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Q: What is a set of numbers made up of rational and irrational numbers?

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The set of real numbers.

Because irrational numbers are defined as those that are not rational. The dichotomy means that every real number belongs to one or the other of these sets, and never to both.

Let x be a rational number and y be an irrational number.Suppose their sum = z, is rational.That is x + y = zThen y = z - xThe set of rational number is closed under addition (and subtraction). Therefore, z - x is rational.Thus you have left hand side (irrational) = right hand side (rational) which is a contradiction.Therefore, by reducio ad absurdum, the supposition that z is rational is false, ie the sum of a rational and an irrational must be irrational.

From the simplest to more complicated, there are:N, the set of Natural or counting numbers [0], 1, 2, 3, ...Z, the set of all integers, ..., -3, -2, -1, 0, 1, 2, 3, ...Q, the set of all rational numbers (Q for quotient). These are numbers of the form p/q where p and q are integers and q is not 0.Irrational numbers, such as sqrt(2), e, cuberoot(171/2) and so on. These cannot be expressed in the form of a ratio and so are not rational.Transcendental numbers are a special type of irrational numbers. Most well known irrational numbers arise as solutions to [non-constant] polynomials with rational coefficients. However, there are numbers such as pi and e (extremely important to mathematics) which are transcendental.R, the set of Real numbers are made up of the rational and irratoinal numbers.C, the set of complex numbers. These have a part that is real (as above) and a part that is iaginary - related to the square root of -1.All these numbers originated in the human mind!

Negative rational numbers; Negative real numbers; Rational numbers; Real numbers. The number also belongs to the set of complex numbers, quaternions and supersets.

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