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
We staid till, it being darkish, we saw the fire as only one entire arch of fire from this to the other side the bridge, and in a bow up the hill for an arch of above a mile long: it made me weep to see it. The churches, houses, and all on fire and … Continue reading Cycloid
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
My mind tends to consider extremes. From thinking I have no options to feeling overwhelmed by options, making choices can be challenging. Counting, when possible, calms my mind and simplifies reality. As summer was beginning, I posted about Pascal’s Triangle, a tool for counting combinations. Considering combinations turns our thinking from a set into a variety of possible subsets; a … Continue reading Combinations
We have opportunities to count everyday. Some of the counting is done for us, such as the number of likes on your favorite social media post. We even celebrate victories when the count, the score, is a desired result. Too often though, numerics bring a gravity—an unfortunate heaviness—to life, especially when we are asked to do … Continue reading Permutations
Pascal’s Triangle gives us a way to count combinations: groupings when the order does not matter. There are other methods of counting worthy of investigation, including occasions when order does not matter and when it does. Sometimes order is important. With lunch lines and horse races, the order matters. If there are a limited number of specials, your spot in … Continue reading When Order Matters
Footpath Math: Chalk it up to a math geek 