Elucubrations on positve, negative & imaginary numbers

[michel123456]

At the risk of being completely idiot here below something I wonder:

Step1: the positive & negative on the number line.

This above is a representation of the number line, the Real numbers going from zero to positives on the right & negative to the left.

Step 2: When it comes to multiplication, we can show the sign rule in the following diagram.

Where we have the product of 2 positives is positive. The product of a positive with a negative is a negative, the product of 2 negatives is a positive.

Step 3: From the diagram above it comes out that the square of a Real number, positive or negative, is always positive. As shown below, the 2 squares are in a positive area.

Step 4: Then we go to imaginary numbers (Complex numbers), in such a way that negative squares can be handled, as shown below.

Where we see that i^2=-1 (the blue square).

Question 1 is: where is the catch? Why is this diagram wrong? Why -i^2 (the red square) shows negative in the diagram? Although the correct answer is that -i^2=1

Question 2 is the following: what is the sign of 2i? Is it positive or negative? Do we follow the rule of the Real numbers, or the rule of Imaginary numbers?

Question 3 is: how can you combine the Real number diagram with the Imaginary number diagram? Where is the common axis? The only common feature is a point: zero.

 

 

[studiot]

1 hour ago, michel123456 said:

Question 1 is: where is the catch? Why is this diagram wrong?

You have shown i and -i appearing on both the real and imaginary axes.

There are no such numbers on either lines.

[michel123456]

1 minute ago, studiot said:

You have shown i and -i appearing on both the real and imaginary axes.

There are no such numbers on either lines.

I have combined 2 imaginary axis. The same way as I have combined 2 Real number axis previously.

It is a different approach than combining Real & Imaginary as perpendicular.

[Prometheus]

\( (-i)^2 = (-1)^2*i^2 = -1 \)

Edited February 18, 2019 by Prometheus
can't work latex

[studiot]

2 hours ago, michel123456 said:

I have combined 2 imaginary axis. The same way as I have combined 2 Real number axis previously.

It is a different approach than combining Real & Imaginary as perpendicular.

My apologies, I didn't read your elucubration properly, but I understand now.
My midnight oil must be dimmer than yours.

3 hours ago, michel123456 said:

Step 4: Then we go to imaginary numbers (Complex numbers), in such a way that negative squares can be handled, as shown below.

No not complex numbers, just imaginary numbers, they are different.

There is no positive and negative with complex numbers.

What you have is two real axes, scaled by an imaginary constant, i.

The positive and negative attach to the real numbers along each axis so, as you correctly observe

+5  *  +10 = +50

but

+5(i) * +10(i) = +50i² = -50

This situation occurs in nature and mathematics.

For instance it could be the vibration mode of a two dimensional membrane (drumskin) with the positive/negative denoting direction of travel so one part of the drumskin is travellong forwards and the other backwards.

Also we have integration above and below the line so the integral of a sine curve from zero to 360^o  is half positive and half negative (above and below the line) an sums to zero.

In order to calculate the area under that graph you have to split into to and reverse the sign of the negative part, before adding.

[michel123456]

9 minutes ago, Prometheus said:

\( (-i)^2 = (-1)^2*i^2 = -1 \)

Thank you.

That is not what I get from a search, even not in Wolfram Alpha.

[studiot]

2 hours ago, michel123456 said:

3 hours ago, Prometheus said:

\( (-i)^2 = (-1)^2*i^2 = -1 \)

Thank you.

That is not what I get from a search, even not in Wolfram Alpha.

 

All numbers, including -1, have two square roots.

-i does not exist but -1 * i does

So

(-1 * i) *(-1 * i) = (-1 * -1) *(i * i) = (+1)*(-1) = -1

So the square roots of -1 are +1i and -1i.

[michel123456]

2 hours ago, studiot said:

 

All numbers, including -1, have two square roots.

