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In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
Positive numbers: Real numbers that are greater than zero. Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal ...
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
Examples are e r and π r, which are transcendental for all nonzero rational r. Because the algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 π + 2, π + √ 2 and e √ 3 are irrational (and even transcendental).
Since the real and complex numbers both form a field and contain the positive reals, they also contain the reciprocals of the magnitudes of all non-zero numbers. This means that any non-zero number may be multiplied with the reciprocal of its magnitude, that is, divided by its magnitude. It is immediate that the quotient of any non-zero real ...
nonzero real numbers with multiplication N Z 2 – abelian R: 1 R + positive real numbers with multiplication N 0 0 abelian R: 1 S 1 = U(1) the circle group: complex numbers of absolute value 1 with multiplication; Y 0 Z: R: abelian, isomorphic to SO(2), Spin(2), and R/Z: R: 1 Aff(1) invertible affine transformations from R to R. N Z 2 –
Example: Let a and b be nonzero real numbers. Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.
The product of positive real numbers = converges to a nonzero real number if and only if the sum = converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products.