Better definition for complex number

[FactChecker]

Sure. But then you have the following results:
- The real numbers contain way more elements than the rational numbers. We don't even understand how much more. We don't even know whether there is a set with cardinality between the rationals and the reals. And we can never know this.
- There are only countably many numbers that we have a complete description for. Most real numbers end up to be not even definable. We can't even define most numbers! We just know they're there. https://en.wikipedia.org/wiki/Definable_real_number Let alone compute the number...
- There are subsets of the real numbers which we can't even measure. Sure, we can take the length of intervals and other easy sets. But a LOT of subsets of the reals don't have a length.

Sure, the real numbers look easy. But a closer look should convince you that you're dealing with a horribly complicated object that mathematicians will probably never figure out completely.

The only difference between real and rational numbers is that one is harder to represent in the units of the other. Take any real number representing a physical quantity, say pi and change the scale so that pi is 1 unit. Now all the rational multiples of pi are easy to represent and all the measurements that used to be rational are irrational in the new units. So the entire issue is just a problem with our choice of units, not a problem with any particular physical amount, quantity, or position on a real line.

 

[lavinia]

That's all true, but it is almost completely irrelevant for practical applications of mathematics in other human activities, such as science and engineering. That's why it is still meaningfull to teach students (except those of pure mathematics) that real numbers are much more intuitive than complex numbers.

In practical applications such as measurement, one uses rational numbers not the real numbers.If you are saying that the rational numbers that appear in measurement are intuitive - I guess because they are practical - that is not the same as saying that the real numbers are intuitive.

 

[micromass]

That's all true, but it is almost completely irrelevant for practical applications of mathematics in other human activities, such as science and engineering.

The full collection of real numbers itself is irrelevant for practical applications of mathematics.

 

[FactChecker]

In practical applications such as measurement, one uses rational numbers not the real numbers.If you are saying that the rational numbers that appear in measurement are intuitive - I guess because they are practical - that is not the same as saying that the real numbers are intuitive.

There are a few irrational constants that are so convenient that they are the standard unit of measure in a particular context. For instance, π is a very convenient unit of angular measurement. 3/4 π is an irrational number, but a rational number of π units.

 

[lavinia]

There are a few irrational constants that are so convenient that they are the standard unit of measure in a particular context. For instance, π is a very convenient unit of angular measurement. 3/4 π is an irrational number, but a rational number of π units.

Right. I didn't want to bring this up in order not to complicate the point.

Also, in practice, π is never used exactly - but only approximated by some rational number. You could also mention the square root of 2.

The idea that the "real world" somehow contains all intuitions, to me, is highly questionable.

 

[FactChecker]

Right. I didn't want to bring this up to complicate the point.

But in practice, π is never used exactly - but only approximated by some rational number. You could also mention the square root of 2.

The idea that the "real world" somehow contains all intuitions, to me, is highly questionable.

An angle of π is used exactly. It is a straight line.

 

[lavinia]

An angle of π is used exactly. It is a straight line.

?

 

[Demystifier]

In practical applications such as measurement, one uses rational numbers not the real numbers.If you are saying that the rational numbers that appear in measurement are intuitive - I guess because they are practical - that is not the same as saying that the real numbers are intuitive.

There are applications beyond measurement. For instance, physicists and engineers use calculus, which is based on real numbers.

 

[FactChecker]

It the context of angles, π is used exactly. In the context of calculus, e is used exactly as the only value of 'a' where the derivative of a^x is a^x. In the context of plane geometry, √2 is used exactly as the exact diagonal of the unit square. I agree that in the context of addition, multiplication, and division of natural numbers, the rationals are the only numbers used exactly. But that is a very limited context.

 

[micromass]

There are applications beyond measurement. For instance, physicists and engineers use calculus, which is based on real numbers.

Sure, the easiest and most elegant theory of calculus definitely uses the reals.
But, it is definitely not impossible to do calculus without having the real numbers available. One route that I really like is synthetic differential geometry. It develops calculus and differential geometry without mentioning the reals at all.

 

[Sangam Swadik]

I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance?

In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0^-) is the ∞. But, 1 is the additive neutral element and it haven't a symbol for its additive inverse (-1), the inverse additive is wrote simply how -1.

Other example: the conic equation is, actually, the vetorial form of the quadratic equation a x² + b x + c = 0. Therefore, the 'correct' form of write it is: a_{ij} : \vec{r}^2 + b_{i} \cdot \vec{r} + c = 0 In matrix form: <br /> \begin{bmatrix}<br /> a_{11} &amp; a_{12}\\<br /> a_{21} &amp; a_{22}<br /> \end{bmatrix}:\begin{bmatrix}<br /> xx &amp; xy \\<br /> yx &amp; yy<br /> \end{bmatrix}<br /> +<br /> \begin{bmatrix}<br /> b_1\\<br /> b_2<br /> \end{bmatrix}\cdot<br /> \begin{bmatrix}<br /> x\\<br /> y<br /> \end{bmatrix}<br /> +c=0 And not this way:

https://en.wikipedia.org/wiki/Conic_section#Matrix_notation

Considering these 'mistakes' and others that I not wrote here, I me asked why the complex numbers are defined how z = x + i y.

Why i is important? Why? Why emphasize the number i ? Why? i is not important!

You known that x² - y² = (x + y) (x - y), all right!? And that x² + y² = (x + i y) (x - i y), correct!? But, you know that x³ + y³ = (x + α y) (x + β y) (x + γ y) ? No? You know that α, β and γ are the roots of ³√(-1) ?

α, β and γ appears in the cubic formula too. Write α, β and γ how the linear combination x + i y always complicates every equation! So, is necessary to define one symbol for α, β and γ too! Or, why not? Why the preconception? Why ²√(-1) has a proper symbol and ³√(-1) haven't? Why ²√(-1) is special?

I thought wrote the complex number like z = x^y, because this notation no emphasize none root to the detriment of other.

But, linear combination of roots appears be a good way of write complex numbers too. In this case, 1 and i are the best base for the lienar combination?

And why the complex numbers are bidimensional numbers? If the root of the linear equation is and unidimensional number and if the roots of the quadratic equation are bidimensional numbers, so, the roots of the cubic equation are tridimensional numbers, correct or not!?

So, what you think about? I'm a little confused and unbeliever...

the concept of complex numbers was introduced , since , negative number , have no square or any kind of roots or cubes etc.. , there may be a point in space , which , is a root of a negative number (say) there fore it was introduced.

 

[HallsofIvy]

That is true but is not a "definition" which is what this thread is about.