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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Euclid defines a ratio as between two quantities of the same type, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other.
In this example, the ratio of adjacent terms in the blue sequence converges to L=1/2. We choose r = (L+1)/2 = 3/4. Then the blue sequence is dominated by the red sequence r k for all n ≥ 2. The red sequence converges, so the blue sequence does as well. Below is a proof of the validity of the generalized ratio test.
For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, [1] [2] when the rectangle is oriented as a "landscape". The aspect ratio is most often expressed as two integer numbers separated by a colon (x:y), less commonly as a simple or decimal fraction. The values x and y do ...
In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So: n 7 = n × n × n × n × n × n × n.. Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth power, or the cube of a number by its fourth power.
The digit ratio is the ratio taken of the lengths of different digits or fingers on a hand. The most commonly studied digit ratio is that of the 2nd ( index finger ) and 4th ( ring finger ), also referred to as the 2D:4D ratio, measured on the palm side.
Age dependency ratio as of 2017 [1] The dependency ratio is an age-population ratio of those typically not in the labor force (the dependent part ages 0 to 14 and 65+) and those typically in the labor force (the productive part ages 15 to 64). It is used to measure the pressure on the productive population.
The term spreadsheet may also refer to one such electronic document. [5] [6] [7] Spreadsheet users can adjust any stored value and observe the effects on calculated values. This makes the spreadsheet useful for "what-if" analysis since many cases can be rapidly investigated without manual recalculation.
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