Tag Archives: algebra

The mean ? Which mean ? With interesting ratios.

Playing around with the Harmonic Mean of two numbers I stumbled on an interesting ratio, and looked at the others as well.

Here are the definitions, for numbers a and b

means

If we use m for the mean, then

for the arithmetic mean we have the ratio (b-m)/(m-a) = 1

for the geometric mean we have b/m = m/a

for the harmonic mean we have (b-m)/(m-a) = b/a

and for the RMS mean we have (b2 – m2)/( m2 – a2) = 1

I am quite sure that there is a way of seeing these which ties them all together, perhaps Mr. Joseph Nebus can find it !

The harmonic mean can be used to explain the harmonic tuning of a keyboard instrument (as opposed to equal temper tuning). I am planning a post on this for later.

The formula I gave for the harmonic mean is not the usual one – use a bit of algebra ! – but it is easier to calculate with.

The RMS mean is used extensively in Statistics, Rigid Body Dynamics and Electrical Engineering. The well known 110 volts in your house electric system is the RMS mean of the alternating voltage actually supplied. The Standard Deviation is the RMS average of the distances of the data values from the arithmetic mean value.

A non formal view of these means (the first three) is that the arithmetic mean is about the positions of the two numbers, the geometric mean is about the absolute sizes of the numbers and the harmonic mean is about the relative sizes of the numbers.

if we take the zero, the two numbers, and the harmonic mean the four values have a cross ratio of -1 (see part 3 of the Christmas Tale)

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Completing the (four sided) square

completing the square

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Pythagoras, triples, 3,4,5, a calculator.

How to generate Pythagorean triples (example: 3,4,5), well one way at least.

Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy

we can write the difference of two squares

(x + y)2  –  (x – y)2 = 4xy

and if we write  x = A2 and y = B2 the right hand side is a square as well.

Thus:

(A2  +  B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2

which can be written as

(A2  +  B2) 2 = (A2 – B2) 2 + (2AB) 2

the Pythagoras form.

Now just put in some integers for A and B

2 and 1 gives 3,4,5

Conjecture1: This process generates ALL the Pythagorean triples.

Conjecture2: Every odd number belongs to some  Pythagorean triple.

Have fun…….

My next post will be about finding the radius of the inscribed circle in a right angled triangle…..

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“Observe and make use of structure”. Observe would be a start. A tale from the chalkface.

Here’s my little story:
It was a class of day release students on a Higher National Certificate course in engineering. I reached a point in one class with a relationship between p and q, p = kq, with k a constant. “What’s its graph look like”, I asked. Deathly silence. “Ok, let’s try x and y”. Result y = kx. Same question, same response. “Well, what about y = 3x ?”. Same question, same response. So I wrote y = 3x + 2. Their eyes lit up, and they unanimously shouted “It’s a straight line!”.

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Subtraction in algebra – let’s use algebra !

I have seen some heavy handed ways of explaining the identity

a – (b + c) = a – b – c

Let us use algebra. Give the left hand side a name, say d .Then

a – (b + c) = d

This is an equation, so add (b+c) to each side and get

a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.

Now subtract  b  from each side

a – b = d + c

Now subtract  c  from each side

a – b – c = d

so  a – (b + c) =  a – b – c

or is this too simple ? Look, no messing with p – q = p + -(q) stuff,

and no appeal to the famous distributive law.

You can do this, and other stuff, with numbers as well.

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Rigid Transformations – Coordinate axes

A simple diagram with original axes in blue.
The coordinates of point E are (1,1)
A translation defined by x -> x + 2, y -> y + 1 moves point E to point D, with coordinates (3,2)
translation v moving axes

If the x axis is moved 2 steps left and the y axis is moved one step down then the coordinates of the original point E in the moved axes are (3,2)

This will be the case for any original point – the coordinates of each one of them will be the same as the coordinates of their new positions under the translation (in the original coordinate system).

This can be seen to be true for the other rigid motions, for example
rotation about the origin through an angle theta is equivalent to a rotation about the origin of the axes through an angle minus theta. So there is a one to one correspondence between rigid motions and change of axes (scales preserved).

On a lighter note, it does seem easier to rotate a pair of axes than rotating the whole plane ! ! !

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Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

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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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Calculus without limits 2

As h approaches zero
I quietly despair.
It really is the limit.
Please don’t take me there.

The funny thing about the calculus is that it was brought into existence by Isaac Newton in 1666 or earlier, and was developed and used without the idea of limits for over 150 years. The first attempt to get rid of the troublesome infinitesimals was by Cauchy in 1821, where he introduced the chord slope (f(x + h) – f(x))/h. The whole business of finding a satisfactory definition of the derivative was finally achieved by Weierstrass in the mid 19th century.

So here we go with cubics, and the same approach can be used for any whole number power of x, even negative ones. You should try it.

calculus2

Next time  sin(x) and cos(x), so no more  sin(h)/h stuff.

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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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