Define a variable in header file works only when extern keyword is not present?

I am learning at moment C and I really do not understand how header files works and to be sure I have two Questions.

1) Let's take a look at the following program: main.c:

#include <stdio.h>
#include <string.h>
#include "functions.h"


int main( void )
{
    printf( "Num = %d\n", number );
    printNumber();
    return 0;
}

functions.c:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "functions.h"

void printNumber( void )
{
    printf("Number = %d\n", number );
}

functions.h:

#ifndef FUNCTIONS
#define FUNCTIONS

int number;
extern void printNumber( void );

#endif // FUNCTIONS

The way the program is in the Header file there is no extern keyword involved so there seems to be reference to number and the program Outputs:

Num = 0
Number = 0

The first Question is, does number get initialized (is number global variable or similar if number is present only in the header file) and is this a legal code/program?

Second scenario, let's take a look at the following code main.c:

#include <stdio.h>
#include <string.h>
#include "functions.h"

int main( void )
{
    printf( "Num = %d\n", number );
    printNumber();
    return 0;
}

functions.c:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "functions.h"

void printNumber( void )
{
    printf("Number = %d\n", number );
}

functions.h:

#ifndef FUNCTIONS
#define FUNCTIONS

extern int number;
extern void printNumber( void );

#endif // FUNCTIONS

Here the program will not compile due the

`undefined reference to number`

Which force me to declare number in main:

#include <stdio.h>
#include <string.h>
#include "functions.h"

int number;

int main( void )
{
    printf( "Num = %d\n", number );
    printNumber();
    return 0;
}

Which is the Right way and why? Last thing, why does exactly not apply to void printNumber( void ) as well. I see that it is working with or without the extern keyword.

Solution

For the first version, in your header file, you define the variable number. That means every translation unit that includes the header file will have a definition of the variable. That's not allowed, you can only have a single definition spread over all translation units.

The solution to that problem is to declare the variable in the header file instead:

extern int number;

The use of the keyword extern marks this as a declaration instead of a definition, the compiler will know that the variable is defined somewhere else.

Then you of course need to define it somewhere. In one single source file put the definition:

int number;

I.e. exactly what you do in your last variant.