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However, the equality of two real numbers given by an expression is known to be undecidable (specifically, real numbers defined by expressions involving the integers, the basic arithmetic operations, the logarithm and the exponential function). In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a {\displaystyle a} and b {\displaystyle b} with a > b > 0 {\displaystyle a>b>0} , a {\displaystyle a} is in a golden ratio to ...
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a / b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.
If the means of the geometric proportion are equal, and the rightmost extreme is equal to the difference between the leftmost extreme and a mean, then such a proportion is called harmonic: [6]: =: (). In this case the ratio : is called golden ratio.
Two functions and () are proportional if their ratio () is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion , e.g., a / b = x / y = ⋯ = k (for details see Ratio ).
For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.
Ratio of a circle's circumference to its diameter. 1900 to 1600 BCE [2] Tau: 6.28318 53071 79586 47692 [3] [OEIS 2] Ratio of a circle's circumference to its radius. Equal to : 1900 to 1600 BCE [2] Square root of 2, Pythagoras constant [4]
See also: Positive real numbers § Ratio scale. The ratio type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit of measurement of the same kind (Michell, 1997, 1999). Most measurement in the physical sciences and engineering is done on ratio scales.