Proportionality

Given an independent variable $x$ and a dependent variable $y$:

• direct proportionality

  • $y$ is directly proportional (or proportional (מתכוּנתי)) to $x$ if $\exists k>0:y=kx$.
  • denoted by $y\propto x$ or $y\sim x$
  • $k$ is called the coefficient of proportionality (or proportionality constant)
  • cf. linear function $f(x)=ax+b$

• inverse proportionality

  • $y$ and $x$ are inversely proportional if $\exists k>0:y=\frac{k}{x}$.
  • denoted $y\propto \frac{1}{x}$ or $y\propto x^{-1}$

• power-law proportionality

  • For $n\in \mathbb{R}$, we say that $y$ is proportional to the $n$-th power of $x$ if $y=k{x}^n$
  • denoted by $y\propto x^n$
  • special cases:
    • $n=1$: direct proportionality
      • (e.g., Ohm’s law: $V=IR$, circumference of a circle: $C=\pi d$, Charles’s law: $V\propto T$)
    • $n=-1$: inverse proportionality
      • (e.g., Boyle’s law: $P\propto \frac{1}{V}$)
    • $n=2$: quadratic proportionality
      • (e.g., kinetic energy: $E_k=\frac{1}{2}m{v}^2$, area of a circle: $A=\pi r^2$, surface area of a sphere: $A=4\pi r^2$)

• similar definitions for two functions $f(x)$ and $g(x)$, or two sequences of numbers $x_n$ and $y_n$