Does a mathematical proof in $\mathbb{C}$ imply a proof in the $\mathbb{R}$?

Question

Does every proof in the complex numbers also prove the statement in the real numbers? I thought it might be true, because the real numbers are part of the complex numbers.

Answer 1

If we show something is true for every complex number $z\in \mathbb{C}$, then we have shown that it is true for every $x\in \mathbb{R}$ since $\mathbb{R}\subset\mathbb{C}$.

However, not "every proof in the complex numbers" is of this form. For instance, consider the following example. We can show that there exists a $z\in \mathbb{C}$ such that $z^2=-1$, but there is no real number that has this property.

Answer 2

This depends on what you mean by a proof in the reals. Not every statement about the complex numbers is a statement about the reals. However, you can think of every complex number as a pair of real numbers (a,b), with a corresponding relationship, where one defines complex addition and multiplication as operations on ordered pairs of real numbers. So any statement you make about the complex numbers does correspond to a statement about real numbers with twice as many variables.

Answer 3

Your reasoning that if something is true about complex numbers it must be true about reals because reals are complex is sound. But I don't think you are thinking through just what sort of statements can be proven.

This is not a complete answer but you need to think about "some" and "all".

If X is true for all complex, it is true for all reals.

If Y is true for some complex, it may or may not be true for some, all, or no reals.

If W is true for no complex, it is not true for any reals.

If A is true for all reals then it is true for some complex. It may or may not be true for all conplex.

If B is true for some reals then it is true for some complex.

And if C is not true for any real it might or might not be true for some complex (but definitely not true for all).

Answer 4

Other answers have already said that the validity of this statement depends on what you mean by a "proof". Here's another example of when something is true in $\mathbb C$ but not in $\mathbb R$: the fundamental theorem of algebra. It states that a polynomial of degree $n$ always has exactly $n$ roots in $\mathbb C$, which is however clearly not true in $\mathbb R$ except degenerate cases. In fact, if FTA were to be true in $\mathbb R$, it would be a strictly stronger statement and would imply FTA in $\mathbb C$, not the other way round.

Answer 5

There is a problem in Kreyszig's Functional Analysis:

Let $X$ be an inner product space over $\mathbb{C}$, and $T:X\to X$ is a linear map. If $\langle x,Tx\rangle =0\:\forall x\in X$, then $T$ is the null transformation.

This is not true for inner product spaces over $\mathbb R$.