-i does not exist but -1 * i does

So

(-1 * i) *(-1 * i) = (-1 * -1) *(i * i) = (+1)*(-1) = -1

So the square roots of -1 are +1i and -1i.

Well understood.

I got confused by https://www.wolframalpha.com/input/?i=-i^2

And https://www.google.com/search?q=-i^2&oq=-i^2&aqs=chrome..69i57j69i64.5694j0j7&sourceid=chrome&ie=UTF-8

3 hours ago, studiot said:

-i does not exist

Why? You can invent it. What is wrong with this concept?

[studiot]

41 minutes ago, michel123456 said:

3 hours ago, studiot said:

-i does not exist

Why? You can invent it. What is wrong with this concept?

Can you?

 

What does minus something mean or plus something?

It means that you can order the magnitude of something

-(something) < 0 < +(something).

6 hours ago, studiot said:

No not complex numbers, just imaginary numbers, they are different.

now consider the complex numbers 4+3i  and 4-3i

Can you do this with these complex numbers?

They both have the same magnitude = square root (4² + 3²) = 5

But what about 4+3i and 4-12i ?

Which comes first now?

You cannot divide complex numbers into positive and negative, as you can for the real numbers.

Purely imaginary numbers (those with no real part ) are really complex numbers of the form 0+ai  or 0-ai , where a is some real number.

And as shown above they have the same magnitude, although you can place -a < 0 < +a along the real line.

[taeto]

It looks like you confused \((-i)^2\) with \(-i^2\). The former is equal to \(-1,\) the latter to \(+1.\)

25 minutes ago, studiot said:

What does minus something mean or plus something?

The complex numbers \(\mathbb{C}\) form a field with addition and multiplication as defined for complex numbers. In particular \( (\mathbb{C},+) \) is a group. In an additive group, the inverse of an element \(x\) is written \(-x.\) Minus something denotes the additive inverse to that thing.  

[studiot]

2 hours ago, taeto said:

It looks like you confused (−i)2 with −i2 . The former is equal to −1, the latter to +1.

True, but I am not sure what you are referring to.

Was it this?

6 hours ago, studiot said:

(-1 * i) *(-1 * i) = (-1 * -1) *(i * i) = (+1)*(-1) = -1 

I was simply using the distributive property to rearrange the brackets.

Some interesting powers

i²  = -1 (definition)

i³ = -i since i³ = i² * i = -1*i = -i

i⁴ =( i²)² = (-1)² = +1

and so on for higher powers.

 

2 hours ago, taeto said:

3 hours ago, studiot said:

What does minus something mean or plus something?

The complex numbers C form a field with addition and multiplication as defined for complex numbers. In particular (C,+) is a group. In an additive group, the inverse of an element x is written −x. Minus something denotes the additive inverse to that thing.  

Yes they form the field of complex numbers.
This is not a fully ordered field, as are the reals.

Further i by itself is not a complex number and therefore needs no additive inverse.

 

The problem with saying that i is positive or -i negative is it leads to a contradiction.

This is a direct consequence of the fact that a complex number, in whatever format, is a two part entity.
So addition and its inverse (and multiplication) needs to consider both parts.

If i > 0 then i² > 0 so -1 > 0 which is a contradiction.

If i = 0 then i² = 0 so -1 = 0 which is a contradiction.

If i < 0 then i² > 0 so -1 > 0 which is a contradiction since the ordered field axioms imply that the square of any nonzero number is positive.

Edited February 18, 2019 by studiot

[michel123456]

12 hours ago, taeto said:

It looks like you confused (−i)2 with −i2 . The former is equal to −1, the latter to +1.

 

Yes, I see the difference now. Thank you for the clarification.

13 hours ago, studiot said:

Can you?

 

What does minus something mean or plus something?

It means that you can order the magnitude of something

-(something) < 0 < +(something).

now consider the complex numbers 4+3i  and 4-3i

Can you do this with these complex numbers?

