Monthly Archives: July 2014

The abstract approach to the abstraction which is “Negative Numbers”

An approach to the formal definition of negative numbers, using the ideas of abstract algebra.

Section 1 – is background. Skip it if you like.

What is a negative number?

1: It’s a number with a “-” in front.

2: It’s the opposite of a positive number.

Well, 1 is very poor, and 2 is no good, as there are no positive numbers until we have  negative numbers, they are just numbers (referred to later as the original numbers).

There is a need to compare numbers, and one way is to ask “What is the difference  between this number and that number. This is easy – the difference between 3 and 7 is  4, and we all learn to write 7 – 3 = 4.

Everything is fine for a while until someone says “But what about 3 – 7 ?”.

“Cannot be calculated. Has no meaning. You cannot take 7 things away from 3 things.  You cannot cut a 7 inch piece of wood off a 3 inch piece.” are the answers.

These original numbers are usable for counting and measurement of quantity, but numbers can also be used to measure position, leading to questions of the form “How far is it  from this number to that number?”. Temperature is the most obvious situation. “How  much warmer is it today, compared to yesterday?”. With numbers we can ask “How far  is it from 3 to 7 ?” and get a response ” 4 “, but we can also ask the question “How far  is it from 7 to 3 ?”. The response is the same, with the extra “but in the opposite  direction”.

Thus there arises a need for numbers capable of dealing fully with this new situation , the  measurement of changes in position. So negative numbers are born (or created), and  we hope they obey the same rules as the original numbers. Playing around seems to  support this position, with a few mysteries, such as (-1) times (-1) equals 1, and two  negatives make a positive.

However, in math we should not be satisfied by “Well, it seems to work OK”.

Section 2

What follows is a formal definition of an extended number system, in which every number has an “opposite”, or an “additive inverse”, and in which every number not equal to zero has a multiplicative inverse, and in which the “properties of operations” are still valid.

The definition only uses brackets (parentheses), commas, and pairs of original numbers. It does NOT use the negative sign, and subtraction  b – a, is only applied where it makes sense, that is when  b>a.

An extended number (or “thing”), written  ( a , b ), is defined as the distance from a to b.

It is immediately obvious that  ( a , b ) = ( a + 1 , b + 1 ) = ( a + 2 , b + 2 ) and so on.

So we can write  ( a , b ) as ( 0 , b – a )  when  b>a, and as  ( b – a , 0 ) )  when  b<a,  using subtraction only  with the original numbers.


Addition needs to match the original number addition, so

(definition)              ( a , b ) +  ( p , q ) =  ( a +p , b + q )                    check it

Zero is the “thing” which when added to anything has no effect, so the zero “thing” is ( p , p ) for any p.

Now we can have the additive inverse of a “thing”, the one which when added to the “thing” gives zero.

(definition)   Additive inverse of   ( a , b )  is   ( b , a )  since   ( a , b )  +   ( b , a )  =  ( a + b , a + b )

Subtraction for “things” can now be defined as addition of the additive inverse.

We can define multiplication of “things” by looking at the product of two differences.

(original number definition)  ( b – a )( d – c ) = bd + ac – ( ad + bc ) , so we have

(definition)              ( a , b ) X ( c , d ) =  ( ad + bc , bd + ac )

For multiplication we need a unit or identity “thing”, and the obvious choice is ( 0 , 1 ), or anything where the second number is one bigger than the first, for example  ( 12 , 13 ).  Using the “multiply” definition we have                                           ( 0 , 1 ) X ( 0 , 1 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X ( 12 , 13 ) = ( 12 , 13 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X (3, 7 ) = (3, 7 )

Division is defined as the inverse of multiplication, so to divide a “thing” by another “thing” we multiply the “thing” by the multiplicative inverse of the “other thing”, which we now define.

(original number definition)       multiplicative inverse of  ( b – a ) is 1/(b-a)

(definition)           Multiplicative inverse of  ( a , b ) is ( 0 , 1/(b-a))  if  b>a  and  ( 1/(a-b) , 0 )  if  b<a

If we multiply a “thing” by its inverse we should get the unit or identity ( 0 , 1 ), and so we do:

( a , b ) X ( 0 , 1/(b-a)) = ( a/(b-a) , b/(b-a) ) = (  0 , b/(b-a) – a/(b-a)  ) = ( 0 , 1 )

We have enough here to show that the new operations of + and X have the same properties as add and multiply in the original numbers. Go on, show it !!!!

