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Proportionality (mathematics)
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For other uses, see Proportionality.
Variable y is directly proportional to the variable x.
In mathematics, two variables are proportional if there is always a constant ratio between them. The constant is called the coefficient of proportionality or proportionality constant.
If one variable is always the product of the other variable and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio y/x is constant.
If the product of the two variables is always a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product xy is constant.
To express the statement "y is directly proportional to x" mathematically, we write an equation y = cx, where c is the proportionality constant. Symbolically, this is written as y ∝ x.
To express the statement "y is inversely proportional to x" mathematically, we write an equation y = c/x. We can equivalently write "y is directly proportional to 1/x".
An equality of two ratios is called a proportion.[dubious – discuss] An example is a/c = b/d where no term is zero.
Contents
[hide]
1 Direct proportionality
1.1 Examples
1.2 Properties
2 Inverse proportionality
3 Hyperbolic coordinates
4 See also
4.1 Growth
5 Notes
6 References
Direct proportionality[edit]
Given two variables x and y, y is directly proportional to x[1] if there is a non-zero constant k such that
y = k x . {\displaystyle y=kx.}
Unicode characters
U+221D ∝ proportional to (HTML ∝ · ∝)
U+007E ~ tilde (HTML ~)
U+223C ∼ tilde operator (HTML ∼ · ∼)
U+223A ∺ GEOMETRIC PROPORTION (HTML ∺)
See also: Equals sign
The relation is often denoted, using the ∝ or ~ symbol, as
y ∝ x , {\displaystyle y\propto x,}
and the constant ratio
k = y x {\displaystyle k={\frac {y}{x}}}
is called the proportionality constant, constant of variation or constant of proportionality. This can also be viewed as a two-variable linear equation with a Y-intercept of 0.
Examples[edit]
If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
On a map drawn to scale, the distance between any two points on the map is directly proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
The force acting on a certain object due to gravity is directly proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.
Properties[edit]
Since
y = k x {\displaystyle y=kx}
is equivalent to
x = ( 1 k ) y , {\displaystyle x=\left({\frac {1}{k}}\right)y,}
it follows that if y is directly proportional to x, with (nonzero) proportionality constant k, then x is also directly proportional to y, with proportionality constant 1/k.
If y is directly proportional to x, then the graph of y as a function of x is a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.
Inverse proportionality[edit]
Inverse proportionality with a function of y = 1/x.
The concept of inverse proportionality can be contrasted with direct proportionality. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality k) is always the same.
Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion, in reciprocal proportion) if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant.[2] It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
y = ( k x ) {\displaystyle y=\left({\frac {k}{x}}\right)}
or equivalently x y = k . {\displaystyle xy=k.} Hence the constant is the product of x and y.
As an example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging.
The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the x and y values of each point on the curve equals the constant of proportionality (k). Since neither x nor y can equal zero (because k is non-zero), the graph never crosses either axis.
Hyperbolic coordinates[edit]
Main article: Hyperbolic coordinates
The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.
See also[edit]
Correlation
Eudoxus of Cnidus
Golden ratio
Proportional font
Ratio
Rule of three (mathematics)
Sample size
Similarity
Basic proportionality theorem
Growth[edit]
Linear growth
Hyperbolic growth
Notes[edit]
Jump up ^ Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource.
Jump up ^ Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource.
References[edit]
Ya.B. Zeldovich, I.M. Yaglom: Higher math for Beginners. pp. 34-35
Brian Burell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, pp. 85-101
Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), pp. 392-96 (JSTOR)
Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, pp. 140-42 (JSTOR)
Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, pp. 187-211 (JSTOR)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Proportionality_(mathematics)&oldid=815016387"
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Red sky at morning
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
This article is about the saying. For other uses, see Red sky at morning (disambiguation).
Red sky at morning, during sunrise
Red sky at night, with dust and clouds moving away to the west
Further information: Weather lore § Red sky at night
The common phrase "red sky at morning" is a line from an ancient rhyme often repeated by mariners:[1]
Red sky at night, sailors' delight.
Red sky at morning, sailors take warning[2][3][4]
The rhyme is a rule of thumb used for weather forecasting during the past two millennia. It is based on the reddish glow of the morning or evening sky, caused by haze or clouds related to storms in the region.[2][3][5] If the morning skies are red, it is because clear skies over the horizon to the east permit the sun to light the undersides of moisture-bearing clouds. The saying assumes that more such clouds are coming in from the west. Conversely, in order to see red clouds in the evening, sunlight must have a clear path from the west, so therefore the prevailing westerly wind must be bringing clear skies.
There are occasions where a storm system might rain itself out before reaching the observer (who had seen the morning red sky). For ships at sea however, the wind and rough seas from an approaching storm system could still be a problem, even without rainfall.
Because of different prevailing wind patterns around the globe, the traditional rhyme is generally not correct at lower latitudes of both hemispheres, where prevailing winds are from east to west. The rhyme is generally correct at mid-latitudes where, due to the rotation of the Earth, prevailing winds travel west to east.

Like a red morn that ever yet betokened,
Wreck to the seaman, tempest to the field,
Sorrow to the shepherds, woe unto the birds,
Gusts and foul flaws to herdmen and to herds.
from Shakespeare's Venus and Adonis
The perils are foreshadowed using the archaic word "betokened"; some versions use the archaic term "Wrack" (for the word "Wreck").
Contents
1 History
2 Other versions
3 See also
4 References
5 External links
History[edit]
In Matthew 16:2b–3, Jesus says:
When it is evening, you say, "It will be fair weather; for the sky is red."
And in the morning, "It will be stormy today, for the sky is red and threatening."
[6]
Other versions[edit]
Red sky at morning, at sea
There are other variations of the wording, including the following version using the plural word "sailors":
Red sky at night, sailors' delight.[1]
Red sky at morning, sailors' warning;
Another version uses the word "shepherds":
Red sky at night, shepherds' delight;
Red sky in the morning, shepherds' warning.
See also[edit]
Diffraction
Weather lore
"Red Skies"
References[edit]
^ Jump up to: a b Kentucky Weather, by Jerry D. Hill, 2005, p.139, web: Books-Google-ikC
^ Jump up to: a b
"GuideLines - Buoy & Marker Messages",
Paddling.net, 2009, webpage: PN-297
^ Jump up to: a b
"Weathervanes and Weather Wisdom. - Weather Station Channel",
www.usedweatherstation.com, 2009, webpage:
UsedWeath-6300
Jump up ^
The Complete Sea Kayaker's Handbook, Shelley Johnson,
2001, p.171, weblink: Books-Google-IC
Jump up ^ "Everyday Mysteries", Library of Congress, February 12, 2009, webpage: LOC-wsailor
Jump up ^ "Bible Gateway passage: Matthew 16:2-3 - New American Standard Bible". Bible Gateway. Retrieved 2017-08-15.
External links[edit]
Media related to Sunrises at Wikimedia Commons
This old saying actually has a scientific explanation National Oceanic & Atmospheric Administration (NOAA) Earth System Research Laboratory
Everyday Mysteries, Library of Congress
Authority control
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