They both have the same magnitude = square root (4² + 3²) = 5

But what about 4+3i and 4-12i ?

Which comes first now?

You cannot divide complex numbers into positive and negative, as you can for the real numbers.

Purely imaginary numbers (those with no real part ) are really complex numbers of the form 0+ai  or 0-ai , where a is some real number.

And as shown above they have the same magnitude, although you can place -a < 0 < +a along the real line.

That is part of my question.

The multiplication rule for i looks to me completely incompatible with the multiplication rule of the Real numbers. As shown in my graph. 4. There is no common ground.

 

[Eise]

Hi Michel,

I do not see your problem. Everytime when one extends the set of numbers you get surprises:

N: 1,2,3,4,...

When multiplying 2 natural numbers the result is always >= both numbers.

Z: ... -2,-1,0,1,2...

Ups, the above rule is not valid anymore:

-2 * 4 = -8: the result is smaller than both numbers!

Same if you use Q⁺:

0.1 * 0.2 = 0.02: again, the result is smaller than both numbers.

Now with imaginary numbers you get the next surprise: where in the above sets at least the square of a number is always positive, this is not so with imaginary numbers. (i)² = -1, as is (-i)². So rules that seem general (plus times plus makes plus, minus times minus makes also plus) for a subset of all numbers (Z, Q, R), is not valid for C anymore. And as Studiot also explained, the principle of ordering (greater, smaller) in C does not work. The question if i is greater than 1 does not make sense. Therefore C is depicted in a two dimensional plane.

[taeto]

12 hours ago, studiot said:

Further i by itself is not a complex number and therefore needs no additive inverse.

     I would think that \(i\) is just a shorter, and commonly used, way of writing the complex number \(0 + 1\cdot i\)?

[studiot]

Thank you Eise for for those useful insights and clear way of putting things. +1

Michel, (and taeto) The whole point is that

Imaginary numbers do not form a Field.

The fail the Field axiom that the Field should be closed under multiplication in the biggest way possible.

The axiom guarantees that the product of two members of the Field will be another member of that Field.

That is if pi and qi are imaginary numbers then pi*qi should also be an imaginary number.
But piqi = -pq, which is not imaginary.

A great deal of work had been put in over the last 150 years to construct axioms systems in Mathematics that allow us to get on with the mathematics to achieve what we want to achieve, secure in the knowledge that the foundations support these requiements.

The Field axioms are the ones that allow us to do arithmetic without the hiccups such as the ones Eise points to and you have found.

Note that the natrual numbers do not form a field either.

The simplest number system to form a field is the rational number system.

 

This is, of course, why Mathematicians went to the bother of constructing the complex numbers, which are a Field.

It is important to distinguish between imaginary numbers and complex numbers because the former do not comply with the normal rules of arithmetic.

For instance none of the interior points in your red and blue squares are imaginary numbers.
So you do not have imaginary squares.

 

Edited February 19, 2019 by studiot

[michel123456]

4 hours ago, Eise said:

Hi Michel,

I do not see your problem. (...)

 

I'll try to explain. I have those 2 graphs:

If I try to superpose the one with the other, even in 3D space, they do not fit together.

There is nothing common in those 2 graphs that would offer the possibility to put one perpendicular to the other. The only common ground is the point zero. All the rest is pure contradiction.

IOW I don't understand how one can put the i axis perpendicular to the Real number axis. The meaning of the product gets ambivalent: does 3i follow the rule of the Real numbers (3i is positive) or does it follow the rule of i (3i is negative)?

Edited February 19, 2019 by michel123456

[studiot]

34 minutes ago, michel123456 said:

IOW I don't understand how one can put the i axis perpendicular to the Real number axis. The meaning of the product gets ambivalent: does 3i follow the rule of the Real numbers (3i is positive) or does it follow the rule of i (3i is negative)?

Numbers are not lines and lines are not numbers.

