero.html">undefined</a></i> </p>
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<p> </p>
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<p>Multiplying by zero gives zero (the <a href="../algebra/zero-product-property.html">Zero Product Property</a>):</p>
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<p class="item"><span class="item-head">Zero Product</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">If ab = 0 then a=0 or b=0, or both</span><span class="item-ex"> </span></p>
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<p class="item"><span class="item-pr">a × 0 = 0 × a = 0</span><span class="item-ex">5 × 0 = 0 × 5 = 0</span> </p>
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<p> </p>
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<p><a href="../multiplying-negatives.html">Multiplying two negatives make a positive</a>, and multiplying a negative and a positive makes a negative:</p>
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<p class="item"><span class="item-head">Negation</span><span class="item-ex"><i>example</i></span></p>
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<p class="item"><span class="item-pr">−1 × (−a) = −(−a) = a</span><span class="item-ex">−1 × (−5) = −(−5) = 5</span></p>
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<p class="item"><span class="item-pr">(−a)(−b) = ab</span><span class="item-ex">(−3)(−6) = 3 × 6 = 18</span></p>
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<p class="item"><span class="item-pr">(−a)(b) = (a)(−b) = −(ab)</span><span class="item-ex">−3 × 6 = 3 × −6 = −18</span></p><p> </p>
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<p> </p>
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<div class="related"><a href="../numbers/real-numbers.html">Real Numbers</a>
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