Tag Archives: minus

Read this : Negative numbers, by A. N. Whitehead

Alfred North Whitehead, professor of mathematics and philosophy, and famous for his collaboration with Bertrand Russell on their joint effort, the Principia Mathematica, also wrote a book, “Introduction to Mathematics”, in 1911, for High School students and others who really wanted to know what math was all about.
The section on negative numbers is so relevant to the teaching of that topic today that it is a MUST READ. Click the link to download this section.

Alfred North Whitehead: Introduction to Mathematics

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Negative numbers: truth, existence, reality, abstraction.

There are things with names which I can pick up, see, feel, or if too radioactive then at least observe and measure…..this is the reality we have (though some philosophers, and neuroscientists would have us believe otherwise). But what with numbers?

I can see three coconuts, I can count them, and match the count to the quantity, but the number THREE is not real, it is an abstraction from all the occurrences of three object that I have seen or can visualize.

Well, if THREE is not real then MINUS THREE hasn’t got a chance at being real, it is in fact a second stage abstraction, as negative numbers were invented by humans to deal with situations not adequately described with “ordinary” numbers. It gets worse, as complex numbers were invented to get over the difficulties with “real” numbers (the positive and negative numbers). It is a shame about the use of the word “real” in this situation (see above). They should have been called “simple numbers”.

Abstraction is also the basis of geometry. Euclid says “A line is that with extent but no breadth”, which does make it difficult to see! 

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-1 x -1 = 1, but some need convincing

Here is a popular argument:   -1 x -1 has to be 1 or -1

If it was equal to -1 then -1 x -1 = -1 x -1 x 1 = -1 x1 and so dividing both sides by -1 we get -1 = 1, which is not a good idea!, hence -1 x -1 = 1

This argument begs so many questions that it is difficult to know where to start.

Here is a much better one, but it does stretch the idea of area a little :

minusonetimesminusone

From the diagram  (a – 1) x (b – 1) = a x b – a – b + 1

Set a = 0 and b = 0 to get  (0 – 1) x ( 0 – 1) = 1, and since 0 – 1 is equal to -1        we get -1 x -1 = 1

This has some connection with evaluating for example  3 x ( 8 – 2)  using the distributive law.

The distributive law is a law for  a(b + c) and says nothing about  a(b – c), but never mind, we go gaily about the common task.

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