Notation: Why do we learn to write the higher powers in an equation first?

Question

I have always written equations in the form $y=ax^2+bx+c$ but after entering an equation into Wolfram Alpha I noticed that the answer was displayed in the form $y=c+bx+ax^2$. I know that there is no mathematical difference between these two forms but was wondering why I learnt to write the higher powers first. I can't think of any immediate advantages of writing it either way.

Answer 1

There's no difference and no general advantage of one way over the other. In some cases one direction may be preferable to the other, but just on psychological ground.

Writing polynomials starting from the higher or the lower degree term can give more prominence to one or the other term. For instance, when explaining division, it's better to start from the highest degree terms, because those are used for the first step. When writing down the division, who uses the Latin script sorts them from the highest degree term too, mainly because of the writing direction. The algorithm proceeds by lowering the degree, but it would be exactly the same if the polynomial were written in the opposite direction.

For series like $$ 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dotsb $$ it's psychologically better writing the terms starting from the “lowest degree”, because the first terms show the pattern. In case of power series it would be no different writing $$ \dots+(-1)^n\frac{x^{2n+1}}{2n+1}+\dots+\frac{x^5}{5}-\frac{x^3}{3}+x $$ or $$ x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\dots+(-1)^n\frac{x^{2n+1}}{2n+1}+\dotsb $$ (there's a reason why I like this series ;-)), but it's quite clear that the latter form is better suited when using left to right writing.

When we write numbers using the Latin script, we start from the most significant digit and this might have influenced the order for polynomials. On the contrary, in Arabic script numbers are written starting from the least significant digit. Thus there is no necessity to prefer “small endian” to "big endian" or conversely.

Of course, the same considerations made for polynomials don't apply to numbers: in numbers we don't have the variable that makes clear what ordering we're using, so always sticking to one and the same ordering is a necessity.

When doing sums or multiplications between numbers, we follow a pattern going from the least significant digit to the most significant one, which is opposite to the direction of writing in Latin script, but in the same direction for the Arabic script. This is just a historic accident: when Fibonacci imported the numbering system he learnt from the Arabs, he didn't take into account the change in the direction of writing, so he continued to write the numbers in the same visual order as the Arabs'. In turn the Arabs didn't change the ordering when they imported the numbering system from the Indians (who write from left to right).

Why didn't they? One factor can be the way numbers are named: indoeuropean languages name the numbers mainly starting from the most significant digit (some exceptions are found, for instance drei und zwanzig in German), Arab instead starts from the least significant digit.

Answer 2

Presumably for the same reason we write one-hundred as $100$ as opposed to $001$. The emphasis ought to be on the most significant digit, and in a polynomial the highest term is the "most powerful" and dominates all the other terms as we go to infinity.

Edit. Actually, it depends what we're trying to do. If we're trying to approximate a function near $x_0$ with a polynomial in the terms $(x-x_0)^n,$ then lower terms are more important. So perhaps in this case we should write

$$a+b(x-x_0)+c(x-x_0)^2$$

for better emphasis.