Descending Order
ArithmeticDescending order means arranging numbers from largest to smallest.
Formula
a_1 \ge a_2 \ge a_3 \ge \ldots
Definition
Descending order means arranging numbers from largest to smallest, going down.
Example
$50, 35, 18, 9, 2$ are in descending order. Each number is smaller than the one before it.
Key Insight
"Descending" means "going down," like walking downstairs. Each number gets smaller as you move along.
Definition
A sequence is in descending (non-increasing) order if each term is less than or equal to the previous: $a_1 \ge a_2 \ge \ldots \ge a_n$. Strictly descending means each term is strictly less: $a_1 > a_2 > \ldots > a_n$.
Example
Arrange in descending order: $0.5, -2, 3, 1/4, -0.1$. Convert to decimals: $3, 0.5, 0.25, -0.1, -2$.
Key Insight
Descending order is often used for rankings (1st place, 2nd place, ...) and for counting down. Any ascending list reversed becomes descending.
Definition
Descending order corresponds to the reverse total order. A bounded descending sequence of real numbers converges by the monotone convergence theorem (it is a bounded ascending sequence under the reverse order). Standard polynomial convention writes terms in descending order of degree: $3x^4 - 2x^2 + x - 5$.
Example
The sequence $1/n$ ($n=1,2,3,\ldots$) is strictly descending and bounded below by $0$; it converges to $0$. The sequence $2, 3/2, 4/3, 5/4, \ldots = 1 + 1/n$ is descending and converges to $1$.
Key Insight
Polynomials written in descending degree order make the leading term and degree immediately visible, which is important for division, asymptotic analysis, and evaluation by Horner's method.