What Are Some Irrational Numbers What Are Some Irrational Numbers In our daily routine life we all are aware that numbers play an important role. In the routine life we use various types of numbers to make continue flow of our life. But we all are not assured that we know each and every number of the number system. In the number system of mathematics various types of numbers are defined which make the task easier to the human being. From that numbers here we are going to discuss about the irrational numbers. Rational and irrational numbers look like a fractional number. The basic difference between the rational and irrational numbers is that irrational numbers can’t be represented in the form of x / y, where y not be equal to zero. Here x is a numerator and y is a denominator. In the general term any integer or whole number can be represented into the form of rational numbers but there are some type of decimal integer values that can’t be represented into the rational form, these numbers are known as irrational numbers.
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In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational. When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. Perhaps the best-known irrational numbers are: the ratio of a circle's circumference to its diameter π, Euler's number e, the golden ratio φ, and the square root of two √2. It has been suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750–690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. However, Boyer states that "...such claims are not well substantiated and unlikely to be true. The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.” Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes,
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Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. “Eudoxus’ theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.”[13] Book 10 is dedicated to classification of irrational magnitudes.
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