You can put them into one-to-one correspondence for some purposes, but they are not the same.

Your question about positive and negative has no mathematical meaning.

3i is not a real number so why should it follow the rules for real numbers?

Have you read the Wiki on imaginary numbers?

It is quite good and offers a useful extended table of the values of powers of i compared to the one I offered earlier including negative powers.

Note that some powers have real values, some have imaginary values.

 

https://en.wikipedia.org/wiki/Imaginary_number

 

Did you understand my point that a real number x a real number always yields another real number.

But an imaginary number x an imaginary number does not yield an imaginary number.

[michel123456]

20 hours ago, studiot said:

Numbers are not lines and lines are not numbers.

You can put them into one-to-one correspondence for some purposes, but they are not the same.

Your question about positive and negative has no mathematical meaning.

3i is not a real number so why should it follow the rules for real numbers?

Have you read the Wiki on imaginary numbers?

It is quite good and offers a useful extended table of the values of powers of i compared to the one I offered earlier including negative powers.

Note that some powers have real values, some have imaginary values.

 

https://en.wikipedia.org/wiki/Imaginary_number

 

Did you understand my point that a real number x a real number always yields another real number.

But an imaginary number x an imaginary number does not yield an imaginary number.

I understand that the surface of my "i diagram" is spread with real numbers.

That the blue square is a "real square" of imaginary side. And so is the red square.

And that those squares are not representing areas but maybe  a weird unauthorized representation.

All of which make the imaginary numbers totally different from the real numbers. Which again makes me wonder how can you consider correct to create a representation with an axis of imaginary numbers orthogonal to an axis of real numbers just as if they were of equivalent nature.

https://en.wikipedia.org/wiki/Imaginary_number#/media/File:Complex_conjugate_picture.svg

In this diagram, how do we infer that iy has a positive value? How do we treat the imaginary line on the same ground with the real number line? Where are the positive & negative?

I mean, one would automatically infer that the positive numbers are on the upper right sector. Is it so? And why?

[studiot]

17 minutes ago, michel123456 said:

I understand that the surface of my "i diagram" is spread with real numbers.

 

How can this be if both axes are imaginary ?

You are showing a plot or a graph of two variables, both of which are imaginary.

So for instance consider the point with x coordiante 6i and y coordinate 9i

How is this real?

It is not the real 'area' 6ix9i = -54 square somethings!

 

This is no different from saying in a pressure volume plot that the point

x =1bar , y = 1 litre is not 1 x 1 unit of work or energy, although the area under a PV diagram is indeed the amount of work done.

And pressure and volume (one on each axis) are entirely different things.

24 minutes ago, michel123456 said:

 

That the blue square is a "real square" of imaginary side. And so is the red square.

And that those squares are not representing areas but maybe  a weird unauthorized representation.

 

Yes I agree, apart from the 'unauthorized'.

Like above it would be better to say that the area of the square rpresents my new property 'shimmerability' or whatever.

26 minutes ago, michel123456 said:

All of which make the imaginary numbers totally different from the real numbers. Which again makes me wonder how can you consider correct to create a representation with an axis of imaginary numbers orthogonal to an axis of real numbers just as if they were of equivalent nature.

 

So yes you can do it (have different quantities on different axes), but a way to avoid real/imaginary issues is to consider the imaginary y axis as a countercloskwise rotation by 90^o of the real x axis, both being real.

This representation is called an Argand Diagram and is entirely consistent with both the i representation of imaginary numbers since two rotations by 90^o is the same as one rotation by 180^o. Two rotations takes every point to its negative - the same as two multiplications by i.
The maths of this is trigonometric and is much used in continuum mechanics for stress and strain and moments of inertia calculations.

 

One final thought,

You were quite right to identify the origin as being special as it is (the only point) regarded as being both imaginary and real at the same time.

 

[michel123456]

1 hour ago, studiot said:

How can this be if both axes are imaginary ?

