Imaginary Bases - TheGrayCuber
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imaginary bases
We can apply the idea of number bases to the imaginary numbers!
To get started, try pressing the die to randomize.<br>For more control, select the view/edit icon, then drag the filled shapes on screen to change<br>the base and digits.
Press the hearts button to cycle through a curated set of systems.<br>This is open for submissions, so if you find a cool setup,<br>use the copy button to get the code, and paste it in a comment<br>on my video about the topic.<br>I'll add my favorite ones to this page :)
Special thanks to Combo Class for introducing me to<br>base i - 1, and Thomas Kaldahl for introducing me to<br>the Eisenstein bases!
into the math
As a quick recap of base systems, let's take a look at our typical system - base 10.<br>To find the value of a number, we can first split the digits apart:
1234 = 1000 + 200 + 30 + 4
Each of these components is the product of a single digit and a power of the base, determined by the number of zeroes:
1000 = 1 * 10³
200 = 2 * 10²
This idea can be applied to other natural bases, such as base 2, or base 12, but what if we make the base imaginary?
base i
Let's replace our base 10 with imaginary number i, which has the property i² = -1. We can calculate the value 1234 as follows:
1234 = 1000 + 200 + 30 + 4
= 1 * i³ + 2 * i² + 3 * i + 4
= -i - 2 + 3i + 4
= 2i + 2
This base technically works, but it has the problem of duplicates:
10000 = 1 * i⁴ = (i²)² = (-1)² = 1
There are multiple ways to write 1. A well-behaved base system should have a unique representation for each number.<br>Instead we can use base -1+i, which does represent each value uniquely. Paste the following code to see the base:
base -1+i with digits 0, and 1
base ω-1
Instead of the imaginary unit, which produce a square grid, we can create a hexagonal grid by using ω (omega).<br>It can be defined using either of the following equations.<br>As an exercise, show that the second implies the first.
ω² = -ω-1
ω = (√3i-1)/2
Just like i, ω does not work well as a base. However, ω-1 works quite well. Use the following code to check it out:
base -1+ω with digits 0, 1, and 1+ω
about me
I am TheGrayCuber, a creator of mathematical websites and videos
This is the first project that I've made since my transition to this new website framework.<br>I welcome suggestions for improvements, so let me know what you think!
My intent behind this page is to offer you a method to easily explore imaginary bases -<br>quickly making adjustments with live feedback - so that you can get the feeling of these<br>systems without worrying about the mathematical details going on behind the scenes.
This reflects my ongoing shift in focus to the experience that we get by engaging with mathematics,<br>as opposed to the precision that we get by doing it 'correctly'.<br>Of course, building models correctly often leads to positive experiences,<br>but I think it's important to allow for messiness and mistakes,<br>because they too can provide interesting and novel insights.
Is it worthwhile for beings to spend their time solving dozens of exercises of the same form?<br>Should I value a theory based on its applications, even though so many of those<br>applications have accelerated the issues that cause pain to so many earthlings?<br>I am trying to stop viewing math as a tool to solve problems,<br>because open communication and critical thinking will generally provide better solutions.<br>Instead, I'll share my love for mathematics as entertainment.<br>A topic that we think about shrimply because it's fun to think about.
made with love, and without AI
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