Now what has this got to do with negative numbers ?  Well, the first thing is that ( 0 , 1 ) has an additive inverse, namely  ( 1 , 0 ), or any of its other representations, say ( 5 , 4 ) for example.

The second thing is that  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) .

The third, and most important thing is that we have an arithmetic for the “How far is it from A to B” quantities which incorporates direction.  When A<B the direction is one way. When A>B the direction is the other way. These directions are coventionally called “the positive direction” and “the negative direction”.

So, finally, we identify distances in the positive direction with the original numbers, and distances in the negative direction with new numbers, each of which is the “opposite” or “additive inverse” of one of the original mumbers.

Using the minus sign for “the additive inverse of” makes it quicker to write, at a cost of some possible confusion.  We see now that for example  ( 3 , 7) is identified with the original number  4 and  ( 7 , 3 ) is identified with the new number  -4.  Also  ( 0 , 1 ) is identified with the original number  1 and  ( 1 , 0 ) is identified with the new number  -1.

So since we have  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) it follows that  -1 x -1 = 1, and there is no mystery about it!


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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 :


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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Math takes a vacation. Hello real world.

This came to me as I was having lunch today:

fish_dinner_large with health warning and text3








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“Why can’t I just get the answer?”

In the days of two add two
little kids knew what to do.
Now adding is an operation
with properties of commutation,
distribution, association.
God help the kid who answers “Four”.
“There’s more. They need an explanation.”

Common Core 2a subtraction

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“I can’t do math”

Not so long ago I was doing substitute (supply) teaching in a school in Yorkshire,England, and this happened:

“I can’t do math” she said to me.
“Oh yes you can and you will see
that you can do it on your own”.
I showed her how the cryptic code
was just a way of writing.
“The meaning’s there, give it a go,
and try to stop the fighting”.
The tunnel end came into view,
the light was bright and shining.
She did another, all alone,
and smiled with satisfaction.
Then turned to me and said again
“I can’t do math”

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Filed under humor, language in math, Uncategorized, verse

Congruence. Transformations. Definitions. Unnecessary.

Once again I take the Common Core to task.

This time their obsession with transformations.

There is nothing wrong with learning about transformations but it is silly to attempt a definition of congruence in terms of transformations, as we will see. Rigid transformations of the plane (translations and rotations) preserve lengths and angles, and so a rigid transformation of a figure gives a new figure which is “the same”. Also, and ignored by CCSS is the vital fact that such a transformation can take an image to ANY position in the plane, where position is identified as “point + direction”.

So this is what is written:

High school geometry: Congruence
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally).
In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. (me – This last is garbage – see at end)

So here goes, an attempt to see if line segment FQ is congruent to BA

congruent line segs rigid transf

We slide BA in the direction B to F until B coincides with F – this is our translation.  A is now at the point K

We then rotate BA around point F until it is on top of  FQ, it is now in position FL. This is all possible as the position of the (simple) figure FQ must be known.

Have we carried BA onto FQ ? Not necessarily. It depends on where point Q is. The only way that L and Q can be the same point is a) if we know the length of segment FQ and b) this length is equal to the length of FL, which is the length of BA.

Therefore the only way that BA can be moved and placed on FQ is if they are the same length.

What have we done?  We have shown that if BA and FQ (or any two line segments) have the same length then there is a rigid transformation carrying one to the other. Hence by the definition of “congruent” BA and FQ are congruent.

In simple words, If BA and FQ have the same length then they are congruent, so simplicity (a highly desirable aim in math) is better served by the simpler definition:


The dragging in of transformations really muddies the water.
Here is part of the High School geometry detail

Understand congruence in terms of rigid motions
6. …..; given two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent.

and here is some dictionary stuff on superposition
 2.  The principle by which the description of the state of a physical system can be broken down into descriptions that are themselves possible states of the system. For example, harmonic motion, as of a violin string, can be analyzed as the sum of harmonic frequencies or harmonics, each of which is itself a kind of harmonic motion; harmonic motion is therefore a superposition of individual harmonics.
3. superposition – (geometry) the placement of one object ideally in the position of another one in order to show that the two coincide
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place
 4. superposition – the placement of one thing on top of another
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place

There is no Principle of Superposition in geometry!


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Pythagoras converse, proof from scratch

There is a website with 100 proofs of the famous theorem of Pythagoras, but when I trawled the net looking for a proof of the converse, they all assume the basic theorem.

Here’s how to do it from scratch, which is considerably more satisfying, and also a simple application of similar triangles and basic algebra:

pythag converse diagram pythag converse text

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July 2, 2014 · 7:21 pm