Rational and irrational numbers are both part of the

Question

real number system
. Rational numbers include
whole numbers
.The square root of a
perfect square
is a rational number since it can be written as a whole number. Fractions written as a ratio of integers can be converted to
terminating or repeating
decimals. Irrational numbers cannot be written as
a ratio of two integers
. Irrational numbers include
the square root of imperfect squares
. The square root of imperfect squares
repeat or terminate
in decimal form. Pi is an example of
an irrational number

Answer 1

You’ve provided a good overview of rational and irrational numbers as they relate to the real number system. Here’s a more structured summary of the key points:

1. Real Numbers: The real number system consists of both rational and irrational numbers.

2. Rational Numbers:

   • Include integers, whole numbers, and fractions.
   • Can be expressed as a ratio of two integers (a/b, where a and b are integers, and b ≠ 0).
   • These can be represented as either terminating or repeating decimals.
   • The square root of a perfect square is a rational number (e.g., √4 = 2, √9 = 3).

3. Irrational Numbers:

   • Cannot be expressed as a ratio of two integers.
   • Include numbers like the square root of imperfect squares (e.g., √2, √3), which result in non-terminating and non-repeating decimals.
   • Are characterized by their decimal expansions that go on forever without repeating.
   • An example is π (pi), which is approximately 3.14159… and continues infinitely without repeating.

In summary, while rational numbers can be neatly categorized and expressed in various forms, irrational numbers introduce complexity and richness to the set of real numbers.