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[4] The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers.
For promotion from Secondary level (Class IX-X) to Senior Secondary level (Class XI-XII), a student must obtain, for all subjects (or best five of six subjects are taken), 33% overall, without any minimum theory mark requirement. Originally, the passing criteria were set such that a student had to get 33% in both the theory and practical ...
The exam usually starts at 10:30 AM IST and the students are usually given 2 to 3 hours time for writing the exam. The question papers are distributed by the overseeing board of education, and their contents are guarded closely until the exam begins. The examinations may include multiple sets of question papers as well.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an ordered field, in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property: Every nonempty subset of ...
Including 0, the set has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to ), and its units (the finite numbers, excluding ) correspond to the positive real numbers.
However, the Term-I examination was criticised by many for having wrong answer keys, tough question papers and wrong or controversial questions, with a question being dropped in Sociology exam of class 12 and a paragraph in the English Language and Literature exam for class 10 by CBSE following which CBSE dropped the experts who set the ...
[3] [4] Similarly, the question of whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable. [3] [4]