# Key difference between two real and single complex variable?

• SVN
In summary, the concept of differentiability for a function of a complex variable is introduced and compared to that of a function of a single real variable. The need for a consistent limit regardless of direction leads to the Cauchy-Riemann equations. While the comparison could be made to functions of two real variables, the use of complex numbers offers mathematical advantages. The split-complex numbers, a special case of complex numbers, do not have a unique inverse and are part of a more general algebra known as Clifford algebras. The coupling in complex numbers comes from the multiplication operation, which allows for a rotation operation in the complex plane. This, in turn, allows for the complex derivative to mimic the properties of the real derivative.
SVN
Notion of differentiability (analyticity) for function of complex variable is normally introduced and illustrated by comparison with function of single real variable.

It is stated that there are infinite number of ways to approach any given point of complex plane where function is defined, not just «left» anf «right» for the case of real variable. That is where necessity for having the same limit no matter which direction is regarded comes into play and the next step is to deduce the Cauchy-Riemann equations from this necessity.

Would not it be more appropriate to compare the case of complex variable with case of two real variables? Here one has x — y plane and different directions as well, but no eqivalent of C.-R. equations, one starts considering partial derivatives instead. Why is that? What is the qualitative difference between pairs (a; b) and (a; ib)? And what about split-complex numbers, where i^2 = +1 from the viewpoint of differentiability of their functions?

Sorry, if this thread is redudant and my question has already been answered in another one. Just point me to the latter in this case, I failed to find it myself.

SVN said:
Would not it be more appropriate to compare the case of complex variable with case of two real variables? Here one has x — y plane and different directions as well, but no eqivalent of C.-R. equations, one starts considering partial derivatives instead. Why is that? What is the qualitative difference between pairs (a; b) and (a; ib)? And what about split-complex numbers, where i^2 = +1 from the viewpoint of differentiability of their functions?

An analytic function of a single complex number is a special case of a function of two real numbers. It's a special case that is much easier to deal with, mathematically.

If you have a function $F(x,y)$ of two real variables, then $\dfrac{\partial F}{\partial x}$ and $\dfrac{\partial F}{\partial y}$ are unrelated. But in the case of a single complex variable: $z = x + i y$, we have the constraint $\dfrac{\partial F}{\partial x} = -i \dfrac{\partial F}{\partial y}$. So we don't need partial derivatives, we can just write $\dfrac{dF}{dz}$. There are lots of mathematical advantages to using complex numbers, and lots of theorems that work for complex functions that don't work (with additional assumptions) for general functions of two variables. For example, complex numbers have an associated product that has an inverse: $z \cdot z^{-1} = 1$.

The split complex numbers and the regular complex numbers can both be thought of as special cases of a more general kind of algebra, called "Clifford algebras". They have some of the nice features of complex numbers, but not all of them (in particular, the split complex numbers don't have a unique inverse, I don't think).

FactChecker
Thanks a lot for reply, it has become much clearer now.

I'd like to be sure I got your idea. Can we consider space of complex numbers to be effectively space ℝ2 equipped with special constraint coupling ℝ and ℝ? That means complex number to be a single entity (in some cases one can find it convenient to deal with it as if it would be two unrelated entities, i. e. normal ℝ2, but the coupling is not to be forgotten at the end). Please, correct me if I am wrong.

But the thing which still puzzles me is where this coupling come from? As far as I know, complex space is essentially a vector space over field of real numbers. But the very idea of vector space is based on fact the basis vectors, no matter which proper basis we use, are linearly independent of each other (they can always be ortogonalised). How does this independence agree with the mentioned coupling (sorry if my questions are naïve, my background in mathematics is not exactly strong).

Multiplication of complex numbers provide a rotation operation in C that regular R2 vector spaces do not have. That is a very important geometric operation to have. There is so much in physics and math that you want to represent by a local scaling and twisting. With that definition of complex multiplication, the complex derivative can mimic the property in R where the derivative is a local multiplication by a single number (dx * slope in R and (scale+rotate) dz in C). The existence of a complex derivative gives analytic functions surprising properties and lead to powerful results.

Last edited:
Thank you for explanation. That answers my question.

## 1. What is the main difference between a real and a complex variable?

The main difference between a real and a complex variable is that a real variable can only take on values from the set of real numbers, while a complex variable can take on values from the set of complex numbers, which includes both real and imaginary numbers. This means that complex variables have two components - a real part and an imaginary part - while real variables only have one component.

## 2. How are the two types of variables used differently in scientific research?

Real variables are typically used to represent physical quantities in the real world, such as distance, time, and temperature. Complex variables, on the other hand, are primarily used in theoretical and mathematical models, particularly in fields such as physics, engineering, and economics, where they provide a more powerful tool for describing and analyzing complex systems.

## 3. Can real and complex variables be used interchangeably?

No, real and complex variables cannot be used interchangeably because they represent different types of numbers and have different properties. For example, complex variables can have multiple solutions to equations, while real variables only have one solution. Additionally, complex variables involve operations such as complex conjugation and the imaginary unit, which are not applicable to real variables.

## 4. How do the domains and ranges of real and complex functions differ?

The domain and range of a real function are both sets of real numbers, while the domain and range of a complex function are sets of complex numbers. Real functions can only take in and output real numbers, while complex functions can take in complex numbers and output complex numbers.

## 5. What are some practical applications of complex variables in science?

Complex variables are used in various fields of science, including physics, engineering, and mathematics. They are particularly useful in solving problems involving electrical circuits, fluid dynamics, and quantum mechanics. They are also used in signal processing, control systems, and image processing. In mathematics, complex variables are used to study functions and analyze complex systems.

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