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181 lines
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181 lines
8.3 KiB
HTML
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<title>Algebra - Definitions</title>
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<meta name="description" content="Basic definitions in Algebra such as equation, coefficient, variable, exponent, etc.">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Algebra - Basic Definitions</h1>
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<p class="center"><i>It may help you to read <a href="introduction.html">Introduction to Algebra</a> first</i></p>
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<h2>What is an Equation</h2>
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<p>An equation says that two things are equal. It will have an equals sign "=" like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"> <b><i>x</i></b></td>
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<td class="huge" align="center" width="50"> <b>+</b></td>
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<td class="huge" align="center" width="50"> <b>2</b></td>
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<td class="huge" align="center" width="50"> <b>=</b></td>
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<td class="huge" align="center" width="50"> <b>6</b></td>
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</tr>
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</tbody></table>
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<p>That equation says: <b>what is on the left (x + 2) is equal to what is on the right (6)</b></p>
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<p>So an equation is like a <b>statement</b> "<i>this</i> equals <i>that</i>"</p>
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<h2>Parts of an Equation</h2>
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<p>So
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people can talk about equations, there are <b>names</b> for different parts (better than saying "that thingy there"!)</p>
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<p>Here we
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have an equation that says <span class="large">4x − 7</span> equals <span class="large">5</span>, and all its parts: </p>
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<br>
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<p class="center"><img src="images/variable-constant.svg" alt="4x-7=5: 4 is coefficient, x is variable, 7 and 5 constant, - is operator"></p>
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<p>A <b>Variable</b> is a symbol for a number we don't know yet. It is usually a letter like x or y.</p>
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<p>A number on its own is called a <b>Constant</b>.</p>
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<p>A <b>Coefficient</b> is a number used to multiply a variable (<b>4x</b> means <b>4</b> times <b>x</b>, so <b>4</b> is a coefficient)</p>
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<p>Variables on their own (without a number next to them) actually have a coefficient of 1 (<b>x</b> is really <b>1x</b>)</p>
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<p>Sometimes a coefficient is a letter like <b>a</b> or <b>b</b> instead of a number:</p>
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<div class="example">
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<h3>Example: ax<sup>2</sup> + bx + c</h3>
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<ul>
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<li><b>x</b> is a variable</li>
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<li><b>a</b> and <b>b</b> are coefficients</li>
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<li><b>c</b> is a constant</li>
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</ul>
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</div>
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<p>An <b>Operator</b> is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).</p>
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<p> </p>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/expression-term.svg" alt="4x-7=5: 4x-7 is expression, 4x, 7 and 5 are terms"></p>
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<p>A <b>Term</b> is either a single number or a variable, or numbers and variables multiplied together.</p>
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<p>An <b>Expression</b> is a group of terms (the terms are separated by + or − signs)</p>
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<div style="clear:both"></div>
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<p>So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"</p>
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<h2>Exponents</h2>
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<p>The<b> </b><a href="../exponent.html">exponent</a> (such as the 2 in x<sup>2</sup>) says <b>how many times </b>to use the value in a multiplication.</p>
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<p style="float:left; margin: 20px 10px 5px 0;"><img src="images/exponent-8-2.svg" alt="8 to the Power 2"></p>
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<div class="example">
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<h3>Examples:</h3>
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<p class="larger">8<sup>2</sup> = 8 × 8 = 64</p>
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<p class="larger">y<sup>3</sup> = y × y × y</p>
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<p class="larger">y<sup>2</sup>z = y × y × z</p>
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</div><p></p>
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<p>Exponents make it easier to write and use many multiplications</p>
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<div class="example">
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<p>Example: <b>y<sup>4</sup>z<sup>2</sup></b> is easier than <b>y × y × y × y × z × z</b></p>
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</div>
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<h2>Polynomial</h2>
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<div class="example">
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<p>Example of a Polynomial: <b>3x<sup>2</sup> + x - 2</b></p>
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</div>
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<p>A <a href="polynomials.html">polynomial</a> can have <b>constants</b>, <b>variables</b> and the <b>exponents 0,1,2,3,...</b></p>
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<p>But it never has division by a variable.</p>
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<p class="center"><img src="images/polynomial.svg" alt="polynomial" style="max-width:100%"></p>
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<h2>Monomial, Binomial, Trinomial</h2>
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<p>There are special names for polynomials with 1, 2 or 3 terms:</p>
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<p class="center"><img src="images/polynomials.svg" alt="monomial, binomial, trinomial" style="max-width:100%"></p>
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<h2>Like Terms</h2>
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<p><a href="like-terms.html">Like Terms</a> are <b>terms</b> whose variables (and their <a href="../exponent.html">exponents</a> such as the 2 in x<sup>2</sup>) are the same.</p>
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<p>In other words, terms that are "like" each other. (Note: the <b>coefficients</b> can be different)</p>
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<div class="example">
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<h3>Example:</h3>
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<table align="center" width="50%" border="0">
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<tbody>
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<tr align="center">
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<td class="larger">(1/3)xy<sup>2</sup></td>
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<td class="larger">−2xy<sup>2</sup> </td>
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<td class="larger">6xy<sup>2</sup> </td>
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</tr>
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</tbody></table>
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<p>Are all <b>like terms</b> because the variables are all <b>xy<sup>2</sup></b></p>
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</div>
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<div class="beach">
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<p> </p>
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</div>
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<div class="related">
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<a href="introduction.html">Introduction to Algebra</a>
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<a href="index.html">Algebra Index</a>
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