Quick Insight: Intuitive Meaning of Division

While working on somemath colorizations我发现了一些关于"除法"的解释。

Multiplication can be repeated addition, scaling, rotating (viaimaginary numbers), and more.

部门呢?让我们一起来看看吧。

Thepermutation formulalets us pick 3 items out of 10, in a specific order. To order 3 items from 10, we have 10 options for the first choice, 9 options for the second, and 8 for the third, giving us 10 _ 9 _ 8 = 720 possibilities.

But how is this process written in most math books?

$3 \ displaystyle {P(10日)= \压裂{10 !} {(10 - 3) !} = 720}$

What's going on?

Well, we just want a portion of the factorial. 10! gives the full sequence (10 _ 9 _ 8 _ 7 _ 6 _ 5 _ 4 _ 3 _ 2 _ 1), but we want to stop it after we hit 8. That means we divide by 7!. The only part that's not removed is 10 _ 9 * 8.

在这种情况下，除法就像一个刹车/边界/过滤器，阻止阶乘疯狂运行。(你知道，他们会失去控制。)

Ah! If writing a software program, you wouldn't actually compute 10!, 7! and do a division. What if we needed 3 choices from 1000 options? (1000 factorial has2568 digitsand will make your computer cry.I told you this would happen!)

如果我们认识到除法作为边界标记的作用，我们只要计算1000 _ 999 _ 998 = 997002000就可以了。

Let's keep going.

假设我们不关心所选商品的顺序:ABC与CBA相同。要做什么吗?

Well, we can apply another division! This time, we don't want a boundary, but want to merge/consolidate/group up similar items. Everything that looks like ABC (e.g., ACB, BAC, BCA, CAB, CBA) should be counted once.

With 3 items there are 3! rearrangements, so the final count is 720/3! = 720/6 = 120 choices.

As a formula:

整洁的,对吧?排列公式中的除法作为边界，组合公式中的除法是一种“群上”。I imagine the variations being merged into a single option:

The words we pick frame how we think about an equation. "Divide" implies we're splitting things apart. If we know alternate meanings (repeated subtraction, boundaries, consolidation), we may pick a better description. Saying "divide by k!" doesn't have the same intuition as "consolidate the reorderings".

I think math concepts are fundamentally simple but their written description may not be. (Ever try to describe how to put on a shirt?) The goal is finding the words to make the idea click.

Happy math.

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