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The ratio of width to height of standard-definition television. In mathematics, a ratio (/ ˈ r eɪ ʃ (i) oʊ /) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3).
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
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]
For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative: 20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0, while 20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0. In this sense, a quotient is the integer part of the ratio of two numbers. [9]
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 ...
The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise. [31] [32] Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified ...
Quadrant 3 (angles from 180 to 270 degrees, or π to 3π/2 radians): Tangent and cotangent functions are positive in this quadrant. Quadrant 4 (angles from 270 to 360 degrees, or 3π/2 to 2π radians): Cosine and secant functions are positive in this quadrant. Other mnemonics include: All Stations To Central [6] All Silly Tom Cats [6]
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.
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