Everything about Aliquot totally explained
In
mathematics, an
aliquot part (or simply
aliquot) of an integer is any of its
integer proper divisors. For instance,
2 is an aliquot of
12. The sum of all the aliquots of an integer
n is the value s(
n) = σ(
n) -
n, where σ(
n) is the
sum of divisors function.
In
chemistry, an aliquot is usually a portion of a total amount of a solution.
The word is derived from the
Latin aliquoties, "several times".
In
pharmaceutics, aliquot refers to a method of measuring ingredients below the sensitivity of a
scale. For example, if a scale is inaccurate for samples under 120 mg, but the
prescription calls for only 40 mg of drug, an aliquot must be done. This involves adding active ingredient and a proportional amount of
diluent to make a "stock" supply. In this case, 120 mg active drug must be weighed and mixed with diluent. Once this stock supply is made, at least 120 mg of this mixture will be taken out and used (as long as this portion contains exactly 40 mg of active drug).
Music
In the construction of
string instruments often aliquot parts of the
scale length are being used to enhance the
timbre of musical instruments. In pianos the
aliquot stringing system is sometimes used. Non-Western traditional instruments with
sympathetic strings make also use of timbre enhancing based on aliqout stringing and
string resonance. The aliquot position (1/7th of the
scale length) of the
bridge on a
violin is also important for the sound of the instrument.
Land Surveying
An aliquot part, in the U.S. Public Land Survey System, is the standard subdivision of area of a section, (a.k.a. a half section, quarter section, or quarter-quarter section). One section of land is a square mile, containing 640 acres (but actual lines as run in the field can produce varying area totals).
Egyptian fraction arithmetic
The aliquot parts of the denominator of the first partition of 2/p conversions to Egyptian fractions was used 25 times in the 1650 BCE
RMP 2/n table. The method was in use as late as the
Liber Abaci, a text written by Fibonacci in 1202 AD. F. Hultsch first noticed the aliquot part use in 1895. E.M. Bruins confirmed its use in 1945. Today, the use of aliquot parts in Egyptian fraction arithmetic is know as the Hultsch-Bruins method.
Aliquant
Differently, an
aliquant part (or simply
aliquant) is an integer that isn't an exact divisor of a given quantity. For instance,
7 is an aliquant of
16. All numbers which are greater than half of a given quantity, except itself, are aliquants of the given quantity.
So, the aliquants of an integer
n include all the positive integers
m smaller than
n that are coprime to
n (for example gcd(m,n) = 1) and all positive integers
m smaller than
n that are not coprime to
n while not dividing
n (for example 1 < gcd(m,n) < m), which means that the aliquants of
n are all the
m with gcd(m,n) < m, the aliquots of
n being all the positive integers
m smaller than
n for which gcd(m,n) = m, that's all the proper divisors of
n. Consequently, the number of aliquants of
n and the number of aliquots of
n sums to
n - 1.
Further Information
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