Chapter 0, Problem 1CCE

Explain the relationships and differences among real numbers, rational numbers, and irrational numbers.

To determine

The relationships and differences among real numbers, rational numbers, and irrational numbers.

Answer to Problem 1CCE

Solution:

Relationship among real numbers, rational numbers, and irrational numbers is that the sets of rational numbers and irrational numbers are subsets of set of real numbers.

Difference between rational and irrational number is that the rational number can be written as $\frac{a}{b}$, $b\ne 0$ ( ratio of two integers) but irrational numbers cannot be written as a fraction of two integers.

Explanation of Solution

Given information:

Set of real numbers, rational numbers, and irrational numbers

Explanation:

Rational numbers are the numbers which can be written as $\frac{a}{b}$, where, $b\ne 0$ and as a finite or infinitely repeating decimal, such as $-\frac{3}{2}=-1.5,\frac{14}{3}=\mathrm{4.6666...}$ etc.

Irrational numbers are the numbers which cannot be written as a ratio of two integers and have non-repeating and non-terminating decimals. For example, $-\sqrt{5}$ or $\pi$.

The rational as well as irrational numbers can be converted to decimals and can be shown on the number line as shown below,

So, rational and irrational numbers are subsets of real numbers.