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The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted throughout mathematics, science, technology and many computer programming languages. It is summarized as: [2] [5] Parentheses; Exponentiation; Multiplication and division; Addition and subtraction
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things. List of mathematical examples; List of algebraic surfaces; List of curves; List of complex reflection groups; List of complexity classes
An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. [122]
Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra. Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add ...
For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches: Conclude that 26 cannot be subtracted from 11; subtraction becomes a partial function. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15.
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