Hamilton's Errors?

[Capiert]

I can't make any sense of "some" of the things (the most famous & respected (irish) mathematician) Hamilton wrote (about Quaternions).

E.g.

 i*j*k can NOT (possibly) be -1,

 because each (i, or j, or k), separately squared, is -1,

 resulting in a "negative" singular (of either i, or j or k),

 i.e. (that'( i)s) the(e) "minus" root, of minus 1;

 NOT (simply) minus 1!

 e.g. -j (instead of -1=(j^2)).

 

His use of accelerating "force" is (also) most profound (=doubtful)!;

 if (that's) NOT a typo (need(ing, the word) "of" (in the sentence),

 as:

 accelerating (of) force).

 

Can anybody clarify (the non_sense, that'( i)s going on)?

..so I can make some (better) sense of it.

2018 02 15 2342_Errata for Hamilton 2018 02 16 0342 PS Wi.pdf

Edited February 16, 2018 by Capiert

[studiot]

Have you heard of Argand Diagrams?

[Capiert]

15 minutes ago, studiot said:

Have you heard of Argand Diagrams?

(Sorry) No.

I guess you mean

 the x,y axii (plot, for a radius & angle)

 that Euler named real & imaginary (axii).

Edited February 16, 2018 by Capiert

[studiot]

Look at this bit of maths (arithmetic) first.

to make a number, say 4, negative multiply by -1.

So 4 * (-1) = -4

To return it to its original multiply again by -1.

So (-4)*(-1) = 4.

 

Now for the clever bit.

Now let's look at this again from thenpoint of view of geometry.

Arrange the numbers along a line (axis) positive to the right, negative to the left as shown in the first sketch..

The first sketch shows how to get from A at +4 to B at-4 geometrically as in the the calculation.

It also shows the second multiplication to get back to the roiginal number or position at A.

This sketch does not have a second axis. It only has one axis.

 

The second sketch shows that if we make four multiplications by a quantity we call i we also get back to where we started.

But in this case the first and third multiplications take us off the original axis.

So we say that multiplication by i = √(-1) has the effect of rotation by 90^o

A second multiplication has the effect of rotation by 180^o

A third multiplication by 270^o

And the fourth multiplication returns us to where we started with a rotation of 360^o.

 

This construction is known as an Argand diagram.

 

That generates the second dimension and axis which is at right angles to the first.

But of course we can have another (third) dimension also at right angles to the first, but different from the second axis.

 

This way of thinking is where the  i,j,k notation came from.

And that is all it is. A notation to represent the fact that we have 2 or 3 or even more dimensions, all at right angles to each other.

We use different letters because we need to distinguish between each of the axes we use, but they are all have the same numerical value as there is only one √(-1).

There is no information in the number √(-1) about which of the infinite number of possible directions we choose. That is up to us to specify in some other way.

Normally, i j and k are enough as we only have three space dimensions.

 

Does this help?

 

Edited February 16, 2018 by studiot

[Capiert]

6 hours ago, studiot said:

Does this help?

Yes Studiot, it helps.

Thank you for taking the time.

I like the way you explained

 because your style has confirmed for me

 similar ideas I have,

 in a way understandable  to me.

Thanks again.

Edited February 16, 2018 by Capiert