Introduction to Rational Numbers

Rational numbers are a fundamental concept in mathematics, forming an essential subset of the real numbers. They are defined as numbers that can be expressed as a fraction a/b, where a and b are integers and b≠0. This article explores the process of finding a series of 5 rational numbers between 0 and 1.5 using the principles of an arithmetic progression (AP).

Understanding Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between any two successive members is a constant known as the common difference, d. The general form of an AP can be represented as:

Tn is the nth term of the sequence. T1 is the first term. The common difference, d, is the difference between any two successive terms.

The nth term of an AP can be defined as:

Tn a (n - 1)d

Solving the Problem: 5 Rational Numbers Between 0 and 1.5

To find 5 rational numbers between 0 and 1.5, we will use the properties of an arithmetic progression. The first term, T1, is given as 0, and we need to find 5 terms, so a1 0. The seventh term, T7, is given as 1.5. The general formula for the nth term of an AP allows us to find the common difference d.

Settings the Equation for the 7th Term

We know that the seventh term T7 can be represented as:

T7 a1 6d

Substituting the known values:

1.5 0 6d

Solving for d gives:

6d 1.5

d 0.25

Calculating the 5 Rational Numbers

Now that we have the common difference d 0.25, we can find the next 5 terms in the arithmetic progression:

T2 0 1*0.25 0.25 T3 0 2*0.25 0.50 T4 0 3*0.25 0.75 T5 0 4*0.25 1.00 T6 0 5*0.25 1.25

These 5 rational numbers between 0 and 1.5 can be expressed as fractions:

0.25 1/4 0.50 1/2 0.75 3/4 1.00 1 1.25 5/4

Conclusion

This exercise in finding rational numbers within a specified range not only demonstrates the application of arithmetic progressions but also highlights the versatility and importance of rational numbers in mathematics. Understanding the properties of APs and rational numbers can be valuable in various mathematical contexts, including problem-solving, data analysis, and real-world applications.