Better definition for complex number

[Bruno Tolentino]

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: $$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c=0$$ 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...

 

[micromass]

0 is the multiplicative neutral element and your multiplicative inverse (0^-) is the ∞.

No it isn't.

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?

Because the three roots to $x^3 = -1$ are not independent.

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

Sure, but I doubt you'll get many people to follow this notation.

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!?

It depends a lot how you define dimension. If you refer to the dimension of $\mathbb{C}$ over $\mathbb{R}$, then the dimension is $2$. The cubic roots are not all independent, so the dimension remains $2$. A more interesting thing is to regard dimension over $\mathbb{Q}$, in which case you'll get into Galois theory.

 

[HallsofIvy]

Frankly, the fact that, in your introductory paragraph, you made one mistake after another about elementary mathematics does not lead to confidence about what the "correct" way to write things is!

 

[mfb]

I was me asking why the complex numbers are defined how z = x + i y !?

They are not. There are many definitions but this is not one of them.

0 is the multiplicative neutral element

That would imply 5*0=5.

But, 1 is the additive neutral element

That would imply 0+1=0.

 

[FactChecker]

The numbers 1 and i form an orthonormal basis for the complex plane. That is a very good choice.

 

[Bruno Tolentino]

That would imply 5*0=5.

That would imply 0+1=0.

Really!

It depends a lot how you define dimension. If you refer to the dimension of $\mathbb{C}$ over $\mathbb{R}$, then the dimension is $2$. The cubic roots are not all independent, so the dimension remains $2$. A more interesting thing is to regard dimension over $\mathbb{Q}$, in which case you'll get into Galois theory.

The numbers 1 and i form an orthonormal basis for the complex plane. That is a very good choice.

This is a good point! 1 and i are orthonormal basis and lidependent linear because 1 were definied like the unit vector of the x-axis and i like the unit vector of the y-axis. If I define ³√(-1) like the unit vector of the y-axis so now any complex number in the form z = x + i y is a linear combination of α + μ β (being α = real number, β = real number and μ = ³√(-1)) and now (1 , μ) is the orthonomal basis for the complex plane and independent linear!

Or not!?

 

[micromass]

This is a good point! 1 and i are orthonormal basis and lidependent linear because 1 were definied like the unit vector of the x-axis and i like the unit vector of the y-axis. If I define ³√(-1) like the unit vector of the y-axis so now any complex number in the form z = x + i y is a linear combination of α + μ β (being α = real number, β = real number and μ = ³√(-1)) and now (1 , μ) is the orthonomal basis for the complex plane and independent linear!

Or not!?

Sure, you can invent some system where $(1,\mu)$ is an orthonormal basis. But the system where $(1,i)$ is orthonormal has simpler formulas, so we prefer that.

 

[martinbn]

Here is one more way to define the complex numbers: $\mathbb R[x]/(x^2+1)$

 

[mathwonk]

it is ironic that we leave students in the US with the idea that complex numbers are exotic when in actuality it is the real numbers that are much more difficult and subtle to define.

 

[HallsofIvy]

I've mentioned this so often that I am a little reluctant. We can (and some textbooks do) define the complex numbers as pairs of real numbers, (a, b) with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication by (a, b)*(c, d)= (ac- bd, ad+ bc). The real numbers themselves can be identified with the pairs (a, 0) and "i" with (0, 1) so that a+ bi is (a, b).

 

[lavinia]

Given two non-colinear vectors in the plane, any other vector is some linear combination of them. Thus one can write any complex number as a linear combination of two fixed vectors that do not lie on the same line. From this point of view there is nothing special about choosing 1 on the positive x-axis and 1 on positive y-axis. Any two vectors will do.

For each choice one would get a different law of multiplication and perhaps a good reason to use 1 and i is that the law of multiplication is particularly simple. and also that the coefficients of a linear combination are just perpendicular projections onto the x and y axes.

