perimeter v. parameter

A perimeter is a curve or boundary enclosing an area, or the length of such a boundary.

A parameter is a measurable factor that determines the specific nature or behavior of a system. In mathematics, a parameter is a constant in a function that can be varied to generate a family of curves or surfaces with the same general form. In the equation y=ax2+bx+c, a, b, and c are parameters. Similarly, a parameter can be a variable in a statistical distribution, or an argument in a computer program.

AWE on perimeter v. parameter

number v. numeral v. digit

A number is the abstract concept of a quantity.

A numeral is a symbol or a group of symbols used to represent a number. The Roman numeral CXXXVII can be expressed more familiarly as one hundred thirty-seven.

A digit is a single character in a numbering system. Decimal uses the digits 0 through 9; binary uses 0 and 1; the 16 hexadecimal digits are 0 through 9, and A, B, C, D, E, and F.

6732 is a 4-digit numeral representing the number six thousand seven hundred thirty-two.

infinite v. transfinite

An infinite quantity is boundlessly large. The mathematician Georg Cantor realized that all infinite sets don’t have the same size, or cardinality. Some infinite sets are countable, meaning that each member of the set can be mapped to a member of the natural numbers (1, 2, 3, …). For example, every odd number can be placed in an ordered list:

1. 1
2. 3
3. 5
4. 7
5. 9

And so on. Even though intuitively, there should be half as many odd numbers as there are natural (counting) numbers, being infinite, both sets are the same size. Natural numbers, odd numbers, multiples of 18, prime numbers, and even rational numbers (fractions) all have the same cardinality.

The set of real numbers includes both rational (expressible as a fraction) and irrational (infinitely long, non-repeating decimal) numbers. It turns out that the real numbers cannot be made to correspond in a 1-to-1 fashion with the natural numbers, making them uncountable. This means that the real set is infinitely larger than the previously mentioned, already infinite sets.

Cantor developed transfinite numbers to distinguish between the sizes of different types of infinite sets. He was able to demonstrate that, in addition to countably and uncountably infinite sets, there are infinitely more degrees of infinity, each, in some sense, larger than the previous one. Each degree of infinity is assigned a separate transfinite cardinal number.

Infinity is symbolized by a lemniscate:. Transfinite numbers are represented by the Hebrew letter aleph, followed by a subscript: , , , …

Platonic Realms on infinity

linear v. polynomial v. exponential v. logarithmic growth

Mathematical functions can be used to describe the change in a quantity over time. In the equations below, y represents the size of a quantity after x time has elapsed. b is some fixed number.

• Linear growth occurs when there is a constant rate of change. The equation for a linear relationship is y = bx, and its graph is a straight line. If you work for an hourly wage, and don’t spend your earnings, your savings will grow linearly.
• When the rate of change increases with time, numbers can grow more quickly. Quadratic and cubic growth can be represented by y = x2 and y = x3, respectively. The general form of this type of relationship can be written y = xb, and is called polynomial growth. The distance traveled by a falling object can be calculated with a quadratic equation.
• Exponential (or geometric) growth is faster still. Here, the rate of growth is proportional to the value of y at any time. Exponential relationships can be expressed as y = bx. Bacterial populations with unlimited food, nuclear chain reactions, and computer processing power all grow exponentially.
• Logarithmic growth is the inverse of exponential growth. Logarithmic phenomena grow very slowly, and have an equation of the form y = logbx. Sound volume and frequency are both perceived logarithmically, allowing humans to detect a huge range of sound levels.

Linear (red), cubic (blue), and exponential growth (green)

Exponential (green) and logarithmic growth (purple)

parabola v. catenary

A blue parabola and a red catenary. Image courtesy of Ask Dr. Math.

Parabola

A parabola is a curve formed by the intersection of a circular cone and a plane parallel to the side of the cone. It can also be defined as the set of points that are the same distance from a line and a point that isn’t on the line. The graph of a quadratic equation of the form  generates a parabola.

Many physical relationships can be formally described by quadratic equations: When air resistance isn’t taken into account, thrown objects follow a parabolic path under the influence of gravity. In addition, parabolic reflectors are used in radio and optical telescopes, microphones, satellite dishes, and car headlights.

Catenary

Stan Wagon riding his square-wheeled trike on an inverted catenary path. Image courtesy of Science News. Catenary curves can look nearly identical to parabolas, but they are defined differently. A catenary is the curve formed by a string suspended from its ends. The general equation for a catenary is

where cosh is the hyperbolic cosine function.

An upside-down catenary is the ideal shape for an arch that only supports its own weight because the arch’s structural elements will experience pure compression, without bending or shear forces. In applications where an arch has to support a large amount of weight, a parabolic curve is preferable. A similar situation applies when the curve opens upwards. Simple suspension footbridges with curved decks are catenaries, while larger suspension bridges with horizontal roadways have parabolic cables.

Ask Dr. Math on parabola v. catenary