Traveling around the world tends to shift my perspective. If for only a moment, I remember our planet is round, not flat. I know it is round, sure, but I often live as if the world is flat. For example, I tend to think of streets as parallel and perpendicular lines. (Surprising, since I live … Continue reading Let digons be bigons.
You may have met the complex numbers. Gauss named them in 1831, but our knowledge of complex numbers was born in the first half of the 16th century in northern Italy with attempts to solve cubic equations. Leveraging knowledge from Tartaglia, Girolama Cardano published his findings in his book Ars Magna in 1545. Complex numbers have a real part and an … Continue reading Polar Form of Complex Numbers
Trigonometric identities are like a set of high quality knives, staying sharp even when asked to slice, cut, and chop day after day. In Sine Squared, I used a Pythagorean identity to coax the value from an integral. An identity is an equality statement. x = x is an identity; x equals x for any value of the variable x. This is distinct from an equation … Continue reading Pythagorean Identities
Life stands up to the downward force of gravity every day. From table legs to your legs, perpendicularity provides balance and strength. Perpendicular lines played an important role in finding the center of the arc of the bridge in the Pedestrian Bridge post. In writing the equation of those perpendicular lines, I used a concept I picked … Continue reading Perpendicularity
Lorenzo Mascheroni was born in 1750 outside of Bergamo in what is now northern Italy. This was before Napoléon Bonaparte was born, much less taking over the world, and Lombardy was under Austrian control. Young Lorenzo grew up in a wealthy, landowning family. Math folk like to think big, really big, infinitely big. Mascheroni was not … Continue reading Mascheroni Construction
High school geometry changed my life. Really. I loved proofs. I loved geometric constructions. Life on the Euclidean plane had order; it made sense, in an amazing way. Need to find the midpoint of a line segment? No problem. Draw two circles (or just parts of them), and then draw the line determined by the … Continue reading Perpendicular Bisector
Just south of Mueller in Austin is the J.J. Seabrook Greenbelt. It is hugged by Pershing Drive and will soon have a trail connected to the trail at Mueller. The creek bed has been re-engineered and, just a few days ago, a bridge arrived ready to span the creek. It was a sight to see the bridge … Continue reading Pershing Pedestrian Bridge
Austin TX 2013/06/16 This post and the previous are sort of farewell gifts to my students who just finished their first year studying calculus. I love creating lectures (as you might imagine), but they hungered for something else. They challenged, inspired, and reminded me what I love about teaching. For me, the gift of teaching is … Continue reading Euler’s Formula
Barton Springs PoolAustin TX2013/06/02 There are so many books, articles, websites, poems and cartoons about π; I considered not chalking it. I have been working on an upcoming post on Taylor series, but π just keeps saying, “Hey, after all I have done for you, will you please just spend some time with me?” So, here … Continue reading π
Austin TX 2013/04/27 This post began as a rather ambitious endeavor. My hope was to introduce infinite series and then gracefully bounce onto power series on the way to Taylor series. It occurred to me this was too much. Like too many toppings on a pizza, the crust could only hold so much. You still … Continue reading Powers of One Half
Footpath Math: Chalk it up to a math geek 