Complex multiplication can be described as adding the angles to the x-axis and multiplying the radii. If one measures angles from the positive x-axis then angle addition is easily computed from the laws for the sine and cosine of the sum of two angles.

One can certainly use linear combinations of two roots of a polynomial. Try to work out the law of multiplication using two cubic roots of unity.

 

[Bruno Tolentino]

Given two non-colinear vectors in the plane, any other vector is some linear combination of them. Thus one can write any complex number as a linear combination of two fixed vectors that do not lie on the same line. From this point of view there is nothing special about choosing 1 on the positive x-axis and 1 on positive y-axis. Any two vectors will do.

For each choice one would get a different law of multiplication and perhaps a good reason to use 1 and i is that the law of multiplication is particularly simple. and also that the coefficients of a linear combination are just perpendicular projections onto the x and y axes.

Complex multiplication can be described as adding the angles to the x-axis and multiplying the radii. If one measures angles from the positive x-axis then angle addition is easily computed from the laws for the sine and cosine of the sum of two angles.

One can certainly use linear combinations of two roots of a polynomial. Try to work out the law of multiplication using two cubic roots of unity.

This is a good argument!

 

[gill1109]

You can *define* the complex numbers as pairs of real numbers with a certain rule for addition, and a certain rule for multiplication. Now you can *define* i = (0, 1) and now you can *prove* i^2 = -1.

But there are plenty of other possible definitions, because there are plenty of theorems which *characterise* the complex numbers. Of the kind: suppose you have a set and you have an addition and a multiplication operation satisfying ..., then what you have got is: the complex numbers. AFAIK all such theorems and all such definitions take it for granted that we already know what the real numbers are.

 

[Demystifier]

it is ironic that we leave students in the US with the idea that complex numbers are exotic when in actuality it is the real numbers that are much more difficult and subtle to define.

Even if real numbers are harder to define formally and rigorously, they are quite intuitive to most humans (including US students). Continuum is quite intuitive. Complex numbers are not that intuitive.

 

[WWGD]

Ultimately, like many other objects in Mathematics (and likely in other areas) Complex Numbers have many different types of structures:

as fields (And a field extension of the Reals), as metric spaces, etc. So depending on the type of structure, once you fix a choice, you will have a suitable definition.

 

[micromass]

they are quite intuitive to most humans

I think the majority of people living before 1900 would want to disagree with you on that. What constitutes a number has been debated for millenia. They're only intuitive to us now because we're taught real numbers at an early age. In reality, a real number is an amazingly difficult object.

 

[mathwonk]

if the argument is that the real numbers are more intuitive because they are represented geometrically as points on a line, aren't the complex numbers just as intuitive when represented as the points of a plane? if the algebraic part is what seems mystifying, try defining rigorously even the product of two arbitrary reals. by contrast, multiplication by i is just a rotation of the plane. How e.g would you explain multiplication of real numbers to a class, like one I had, that firmly believed multiplication to be repeated addition?

And if you have not had my experience teaching calculus to US students, you may be surprised to find that many of them actually assume all real numbers are integers, when interpreting a general statement about them. I.e. after reading that the derivative is linear, e.g that (af+bg)' = af'+bg' for all real numbers a and b, my students only thought of a and b as being whole numbers, certainly not reals like e or π. Lack of intuition about reals may stem from thinking all numbers appear in the display of a calculator. I.e. other students think all reals are finite decimals., or that π = 22/7, or 3.1416. Even among professional mathematicians there is no clear concept of the real number continuum until maybe Dedekind, as micromass suggests.

 

[WWGD]

I think the majority of people living before 1900 would want to disagree with you on that. What constitutes a number has been debated for millenia. They're only intuitive to us now because we're taught real numbers at an early age. In reality, a real number is an amazingly difficult object.

