Definition of monomial

[poloplayersh]

I thought I completely understood a monomial to be a single algebraic term, but then I saw somewhere that 2/x is not a monomial. Why?

 

[Deleted member 4993]

I thought I completely understood a monomial to be a single algebraic term, but then I saw somewhere that 2/x is not a monomial. Why?

2/x is NOT a function of 'x' with Non-negative integer exponent.

 

[Ishuda]

I thought I completely understood a monomial to be a single algebraic term, but then I saw somewhere that 2/x is not a monomial. Why?

As implied by Subhotosh Khan, in general a (muti-dimensional) monomial is a single term consisting of a coefficient c and an expression
x₁^n₁ x₂^n₂ x₃^n₃ ... x_m^n_m
where the x_j are different variables and the n_j are non-negative integers.

However, a term such as 2/x = 2 x^-1 would be called a Laurent monomial [note the special designation], i.e. the n_j are allowed to be negative integers. In fact a term with rational n_j also has the special name Puiseux monomial [again, note the special designation].

 

[Steven G]

I thought I completely understood a monomial to be a single algebraic term, but then I saw somewhere that 2/x is not a monomial. Why?

A monomial in x will be in the form of cx^n where c is any real number (or complex number) n>=0. A monomial in x and y is of the form c(x^n)*(y^m) where n,m are zero or above (non negative) and c is a number. You can certainly have a monomial with even more variables and the same constraints.

Since in the monomial 5x^n, n can be 0 we have that 5 is a monomial. The reason is that 5 = 5*1 = 5*n^0 =5n^0. In fact any real (complex) number is a monomial.

EDIT: n and m must be integers

 

[Deleted member 4993]

A monomial in x will be in the form of cx^n where c is any real number (or complex number) n>=0. A monomial in x and y is of the form c(x^n)*(y^m) where n,m are zero or above (non negative) and c is a number. You can certainly have a monomial with even more variables and the same constraints.

Since in the monomial 5x^n, n can be 0 we have that 5 is a monomial. The reason is that 5 = 5*1 = 5*n^0 =5n^0. In fact any real (complex) number is a monomial.

The exponents (n and m) has to be integer (non-negative).

 

[Steven G]

The exponents (n and m) has to be integer (non-negative).

Thank you!