DISCOVERING NEW NUMBERS
Editors note: This article by Alex Stutt (’16) is an assignment she was given in Mr. Shah’s precalculus class in which students had to explore an interesting aspect of math. Alex’s submission discusses some of the things she found interesting in a Times article she read about humans discovering numbers that previously seemed impossible. The Times article can be read here.
In the Times Opinion pages I found a really interesting article called Finding Your Roots, by Steven Strogatz on the history of i and complex numbers, and of course finding roots. The very beginning discusses the idea of a mathematical operation being pushed to its limits until it no longer seems plausible, and this leading to the creation of new numbers. Subtracting bigger numbers from smaller ones led to negative numbers, dividing led to fractions, and taking the square root of negative numbers eventually led to the creation of i. Until the 1700s mathematicians did not believe the square roots of negative numbers even existed. However, unlike negative numbers and fractions, i has no place on the number line, instead i can be visualized on a plane. By putting a normal number line on the x-axis and imaginary numbers on the y-axis, you get a plane of complex numbers. The sum of the real coordinate and the imaginary coordinate gives you the complex number a point represents. This plane also shows rotations, for example take 5 on the real number axis; multiply it by i to get 5i; multiplying by i just gave you a 90 degree rotation counterclockwise, and multiplying by i again would get you to -5. Multiplying by i2 is a 180 degree counterclockwise rotation, and multiplying by -1 is also a 180 degree counterclockwise rotation! 5*-1=-5, 5*i2=-5, therefore -1=i2. This easy representation of a 90 degree rotation along with other properties also leads to complex numbers being incredibly useful to all kinds of engineers. An algorithm called Newton’s method finds the roots of equations in complex planes. A friend of Strogatzs’, John Hubbard, wondered about things with more than one root, and what root the algorithm would find. He programmed it, and ended up with this image:
You may not be able to see it in this little picture, but the article contains a little video that more clearly shows that this pattern is a fractal. I just love how everything ends up coming together and connecting back to finding roots, and complex numbers!