Notice the discussion on 0.9999...=1 and the inability to understand the definition of a number as a rep. of a class, and not as an object to start with. Then other things add up to the difficulty. The dense ordering implying the lack of a successor does not help either, as I can't think of a case outside of the Reals where this happens.

 

[jbriggs444]

Notice the discussion on 0.9999...=1 and the inability to understand the definition of a number as a rep. of a class, and not as an object to start with. Then other things add up to the difficulty. The dense ordering implying the lack of a successor does not help either, as I can't think of a case outside of the Reals where this happens.

The standard ordering of the rationals is a dense ordering as well.

 

[WWGD]

The standard ordering of the rationals is a dense ordering as well.

Your right, and this follows from Rationals being Reals, what I meant to say (but did not) is that this is not seen outside of Mathematics, AFAIK.

 

[Shehbaj singh]

I think I got a better definition. Complex numbers is the universal set of all the values in mathematics. Because if coefficient of I is zero, we get real numbers and rational numbers is a subset of real numbers and natural numbers is subset of it and so on.

 

[micromass]

I think I got a better definition. Complex numbers is the universal set of all the values in mathematics. Because if coefficient of I is zero, we get real numbers and rational numbers is a subset of real numbers and natural numbers is subset of it and so on.

There are way more possibly numbers other than the complex numbers!

 

[WWGD]

I think I got a better definition. Complex numbers is the universal set of all the values in mathematics. Because if coefficient of I is zero, we get real numbers and rational numbers is a subset of real numbers and natural numbers is subset of it and so on.

Besides, a definition of a set of numbers should define the properties of the numbers other than their basic structure as sets alone.

 

[pwsnafu]

I think I got a better definition. Complex numbers is the universal set of all the values in mathematics. Because if coefficient of I is zero, we get real numbers and rational numbers is a subset of real numbers and natural numbers is subset of it and so on.

Quaternions, Octonions, split-complex numbers, bicomplex numbers, split-quaternions, split-octonians, dual numbers, surreal numbers, hyperreals, extended reals, real projective line.
I'll stop.

 

[lavinia]

I think I got a better definition. Complex numbers is the universal set of all the values in mathematics. Because if coefficient of I is zero, we get real numbers and rational numbers is a subset of real numbers and natural numbers is subset of it and so on.

In Mathematics, there are many algebraic objects that are used as "values" other than algebras or fields containing the real numbers. For instance, in Algebraic Topology, the field $Z_2$ is important in the study of manifolds. More generally in homology theory, values are taken in modules over commutative rings e.g. $Z_4$ or $Z$ or any abelian group e.g. $R/Z$ or the lattice, $Z^n$.

The complex numbers though are still maximal in the sense that they are the algebraic closure of the real numbers.

 

[Demystifier]

In reality, a real number is an amazingly difficult object.

Would you believe me if I tell you that real numbers, visualized as a continuum line, are more intuitive to me than rational numbers?

 

[pwsnafu]

Would you believe me if I tell you that real numbers, visualized as a continuum line, are more intuitive to me than rational numbers?

I also used to be like this, then I discovered that the long line is also a linear continuum, then I was really confused.
The intuition of continuum gives you the local structure, but the reals are more than just that.

 

[WWGD]

I also used to be like this, then I discovered that the long line is also a linear continuum, then I was really confused.
The intuition of continuum gives you the local structure, but the reals are more than just that.

I think looking into some results of descriptive set theory (actually, plain-old Real analysis) will make most think twice about claiming the Reals are intuitively understandable

 

[micromass]

Would you believe me if I tell you that real numbers, visualized as a continuum line, are more intuitive to me than rational numbers?

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.

 

[Demystifier]

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.

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.

One comment from the point of view of pure mathematics. I don't agree that "we can never know whether there is a set with cardinality between the rationals and the reals". We can never know this from the ZFC axioms, but it seems quite reasonable to me that in the future mathematicians might find a more useful system of axioms on which all mathematics can be based.

 

[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.