{"id":269,"date":"2019-06-26T23:17:14","date_gmt":"2019-06-26T23:17:14","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=269"},"modified":"2019-11-18T22:35:42","modified_gmt":"2019-11-18T22:35:42","slug":"complex-roots-made-visible","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/complex-roots-made-visible\/","title":{"rendered":"Complex Roots Made Visible"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Everyone learns that the roots of a polynomial have a graphical interpretation: they’re the places where the function crosses the x-axis. But what happens when the equation has only imaginary roots? Do those have a graphical interpretation as well?<\/p>\n\n\n\n

Here’s an interpretation that works for quadratics. Take a quadratic, such as 2x2<\/sup> – 8x + 10, and graph it. In the Figure 1, it is shown in red. Because it lies entirely above the x-axis, we know it has no real roots.<\/p>\n\n\n\n

Now, reflect the graph of this quadratic through its bottom-most point, and find the x-intercepts of this new graph, shown in green. Finally, treat these intercepts as if they were on opposite sides of a perfect circle, and rotate them both exactly 90 degrees. These new points are shown in blue.<\/p>\n\n\n\n

If interpreted as points in the complex plane, the blue points are exactly the roots of the original equation! (In our example, they are 2+i and 2-i.)<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Use different colors of chalk to represent various parts of the diagram.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
You can prove it by expanding the generic equations: 
(x-a)2<\/sup> + b and -(x-a)2<\/sup> + b 
and then comparing their roots (using the quadratic equation).<\/p>\n\n\n\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Complex Roots Made Visible.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
A. Norton and B. Lotto, “Complex Roots made Visible”, College Math J., 1984, number 3, pp. 248-249.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Dominic Mazzoni<\/p>\n\n\n\n


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