Question

what are the possible kinds of decimal expansions of a rational number and an irrational number explain with examples​

Answer 1

Answer:

The real numbers which are recurring or terminating in nature are generally rational numbers.

Decimal - Expansion Of Rational Numbers

For example, consider the number 33.33333……. It is a rational number as it can be represented in the form of 100/3. It can be seen that the decimal part .333…… is the non-terminating repeating part, i .e. it is a recurring decimal number.

Also the terminating decimals such as 0.375, 0.6 etc. which satisfy the condition of being rational (0.375 = 323 ,0.6 = 35).

Consider any decimal number. For e.g. 0.567. It can be written as 567/1000 or 567103 . Similarly, the numbers 0.6689,0.032 and .45 can be written as 6689104 ,32103 and 45102 respectively in fractional form.

Thus, it can be seen that any decimal number can be represented as a fraction which has denominator in powers of 10. We know that prime factors of 10 are 2 and 5, it can be concluded that any decimal rational number can be easily represented in the form of pq, such that p and q are integers and the prime factorization of q is of the form 2x 5y, where x and y are non-negative integers.

Decimal - expansion of irrariotional numbers

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. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two; in fact all square roots of natural numbers, other than of perfect squares, are irrational.

Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.

Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.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.HOPE IT HELPS YOU

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