Tag Archives: embedding

Operators, Functions, and Properties – part 13

Posted on 2010/07/28 by Bert Speelpenning

In this series, we have been looking at little machines that do things like “add 2”, and we’ve been chaining such machines together by connecting the output of one to the input of the next.  We’ve been exploring the behavior … Continue reading →

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Representations – Black Boxes – Equivalence

Posted on 2009/09/10 by Bert Speelpenning

I introduced the notion of a black box in an earlier post as some thing that has an internal organization that drives its behavior, and though we can see the behavior, we don’t have perfect knowledge of that internal organization.  … Continue reading →

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Representations – Formulas and Some Alternatives

Posted on 2009/08/23 by Bert Speelpenning

There are systems of notation for mathematical expressions that are in wide use.  One of them is so widely used and so well-known that we often think of it as the only one, the real one, the true one and … Continue reading →

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Quantity – Different Kinds of Numbers: Classes

Posted on 2009/06/12 by Bert Speelpenning

Let me recap what this series of blog posts has been about.  I’ve been looking at different kinds of quantities, different kinds of numbers.  I started with observations like: it’s meaningless to add 1951 AD to 2009 AD but completely … Continue reading →

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Quantity – Different Kinds of Numbers: Vectors

Posted on 2009/06/09 by Bert Speelpenning

Sometimes, when you assign a number to a quantity, a single number really isn’t enough.  Take blue jeans sizes, as a simple example.  Jeans sizes aren’t expressed with a single number, from small to large, like shoes.  Since at least … Continue reading →

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Quantity – Different Kinds of Numbers: Keys

Posted on 2009/06/04 by Bert Speelpenning

I introduced this series by noting that there are different kinds of numbers – more precisely, different kinds of quantities represented by numbers – and that each allows you to draw different kinds of conclusions. The most basic kind of … Continue reading →

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Quantity – Different Kinds of Numbers

Posted on 2009/06/03 by Bert Speelpenning

We’re very used to representing quantities by numbers.   The cash register will tell us how much we have to pay, and the pump at the gas station will tell us how much gasoline we poured into our tank, and the … Continue reading →

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What is Multiplication – the Series

Posted on 2009/05/06 by Bert Speelpenning

Here’s a summary of the series (thus far) of posts on multiplication, with links. This allows you to read them in order, from oldest to most recent. The series grew from a concern raised in the post Middle School Math … Continue reading →

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What is Multiplication – part XIII

Posted on 2009/05/05 by Bert Speelpenning

Time for some wrap-up and some pulling together, after twelve prior installments in this series.  This doesn’t mean that I think I have arrived at some kind of eternal truth about multiplication.  Instead, the whole premise of this work is … Continue reading →

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What is Multiplication – part XII

Posted on 2009/05/04 by Bert Speelpenning

In this post I sliced the various ways we’ve looked at multiplication into two basic types, which I called scalar multiplication and unit multiplication.  This is not the only useful way to slice and dice the various multiplicative structures we’ve … Continue reading →

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