Homogeneity of degree zero and normalization
Demand $x(p, m)$ is the solution to the utility maximization problem:

$\max\limits_{x\in\mathbb{R}^n_+} \ \ u(x) \\ \text{s.t.} \ \  p\cdot x \leq m$

where $p\in\mathbb{R}^n_{++}$ is price vector, and $m$ is the income.

When we multiply both sides of the constraint in problem above by $\lambda &gt; 0$, and look at the revised problem we get

$\max\limits_{x\in\mathbb{R}^n_+} \ \ u(x) \\ \text{s.t.} \ \  \lambda p\cdot x \leq \lambda m$

Since this operation does not affect the constraint, the solution remains unaffected i.e. demand satisfy $x(\lambda p, \lambda m) = x(p,  m) $ which shows that demand is homogeneous of degree 0 in $(p, m)$. So, this is always true for demand function. Given that $p_1 &gt; 0$, we can take $\lambda  = \frac{1}{p_1}$, and find $x\left(\frac{p}{p_1}, \frac{m}{p_1}\right)$ to get $x(p, m)$. 

It is helpful to note that for any function $f(p)$ that is homogeneous of degree $k &gt; 0$, it is the case that $f(\lambda p) = \lambda^k f(p) \neq f(p)$ for $\lambda \neq 1$.