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However, there is a second definition of an irrational number used in constructive mathematics, that a real number is an irrational number if it is apart from every rational number, or equivalently, if the distance | | between and every rational number is positive. This definition is stronger than the traditional definition of an irrational number.
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 to zero. Thus a non-negative number is either zero or positive.
The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.
For rational , the sequence terminates with an exact fractional expression of , and for quadratic irrational, the sequence becomes a periodic continued fraction. The ordered set ( R > 0 , > ) {\displaystyle \left(\mathbb {R} _{>0},>\right)} forms a total order but is not a well-ordered set .
Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (including 0), the result is defined as the set of integers, Z also written .
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.
Approximating an irrational number by a fraction π: 22/7 1-digit-denominator Approximating a rational number by a fraction with smaller denominator 399 / 941 3 / 7 1-digit-denominator Approximating a fraction by a fractional decimal number: 5 / 3 1.6667: 4 decimal places: Approximating a fractional decimal number by one with fewer digits 2.1784
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.