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divider"></i></li><li><span class="gray">Numbers</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 5 minutes"><i class="fa fa-clock-o"></i> 5'</span> </ol> <article id="main-article"> <p><img alt="" src="../../maths/math-2.jpg" style="width: 250px; height: 167px; float: left;">In physics, data collected from experiments and answers to problems are rarely nice easy whole numbers.</p> <p>In IB physics, you can no longer simply write a number on the page - there is lots to consider. Think about significant figures, orders of magnitude and standard notation.</p> <p>Later, we'll get into error and uncertainties too.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Significant figures</p> </div> </div> <div class="panel-body"> <div> <p>When writing down a value from an experiment, we should generally write the number to the best possible precision for the instrument, for example:</p> <ul> <li>12.2 cm on a ruler</li> <li>-57.0<sup>o</sup> on a thermometer</li> </ul> <p>Both of these numbers are to 3 <em>significant figures</em>.</p> <p>In a calculation, we should write our final answer to the smallest number of significant figures provided to us. This is usually 2 or 3 significant figures and will often involve rounding.</p> <div class="magenta"> <p>To state the number of signficant figures for a given number, follow the appropriate step:</p> <ul> <li>For a value of magnitude greater than 1, count the numbers visible to you after any initial zeros: 9.4651 is to 5 sf, 00011 is to 2 sf.</li> <li>For a value of magnitude less than 1, we continue to count the numbers visible to you after any initial zeros: 0.8 is to 1 sf, 0.00011 is to 2 sf.</li> </ul> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Orders of magnitude</p> </div> </div> <div class="panel-body"> <div> <p>The <em>order of magnitude</em> of a value is its approximate number of digits and is the nearest power of 10.</p> <ul> <li>2 is of order of magnitude 10<sup>0</sup> (1)</li> <li>9 is of order of magnitude 10</li> <li>11 is of order of magnitude 10</li> <li>86 is of order of magntude 10<sup>2</sup> (100)</li> <li>The radius of the earth is of order of magnitude 10<sup>7</sup> m</li> <li>The charge on an electron is of order of magnitude 10<sup>-19</sup> C</li> </ul> <p>Note that, unlike rounding on a linear scale (5 or above), the next order of magnitude starts from <span class="math-tex">\(\approx\)</span>4 because of the logarithmic scale.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Mean</p> </div> </div> <div class="panel-body"> <div> <p>When collecting data in an experiment, you might conduct repeated readings. Anomalous values can be discarded and an average can be calculated, which improves the accuracy of the result. In physics, we calculate the mean average:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(\bar{x}=\frac{x_1+x_2+x_3...+x_n}{n}\)</span></p> <p><span class="math-tex">\(n\)</span> represents the number of readings</p> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Standard notation</p> </div> </div> <div class="panel-body"> <div> <p>Because physicists can rarely quote values to more than 2 or 3 signficant figures, very large or very small numbers are expressed in <em>standard notation</em>.</p> <ul> <li><span class="math-tex">\(123 000 000=1.23 \times 10^8\)</span>(count numbers visible after 1st digit)</li> <li><span class="math-tex">\(0.000321 = 3.21 \times 10^{-4}\)</span>(count zeros after the decimal place and one before - note the minus sign)</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="670"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Numbers</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save 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