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Starting with the logical roots of arithmetic with negative numbers, Marcus du Sautoy explains how mathematicians created imaginary numbers by ‘imagining’ the square root of -1.
- Really Understanding Negative Numbers
- Enter Imaginary Numbers
- Visual Understanding of Negative and Complex Numbers
- Finding Patterns
- Understanding Complex Numbers
- A Real Example: Rotations
- Complex Numbers Aren’T
- Epilogue: But They’Re Still Strange!
- Other Posts in This Series
Negative numbers aren’t easy. Imagine you’re a European mathematician in the 1700s. You have 3 and 4, and know you can write 4 – 3 = 1. Simple. But what about 3-4? What, exactly, does that mean? How can you take 4 cows from 3? How could you have less than nothing? Negatives were considered absurd, something that “darkened the very whole doctrines o...
Imaginary numbers have a similar story. We can solve equations like this all day long: The answers are 3 and -3. But suppose some wiseguy puts in a teensy, tiny minus sign: Uh oh. This question makes most people cringe the first time they see it. You want the square root of a number less than zero? That’s absurd!(Historically, there were real quest...
As we saw last time, the equation x2=9really means: or What transformation x, when applied twice, turns 1 to 9? The two answers are “x = 3” and “x = -3”: That is, you can “scale by” 3 or “scale by 3 and flip” (flipping or taking the opposite is one interpretation of multiplying by a negative). Now let’s think about x2=−1, which is really What trans...
Let’s dive into the details a bit. When multiplying negative numbers (like -1), you get a pattern: 1. 1, -1, 1, -1, 1, -1, 1, -1 Since -1 doesn’t change the size of a number, just the sign, you flip back and forth. For some number “x”, you’d get: 1. x, -x, x, -x, x, -x… This idea is useful. The number “x” can represent a good or bad hair week. Supp...
There’s another detail to cover: can a number be both “real” and “imaginary”? You bet. Who says we have to rotate the entire 90 degrees? If we keep 1 foot in the “real” dimension and another in the imaginary one, it looks like this: We’re at a 45 degree angle, with equal parts in the real and imaginary (1 + i). It’s like a hotdog with both mustard ...
We’re not going to wait until college physics to use imaginary numbers. Let’s try them out today. There’s much more to say about complex multiplication, but keep this in mind: 1. Multiplying by a complex number rotates by its angle Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my ...
That was a whirlwind tour of my basic insights. Take a look at the first chart — it should make sense now. There’s so much more to these beautiful, zany numbers, but my brain is tired. My goals were simple: 1. Convince you that complex numbers were considered “crazy” but can be useful (just like negative numbers were) 2. Show how complex numbers ca...
I know, they’re still strange to me too. I try to put myself in the mind of the first person to discover zero. Zero is such a weird idea, having “something” represent “nothing”, and it eluded the Romans. Complex numbers are similar — it’s a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, ne...
Sep 2, 2019 · adding, subtracting, arithmetic, negative numbers, addition, subtraction, add, subtract, multiplying, multiplication, dividing, division
I can solve problems involving positive and negative numbers in a range of contexts.
Understanding patterns will open doors for your students to understand more complex mathematical operations, but also, improve their ability to understand the world. This is why, in this article, we’ll share a number of tips on how to teach your students number sequences and patterns.
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Sep 9, 2019 · words, figures, numbers. Practice Questions. Previous: Flow Charts Practice Questions
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