![]() What transformation x, when applied twice, turns 1 to 9? Visual Understanding of Negative and Complex NumbersĪs we saw last time, the equation $x^2 = 9$ really means: The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it. I dislike the term “imaginary number” - it was considered an insult, a slur, designed to hurt i‘s feelings. But as the negatives showed us, strange concepts can still be useful. New, brain-twisting concepts are hard and they don’t make sense immediately, even for Euler. You may not believe in i, just like those fuddy old mathematicians didn’t believe in -1. New relationships emerge that we can describe with ease. But playing the “Let’s pretend i exists” game actually makes math easier and more elegant. ![]() That is, you multiply i by itself to get -1. 3, and 0 “exist”, let’s assume some number i exists where: So-called “imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world. There’s no “real” meaning to this question, right? It seems crazy, just like negatives, zero, and irrationals (non-repeating numbers) must have seemed crazy at first. You want the square root of a number less than zero? That’s absurd! (Historically, there were real questions to answer, but I like to imagine a wiseguy.) ![]() This question makes most people cringe the first time they see it. But suppose some wiseguy puts in a teensy, tiny minus sign: We can solve equations like this all day long: It’s a testament to our mental potential that today’s children are expected to understand ideas that once confounded ancient mathematicians. They were considered “meaningless” results (he later made up for this in style). Even Euler, the genius who discovered e and much more, didn’t understand negatives as we do today. Today you’d call someone obscene names if they didn’t “get” negatives.īut let’s not be smug about the struggle: negative numbers were a huge mental shift. It didn’t matter if negatives were “tangible” - they had useful properties, and we used them until they became everyday items. The positive and negative signs automatically keep track of the direction - you don’t need a sentence to describe the impact of each transaction. I have +70 afterwards, which means I’m in the clear. If I earn money and pay my debts (-30 + 100 = 70), I can record the transaction easily. Rather than saying “I owe you 30” and reading words to see if I’m up or down, I can write “-30” and know it means I’m in the hole. Negatives aren’t something we can touch or hold, but they describe certain relationships well (like debt). What happened? We invented a theoretical number that had useful properties. Try asking your teacher whether negatives corrupt the very foundations of math. Yet today, it’d be absurd to think negatives aren’t logical or useful. Negatives were considered absurd, something that “darkened the very whole doctrines of the equations” ( Francis Maseres, 1759). Simple.īut what about 3-4? What, exactly, does that mean? How can you take 4 cows from 3? How could you have less than nothing? You have 3 and 4, and know you can write 4 – 3 = 1. Imagine you’re a European mathematician in the 1700s. Video Walkthrough: Really Understanding Negative Numbers By the end we’ll hunt down i and put it in a headlock, instead of the reverse. It doesn’t make sense yet, but hang in there. We’ll approach imaginary numbers by observing its ancestor, the negatives. ![]() Using visual diagrams, not just text, to understand the idea.Īnd our secret weapon: learning by analogy.Seeing complex numbers as an upgrade to our number system, just like zero, decimals and negatives were.Focusing on relationships, not mechanical formulas.Gee, what a great way to encourage math in kids! Today we’ll assault this topic with our favorite tools: It’s used in advanced physics, trust us.It’s a mathematical abstraction, and the equations work out.Like understanding e, most explanations fell into one of two categories:
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