Rational Numbers Properties Rational Numbers Properties
What is a Rational Number While studying the rational numbers first question in are mind, that is, what are rational numbers. Every repeating decimal can be written as a rational number, pq, where p, q are integers and q ≠ 0. If the rational number is represented as fraction, it is not unique. Suppose the rational numbers are used in mathematical operations such as addition, subtraction, multiplication and division. A number in the form of a ratio a /b, where a and b are integers , and b is not equal to 0, is called a rational number. The rational numbers are a subset of the real numbers, and every rational number can be expressed as a fraction or as a decimal form that either terminates or repeats. Know More About Rational Numbers Worksheets
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Conversely, every decimal expansion that either terminates or repeats represents a rational number. Rational numbers can be written in several different forms using equivalent fractions. For example, . There are an infinite number of ways to write 1, ¼ or by multiplying both the numerator and denominator by the same nonzero integer. Therefore, there are an infinite number of ways to write every rational number in terms of its equivalent fraction. The following example shows how to find the ratio of integers that represents a repeating decimal. One way to compare two rational numbers is to convert them into a decimal form. Dividing the numerator by the denominator results in the decimal equivalent. If the division has no remainder, then the decimal is called a terminating decimal. For example, ½ = 0.5, , and . Although some decimals do not terminate, they do repeat because at some point a digit, or group of digits, repeats in a regular fashion. Examples of repeating decimals are ⅓ = 0.333…,, and . A bar written over the digits or group of digits that repeat shows that the decimal is repeating: , and . Rational numbers satisfy the following properties. see also Integers; Numbers, Irrational; Numbers, Real; Numbers, Whole. Example of Rational Numbers The rational function f(x) = \frac{x^3-2x}{2(x^2-5)} is not defined at x^2=5 \Leftrightarrow x=\pm \sqrt{5}. Read More About Rational Number Worksheets
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The rational function f(x) = \frac{x^2 + 2}{x^2 + 1} is defined for all real numbers, but not for all complex numbers, since if x were the square root of -1 (i.e. the imaginary unit) or its negative, then formal evaluation would lead to division by zero: f(i) = \frac{i^2 + 2}{i^2 + 1} = \frac{-1 + 2}{-1 + 1} = \frac{1}{0}, which is undefined. The rational function f(x) = \frac{x^3-2x}{2(x^2-5)}, as x approaches infinity, is asymptotic to \frac{x}{2}. A constant function such as f(x) = π is a rational function since constants are polynomials. Note that the function itself is rational, even though f(x) is irrational for all x. The rational function f(x) = \frac{x}{x} is equal to 1 for all x except 0, where there is a removable discontinuity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function: however, the process of reduction to standard form may inadvertently result in the removing of such discontinuities unless care is taken.
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