For almost fifty years I have explored and taught mathematics, at a variety of levels. The study of science often requires a significant amount of mathematical reasoning. Yet students consistently struggle in, and sometimes even dread, their math classes. In this talk I will share some thoughts on the beauty and importance of mathematics, with an emphasis on some simple steps to student success in this important field.
I want to tell you how to be successful in mathematics. But before I do, I want to tell you why you should care.
1. Mathematics and mathematical exploration is good for the human race!
I am sure you have heard many arguments about mathematics improving science or technology but here are some small examples of which you might not be aware.
Mathematics helps brew good beer. Indeed, a century ago, an engineer at Guinness in Ireland, working on developing quality control, discovered what is now called the Student t-distribution. (The specialist published his results under the pseudonym August Student.) Good beer led to better math led to better beer!
Artists in the Italian Renaissance developed the mathematics of perspective. Filippo Brunelleschi and Leon Battista Alberti were leaders in this field; Leonardo daVinci displays his knowledge of perspective by including the perspective lines in paintings such as The Last Supper.
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| Piera della Francisco's Ideal City |
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| da Vinci's Last Supper |
This study led to projective geometry which violated classical Euclidean geometry.
And non-euclidean geometry is the background for the theory of relativity. The theory of relativity is used to precisely set the clocks in orbiting satellites so that our GPS coordinates are accurate. There is about a 25,000 nanosecond difference (per day) in the clocks in the standard GPS satellite; if this were not corrected by these clock setting, the GPS coordinates would be off by 25,000 feet (8 kilometers) per day!
Your GPS guide not only relies on the geometry of mathematics but the geometry of relativity! (Where the sum of angles of a triangle is NOT 180 degrees and the Pythagorean theorem does not hold!)
Other examples:
- The mathematics behind microbiology has helped us develop antibiotics and vaccines and led to fairly rapid development of vaccines during the COVID pandemic.
- Linear algebra, especially algorithms (a 2-D Fourier transform) are used to find inverse of matrices and more complicated mathematical systems has led to the science of Magnetic Resonance Imaging (MRI).
- The seismic imaging process in geology uses a similar inverse problem to map out the geology of a region (such as Mount Vesuvius at the Vesuvius Observatory) without being able to “see” the internal formations.
- Binary sequences have led to the use of objects like Barker sequences (a Barker sequence of length 11) is used as BPSK signal modulation in many wifi routers.
- Pseudorandom sequences are behind most computer games, assuring that each gaming experience is different than the last.
- Number theory that dates back to Fermat has led to public key exchange algorithms behind internet commerce. If you have ever used a credit card to purchase an item online, you have used one of those key exchanges.
