What is the reciprocal of the reciprocal of any positive or negative number? Is the repeating decimal 2.72 rational or irrational?
The correct answer and explanation is:
The reciprocal of the reciprocal of any positive or negative number is the original number itself.
To understand why, consider a number xx (where x≠0x \neq 0). The reciprocal of xx is 1x\frac{1}{x}. Taking the reciprocal again means finding the reciprocal of 1x\frac{1}{x}, which is 11x\frac{1}{\frac{1}{x}}. Simplifying this expression results in xx. This works for any nonzero number, whether positive or negative. The key point is that taking the reciprocal is an operation that reverses itself when done twice.
Regarding the repeating decimal 2.72, whether it is rational or irrational depends on whether the decimal is actually repeating or terminating. If “2.72” means the decimal terminates exactly at two decimal places, then it is a rational number. Rational numbers are numbers that can be expressed as a fraction of two integers, such as 272100\frac{272}{100}, which simplifies to 6825\frac{68}{25}. Since 2.72 can be written as this fraction, it is rational.
However, if “2.72” is meant to indicate a repeating decimal, for example 2.727272…2.727272… with the digits “72” repeating infinitely, then it is still rational. This is because all repeating decimals can be expressed as fractions of two integers. For example, 2.727272…2.727272… equals 27099\frac{270}{99} or simplified further. Any decimal with a repeating pattern represents a rational number.
In contrast, irrational numbers are decimals that neither terminate nor repeat. Examples include numbers like π\pi, ee, or the square root of 2, which have non-repeating, infinite decimal expansions.
In conclusion, the reciprocal of the reciprocal of any nonzero number returns the original number, and a repeating decimal like 2.72 (if repeating) or the terminating decimal 2.72 is rational because it can be expressed as a fraction of integers.