What set of numbers are scalers precisely? - answer

What set of numbers are scalers precisely? 

Most people often assume scalers to be real numbers only forgetting that a vector space can be defined over any field of numbers including complex numbers, rational numbers and generally any field.

Remember that a vector space (linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars and forming a commutative group under addition. Study the properties below:

For any vector space $u,v\in{V}$ and $a,b\in{F}$

1.  $a(u+v) = au+av$ scalers are distributive of vector addition 
2.  $(a+b)v=av+bv$ field addition distributes over scalers. 
3.  $a(bv)=(ab)v$ scalers are consistent with field multiplication. 

Therefore, scalers can be defined over any set of numbers.