You are showing a plot or a graph of two variables, both of which are imaginary.

So for instance consider the point with x coordiante 6i and y coordinate 9i

How is this real?

It is not the real 'area' 6ix9i = -54 square somethings!

I guess it represents the number -54.

As if the area were representing a distance upon the number line.

[studiot]

2 hours ago, michel123456 said:

I guess it represents the number -54.

As if the area were representing a distance upon the number line.

 -54??    Surely not.

That is the point A on my diagram (ignoring the negative for the moment)

But what about points B and C?

B is x = 1, y = 54

C is x = 54, y = 1.

Some number products  (eg 60) offer many more possibilities.

I hesitate to offer what we normally do when we have what is known as functions of a complex variable.

In real analysis a circle (for instance) or a straight line or a parabola is obtained by have one independent variable (x) and plotting the values for y obtained from some formula or function.

So we have two variables, each one  or single dimensional, which together generate a plane.

We can draw a plane (or part of it) on a piece of paper.

Now a complex number has two dimensions so we need our piece of paper just to draw the plane described by the two independent variables contained in every complex number.

To perform the equivalent plot to real analysis we plot from two variables to another complex number and therefore we require four dimensions in total.

We cannot do this in our real three dimensional world.

However there is a trick we use called conformal mapping whereby we can use two sheets of paper, one for the independant (complex) variable, usually called Z and the other for the dependant (complex) variable, usually called W. Sometimes U and V are also used.

It is possible that you are struggling your way towards this subject, which is used in aeronautics and other fluid flows, stress analysis and similar fields.

 

[michel123456]

15 hours ago, studiot said:

 -54??    Surely not.

That is the point A on my diagram (ignoring the negative for the moment)

But what about points B and C?

B is x = 1, y = 54

C is x = 54, y = 1.

Some number products  (eg 60) offer many more possibilities.

I hesitate to offer what we normally do when we have what is known as functions of a complex variable.

In real analysis a circle (for instance) or a straight line or a parabola is obtained by have one independent variable (x) and plotting the values for y obtained from some formula or function.

So we have two variables, each one  or single dimensional, which together generate a plane.

We can draw a plane (or part of it) on a piece of paper.

Now a complex number has two dimensions so we need our piece of paper just to draw the plane described by the two independent variables contained in every complex number.

To perform the equivalent plot to real analysis we plot from two variables to another complex number and therefore we require four dimensions in total.

We cannot do this in our real three dimensional world.

However there is a trick we use called conformal mapping whereby we can use two sheets of paper, one for the independant (complex) variable, usually called Z and the other for the dependant (complex) variable, usually called W. Sometimes U and V are also used.

It is possible that you are struggling your way towards this subject, which is used in aeronautics and other fluid flows, stress analysis and similar fields.

 

Could you please label your axis? I don't follow.

[studiot]

1 hour ago, michel123456 said:

16 hours ago, studiot said:

 

Could you please label your axis? I don't follow.

Which axis or axes?

The bottom conformal map or the top hand drawn one?

The top one is simply your crossed imaginary axes.
I haven't bothered with the i since every number on those axes is multiplied by i.

[michel123456]

2 hours ago, studiot said:

Which axis or axes?

The bottom conformal map or the top hand drawn one?

The top one is simply your crossed imaginary axes.
I haven't bothered with the i since every number on those axes is multiplied by i.

This one, both axes

 

If they are labelled i on both axes we have

Point A has coordinates 6i, 9i

The area of the green rectangle is 6i.9i=54i^2=-54

It has area the number -54, that you can also put on the number line at position -54.

[studiot]

I apologise I was rushing when I posted.

I should have made clear what I was doing.

At least you have picked it up correctly.

A is the point (6i, 9i) so the product is -54

B is the point (1i, 54i) so the product is -54

C is the point (54i, 1i) so the product is -54

So my point still stands

What does the -54 mean since there are many different points, each with this value of product?