Understanding Rational Numbers in Real Numbers

Study Notes

Math Real Numbers: Understanding Rational Numbers

Real numbers are a fundamental concept in mathematics, and they include both rational and irrational numbers. Rational numbers, in particular, are numbers that can be expressed as a ratio of two integers, such as 3/4 or 2.5. In this article, we'll explore the concept of real numbers and focus on rational numbers.

Real Numbers

Real numbers are numbers that can be plotted on the real number line. They include integers (positive and negative whole numbers), rational numbers (fractions and terminating decimals), and irrational numbers (non-terminating non-repeating decimals). Real numbers can be either positive or negative, and they can be expressed as fractions, decimals, or mixed numbers.

Rational Numbers

Rational numbers are a subset of real numbers that can be expressed as a ratio of two integers. They can be represented as fractions, terminating decimals, or as a combination of both. Some examples of rational numbers are:

3/4
1.5
0.75

Rational numbers have two important properties:

Terminating Decimals: Rational numbers have terminating decimals, meaning their decimal representation ends after a finite number of digits.
Closed Under Addition and Multiplication: Rational numbers are closed under addition and multiplication, meaning that when you add or multiply two rational numbers, the result is also a rational number.

Properties of Rational Numbers

Rational numbers have several important properties, which include:

Ordered Field: Rational numbers form an ordered field under addition and multiplication. This means that you can add and multiply rational numbers in a specific order, and their properties are well-defined.
Infinite Subsets: There are infinitely many subsets of rational numbers. For example, the set of all positive rational numbers, the set of all negative rational numbers, and the set of all rational numbers that are greater than 0 but less than 1.
Dense: Rational numbers are dense in the real number line. This means that between any two real numbers, there exists at least one rational number.

Representation of Rational Numbers

Rational numbers can be represented in various ways:

Fractions: Rational numbers can be represented as fractions, such as 3/4 or 1.5.
Terminating Decimals: Rational numbers can be represented as terminating decimals, such as 0.75 or 2.5.
Mixed Numbers: Rational numbers can also be represented as mixed numbers, such as 1 1/3 or 3 2/5.

Operations with Rational Numbers

Rational numbers can be added, subtracted, multiplied, and divided. When performing these operations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

For example, to add two rational numbers like 3/4 and 1.5, you would first convert them to the same form. 1.5 can be converted to 15/10. Then, you would add the fractions:

$$\frac{3}{4} + \frac{15}{10} = \frac{3 + 15}{4 + 10} = \frac{18}{14}$$

Conclusion

Rational numbers are a crucial part of the real number system. They are numbers that can be expressed as a ratio of two integers and have various representations, such as fractions, terminating decimals, and mixed numbers. Rational numbers have essential properties, like being an ordered field, having infinite subsets, and being dense in the real number line. They can be operated upon through addition, subtraction, multiplication, and division, following the order of operations. Understanding rational numbers is a fundamental step in mastering the real number system.