# Identity true in the reals, not in complex?

• Char. Limit
In summary, there are several identities that are true for all real numbers but not for all complex numbers. One example is the identity \sqrt{ab}=\sqrt{a}\sqrt{b}, which is only true if both a and b are positive. Another example is the inequality x<y or x=y or x>y, which is not allowed in the complex field. However, equalities are allowed, such as |x| != |y| => x = -y. Additionally, the law of trichotomy, which states that for all x,y, x<y or x=y or x>y, does not hold in the complex field. Finally, Cauchy's integral formula is an example of a formula that is different for
Char. Limit
Gold Member
Are there any identities that are true for all real numbers, but not for all complex numbers? The only one I can think of is...

$$\sqrt{ab}=\sqrt{a}\sqrt{b}$$

Which is only true if a and b are POSITIVE, not real. But is there any identity that works for all real numbers, but fails for complex numbers?

The most obvious is Re(x) = 0.

$$x^2\neq-1$$

Hurkyl said:
The most obvious is Re(x) = 0.
And, of course, by Re I mean Im.

Pretty much all the formulas concerning multivalued functions must be treated with caution.

For all x,y x<y or x=y or x>y

Bill Simpson said:
For all x,y x<y or x=y or x>y

Inequalities aren't allowed in the complex field, but equalities are.

|x| != |y| => x = -y

the law of trichotomy

Last edited:
I don't know that it's the same thing as the OP asks but cauchy's integral formula is something else that's different. the value of a real function doesn't have anything to do with the value of its derivative but with complex functions it does.

## What is meant by "identity true in the reals, not in complex"?

This phrase refers to an identity or mathematical equation that holds true when using only real numbers, but does not hold true when using complex numbers. Complex numbers include both real and imaginary components, while real numbers only include positive and negative whole numbers and fractions.

## Why is it important to understand the difference between real and complex identities?

Understanding the difference between real and complex identities is important for accurately solving mathematical equations and understanding their solutions. It also helps to understand the limitations of using only real numbers in mathematical calculations.

## Can you provide an example of an identity that is true in the reals, not in complex?

One example is the identity (a + b)^2 = a^2 + 2ab + b^2. This identity holds true when using real numbers, but when using complex numbers, the middle term 2ab becomes 2abi and the equation is no longer true.

## What are the implications of using complex numbers in mathematical equations?

Using complex numbers allows us to solve more complex equations and problems, but it also adds another level of complexity and can lead to different solutions or outcomes compared to using only real numbers.

## How does this concept relate to real-world applications?

In real-world applications, understanding the difference between real and complex identities is important in fields such as engineering, physics, and economics, where complex numbers are often used in calculations. It also has implications in computer science and signal processing.

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