How do we know if non terminating non repeating
number is a rational or irrational?
1
Here I'll sketch a reply, but not provide all of the
supporting details (these are covered in the links
sprinkling my reply). The structure of the argument !
present is as follows:
1. Define Rational Numbers
2. Define Real Numbers
3. Not all Real Numbers are Rational
4. Define Irrational Numbers
5. Decimal expansions and Rational Numbers
6. Decimal expansions and Irrational Numbers
Answers
A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result are irrational numbers. Pi is a non-terminating, non-repeating decimal.
1. In mathematics, a rational number is a number such as -3/7 that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Every integer is a rational number: for example, 5 = 5/1.
2. In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials
3. In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.
4. An irrational number is a number that cannot be expressed as a fraction for any integers and. . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.
5.Rational number 3/6 results in a terminating decimal. Example: Express 5/13 in decimal form. A rational number gives either terminating or non-terminating recurring decimal expansion. Thus, we can say that a number whose decimal expansion is terminating or non-terminating recurring is rational.
6. Decimal expansions
The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), unlike any rational number. The same is true for binary, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.