Formal definition of multiplication for real and complex numbers

[logicgate]

TL;DR Summary

What is the formal definition of multiplication for real and complex numbers ?

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e times ? So it is very clear that the definition of repeated addition for multiplication doesn't work for all real numbers. So how is multiplication defined for real and complex numbers ?

 

[PeroK]

TL;DR Summary: What is the formal definition of multiplication for real and complex numbers ?

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e times ? So it is very clear that the definition of repeated addition for multiplication doesn't work for all real numbers. So how is multiplication defined for real and complex numbers ?

This is part of the wider question of how the real numbers are developed in the first place. The starting point is the Peano Axioms:

https://en.wikipedia.org/wiki/Peano_axioms

From this, the properties of addition and multiplication of natural numbers can be formally developed from first principles.

Next comes the development of the rational numbers, from the naural numbers. With addition and multiplication defined as:
$$\frac a b + \frac c d = \frac{ad + bc}{bd}, \ \text{and}\ \frac a b \cdot \frac c d = \frac{ac}{bd}$$Finally, the real numbers can be developed essentially as a completion of the rationals. Each real number can be defined as the limit of some sequence of rationals, And addition and multiplication of real numbers can then be defined using addition and multiplication for these rational sequences.

It probably takes a semester or more to go through all this formally.

PS addition and multiplication of complex numbers is defined directly using addition and multiplication of real numbers. This is actually the only easy bit!

 

[Erland]

You can also view it this way:

$3\times 2=6$
$3.1\times 2.7=8.37$
$3.14\times 2.71=8.5094$
$3.141\times 2.718=8.527238$
$3.1415\times 2.7182=8.5392253$
etc.

The limit of this sequence is $\pi\times e = 8.539734222673567\dots$

We can also do the same from above, getting successively smaller intervals which contain the value:

$4\times 3=12$
$3.2\times 2.8=8.96$
$3.15\times 2.72=8.568$
$3.142\times 2.719=8.543098$
$3.1416\times 2.7183=8.53981128$
etc.