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Complex Numbers A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number. The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi. Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary. Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty. ![Merit Badge in Quill Award
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| Hey, look, a headline question to which the answer isn't "No." Are All Brains Good at Math?  Math provokes dread in so many people—yet we are all born with a sense for numbers. Well... okay. Technically, the answer is "No." The headline doesn't specify human brains, and while most nonhuman animals have brains, it's not clear that they all have a sense of number. No, all y'all for whom math provokes dread, you only think you're bad at it. It becomes a self-fulfilling prophecy, like when I say I'm not creative. Ken Ono gets excited when he talks about a particular formula for pi, the famous and enigmatic ratio of a circle’s circumference to its diameter. He shows me a clip from a National Geographic show where Neil Degrasse Tyson asked him how he would convey the beauty of math to the average person on the street. In reply, Ono showed Tyson, and later me, a so-called continued fraction for pi, which is a little bit like a mathematical fun house hallway of mirrors. Instead of a single number in the numerator and one in the denominator, the denominator of the fraction also contains a fraction, and the denominator of that fraction has a fraction in it, too, and so on and so forth, ad infinitum. Written out, the formula looks like a staircase that narrows as you descend its rungs in pursuit of the elusive pi. The calculation—credited independently to British mathematician Leonard Jay Rogers and self-taught Indian mathematician Srinivasa Ramanujan—doesn’t involve anything more complicated than adding, dividing, and squaring numbers. “How could you not say that’s amazing?” Ono, chair of the mathematics department at the University of Virginia, asks me over Zoom. On the other hand, maybe I'm just linking this because it calls out my alma mater. But not everyone sees beauty in fractions, or in math generally. In fact, here in the United States, math often inspires more dread than awe. Story idea: Horror novel, but the antagonist is math. Math anxiety seems generally correlated with worse math performance worldwide, according to one 2020 study from Stanford and the University of Chicago. Seriously? Someone funded a study about that? Some scholars argue that American culture’s saturation with negative stereotypes around math in combination with current approaches to teaching the subject are perpetuating anxiety, making some kids think they are bad at math, and preventing them from excelling. I can't forget when they put out a Barbie doll that voiced her frustration with math. And several other phrases promoting stereotypes and gender roles. Yes, that was already a thing in the early 90s.In other words, we are primed to do basic math, but culture gets in the way. This is one of the few areas where I genuinely believe the phrase "You can do anything you set your mind to." If you think you're good at math, you'll be good at math. At least enough to do some basic calculations, like tips and imperial-to-metric conversions. This ability to estimate and understand quantity may have evolved as a basic survival skill. It could have helped our ancestors and members of other species quickly assess whether they were outnumbered by predators, for example, or to forage in places with more available food relative to others. And then the article has to go into evolutionary speculation. It's just that: speculation. At least the author acknowledges that. There seems to be consensus among scientists, however, that only humans mentally represent numbers precisely and with symbols, and that we need some kind of education to do so. This is potentially because many higher math skills, including arithmetic, depend on the use of language—a symbols-based system—where quantity-based judgments are pre-verbal. I mean, when you think about it, numbers are quite abstract. If you have a small group of trees, and the same number of hogs, it's a pretty wild mental leap to say that the number "3" (or whatever) can apply to both of them. Indeed, arithmetic is difficult to do if one does not have the language for it. Just ask a Roman. If we have the innate ability to understand math and acknowledge its importance in making sense of the world, why are so many—in Western culture, mainly—averse to it? Scholars who study math and math education have a hard time answering this question. Oh, but I bet we're going to try. Perhaps it is related to the way we in the U.S. tend to conceptualize math ability. People in the West often say, “I’m bad at math,” as though it were a personality trait or even a badge of honor. Of all the things that piss me off about people, one of the biggest is when they make ignorance a positive trait. Yeah, yeah, I know; I've done it myself, like when I proclaimed I've never seen Titanic and never will. The difference is, Titanic isn't going to help me keep a budget or measure quantities in a recipe. Plus, I never claimed to be internally consistent. No one is. Of course, some kids do have clinical learning disabilities that are not caused by anxiety, cultural myths, or poor teaching. And none of what I'm saying here is meant to rag on them. Learning disabilities are what they are; willful ignorance is the problem. Evidence suggests that cultural assumptions that women are less skilled at math than men may account for much of the gender gap in math performance. While one data point is meaningless, the top math-doer in my high school class was a girl. So I never bought into the idea that there's an innate gender difference. What, estrogen means you can't count? Come on. No. Math anxiety may be transmitted from teachers and parents to kids, too. “Evidence from the United States suggests that children who interact with high-math-anxiety adults show impaired math performance relative to their peers,” the authors of a 2017 study wrote. I will say this, though: math education has changed since I was a kid. Now it's all that common-core stuff. I'm not saying it's wrong or bad; just that it wasn't the way I learned. So when a friend of mine got me to help her kid with their math homework once, I took one look at the common-core textbook and noped out. It's not that I can't learn it. It's just that the kid already knew more than I did about it. The article touches on that subject. Then: A seemingly opposing perspective comes from Barbara Oakley, author of A Mind For Numbers: How to Excel at Math and Science, who argues that, ultimately, learning math is more like learning a language or learning music than it is like some other forms of learning. I tend to agree with this. Mathematics is a language. Conversely, language is math. Ultimately, individual kids have specific needs, and parents can work with teachers on figuring out tailored approaches. But, generally parents can create a positive attitude toward math by refraining from sharing their own math anxieties and encouraging creative problem solving. And while that's important, I'd like to see a push toward convincing adults that, no, math isn't going to pull them down into the gutter and suck out their blood or whatever. So, maybe don't write that horror story after all. Meanwhile, can we please stop pandering to the "when are we going to use this?" crowd? Let's try to instill a love of learning for the sake of learning. There's no such thing as useless information—except maybe what some internet influenza is trying to tell us. |
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