The order of terms matters even when they commute

When writing an expression that consists of several terms, the conventions regarding their order appear arbitrary.

Multiplication

It is usual to write:

• \(xy\) and \(yx\) in either order;
• \(5t\) but not \(t5\) (to avoid confusion with \(t_5\) or \(t^5\));
• \(x\sqrt{2}\) but not \(\sqrt{2}x\) (to avoid confusion with \(\sqrt{2x}\)). (See Something on the right of a radical)
• \(\sqrt{2}\sin x\) but not \(\sin x \sqrt{2}\) (to avoid confusion with \(\sin \left(x\sqrt{2}\right)\)). (See Function application without parentheses)

Addition

People sometimes rearrange a sum to avoid a leading unary minus, even when this contradicts the convention of writing terms in decreasing order of degree:

\[ 1 - x \]

instead of

\[ -x + 1 \]

Polynomials

When writing a polynomial on its own, the usual convention is to write the terms in decreasing order of degree:

\[ x^3 - 32x^2 + 3x -1 \]

But for a series expansion, where higher-order powers are often omitted, it makes more sense to start with the lowest-degree term:

\[ -1 + 3x - 32x^2 + x^3 \]