Multipicative group and Additive groups Maybe it's a silly question but how exactly do you find a multiplicative group and additive group? Is it always a modulo function? 

How do you find the multiplicative group and additive group of real numbers? 

An example or maybe a few, would be very helpful! Thank you!
We tend to call the group operation multiplication regardless of whether it is multiplication or not.

Addition modulo $n$ will form a group.

Multiplication modulo $n$ is a little trickier.  If we take the integers that are co-prime with $n$ we can form a group with the multiplication as the group operation.

But, our group operation may not be addition or multiplication.  It may be something else entirely, it may be function composition, for example.

i.e. the set of bijective functions from $A\to A$ forms a group, with group operation as composition.

The group may represent geometry.  The set of rigid motions in the plane forms a group.
Any set with a binary operation that satisfies the group axioms is a group.