backwards to i

_drafts/backwards-to-i.md

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+---
+layout: post
+title: "square root of negative one"
+---
+
+a wheel is turning around its axle at a steady rate. the wheel's radius is one and a spot on its edge has coordinates x = cos(t) and y = sin(t). the distance of the spot from the axle is x^2 + y^2 = 1. 
+
+rather than represent x and y separately, can they be at least somewhat unified? for example, a tuple (x, y). or a vector. or, even better, a unit-vector. or an arithmetic expression. one possibility is x + k * y. the factor 'k' can be plus or minus itself and x - k * y is in some sense equivalent, and leads onward to writing down and contemplating some algebra.
+
+    (x + k * y)(x - k * y) = x^2 - k^2 * y^2 
+
+if this expression equals the wheel's radius of one, then 'k' is the square root of negative one, and the symbol 'i' has appeared as a scaling factor that makes everything work within the geometry of a spot on a steadily turning wheel.
+
+this is a 'backwards' approach to what feynman is discussing towards the [end of this chapter](https://www.feynmanlectures.caltech.edu/I_22.html). basically, starting at the end of the chapter and 'reverse engineering' back towards the beginning.
+
+to start at the beginning and go forwards, instead of backwards, what feynman did was to gradually and step-by-step get to an imaginary exponent. in particular, ten to the 'i' times 's', 10^is. he then noted that 10^is = x + i * y, and that 10^-is = x - i * y, and that their product is one.
+
+    (x + i * y)(x - i * y) = x^2 + y^2 = 1
+
+in the end, after changing the base from ten to 'e', he arrived at the final destination.
+
+    e^is = cos(s) + i * sin(s)

_posts/2024-08-17-backwards-to-i.md

@@ -0,0 +1,22 @@
+---
+layout: post
+title: "square root of negative one"
+---
+
+a wheel is turning around its axle at a steady rate. the wheel's radius is one and a spot on its edge has coordinates x = cos(t) and y = sin(t). the distance of the spot from the axle is x^2 + y^2 = 1. 
+
+rather than represent x and y separately, can they be somewhat unified? for example, a tuple (x, y). or a vector. or, even better, a unit-vector. or an arithmetic expression. one possibility is x + k * y. the factor 'k' can be plus or minus itself and x - k * y is in some sense equivalent, and leads onward to writing down and contemplating some algebra.
+
+    (x + k * y)(x - k * y) = x^2 - k^2 * y^2 
+
+if this expression equals the wheel's radius of one, then 'k' is the square root of negative one, and the symbol 'i' has appeared as a scaling factor that 'makes the geometry work'.
+
+this is a 'backwards' approach to what feynman is discussing towards the [end of this chapter](https://www.feynmanlectures.caltech.edu/I_22.html). basically, starting at the end of the chapter and 'reverse engineering' back towards the beginning.
+
+to start at the beginning and go forwards instead of backwards, what feynman did was to gradually and step-by-step get to an imaginary exponent. in particular, ten to the 'i' times 's', 10^is. he then noted that 10^is = x + i * y, and that 10^-is = x - i * y. he was then able to write some algebra.
+
+    (x + i * y)(x - i * y) = x^2 + y^2 = 1
+
+from there, he soon arrived at the final destination, after changing the base from ten to 'e'.
+
+    e^is = cos(s) + i * sin(s)