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Miscibility (/ ˌ m ɪ s ɪ ˈ b ɪ l ɪ t i /) is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration), forming a homogeneous mixture (a solution). Such substances are said to be miscible (etymologically equivalent to the common term "mixable").
The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...
Some polymer solutions also have a lower critical solution temperature (LCST) or lower bound to a temperature range of partial miscibility. As shown in the diagram, for polymer solutions the LCST is higher than the UCST, so that there is a temperature interval of complete miscibility, with partial miscibility at both higher and lower temperatures.
For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. However, operations such as function composition and matrix multiplication are associative, but not (generally) commutative.
For example, take A to be the ring of continuous functions on a compact group G. Then, not only A is an associative algebra, but it also comes with the co-multiplication Δ(f)(g, h) = f (gh) and co-unit ε(f) = f (1). [1] The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom.
For example (n/p), the Legendre symbol, considered as a function of n where p is a fixed prime number. An example of a non-multiplicative function is the arithmetic function r 2 (n) - the number of representations of n as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is ...
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]