Tag Archives: formal

What is a number? Particularly √2

After the previous post, on the reality or otherwise of the square root of -1, I thought that the square root of 2 might benefit from a similar inquiry. After all, what can we actually say about √2 ? The answer to this question is very simple. “Not a lot !”.

In the real world of engineering, architecture, mathematical modelling, business, medicine and so on numbers are either counts (1,2,3,4,…) or absolute or relative measurements (1.20cm, 240 secs, 15 mins, 4096 ft, 35.7 mph,….). The first group is the natural or counting numbers, the second group is the rational numbers, and not so many of them. In practice it is rare for the size of a quantity to be expressed with more than four significant figures. So every practical quantity has a measurement in the form of a rational number, and most importantly IT CAN BE WRITTEN DOWN. I am going to call this the VALUE of the number.

The only thing is an assertion that there is a sort of number which when multiplied by itself produces the value 2.

So where does that leave √2 ?  It cannot be written down in the form of a rational number, so it has NO value in the above sense.
Ok, I can write  1.41422 < 2 < 1.41432 but neither of the two values shown is the value of √2. I could go on and get more digits in the two numbers and this would still be true.

This all started with the ancient Greeks, who found out that the length of the diagonal of a unit square was a quantity very different from quantities which could be measured using the side length of a unit square as the measurement unit. They described this state of affairs as “The side length and the diagonal length of a square are incommensurable”, which is a nice long word.

In passing I have to say that the Common Core math makes a real pig’s ear of this stuff.

So the Greeks were happy with the idea that every line segment has a length, and that the length is expressed as a number, but this wasn’t good enough for the nineteenth century mathematicians. I may write about this later, but for now we should be seeing if √2 can reasonably be “joined ” to the rational number system in a non magical, non wishful thinking way.

Let’s pretend that √2 is a sort of number, and that new numbers can be formed by a rational number “a” plus a rational amount “b” of √2, and write this as a + b√2

Then the sum of two these comes in as
(a + b√2) + (p + q√2) = (a + p) + (b + q)√2

and the product comes in as 
(a + b√2)(p + q√2) = (ap + 2bq) + (aq + bp)√2

In each case we have another of the “new” numbers.

One tricky question remains. What about division ?

If I multiply a + b√2 by a – b√2 I get a2 – 2b2 which has no √2 in it, it is a normal rational number, and it is only zero if BOTH a and b are zero.
This is called the root(2) conjugate.
In a division, if the divisor has its b not zero then I can multiply the top and the bottom (the divisor and the dividend) by the conjugate of the bottom, and the only √2’s are then on top.

(3+2√2)/(4-√2) = (3+2√2)(4+√2)/((4-√2)(4+√2)) = (16+11√2)/(16-2) …

As with the square root of -1 we can see that this is all about pairs of rational numbers, and the √2 symbol just keeps the members of each pair in order.

So rewriting the multiplication we get (a,b)(p,q) = (ap + 2bq, aq + bp)
and all the rules for operations can be expressed in this way and be seen to work.

We have ended up with a totally valid extension of the rational numbers by √2.
It is quite amusing to represent these pairs on an xy grid, and see the effect of multiplication.

But √2 still does not have a value ! ! ! ! !
 

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Filed under extension, irrational numbers, math, square roots

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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Filed under abstract, algebra, arithmetic