find five rational number between 1 and 2
Find Five Rational Numbers Between 1 and 2
Understanding Rational Numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In other words, a number q is rational if it can be written in the form \frac{a}{b}, where a and b are integers and b \neq 0. Rational numbers can be positive, negative, or zero, and they include fractions as well as whole numbers.
Steps to Find Rational Numbers Between 1 and 2
To find rational numbers between two given numbers, 1 and 2, we can think of any numbers that satisfy the inequality 1 < \frac{a}{b} < 2. Here’s how we can discover some examples:
1. Express 1 and 2 with a Common Denominator
To simplify selecting numbers between 1 and 2, express both with a common denominator. A simple way to do this is:
- Express 1 as \frac{10}{10}.
- Express 2 as \frac{20}{10}.
Using this common denominator allows us to easily see numbers between them.
2. Identify Rational Numbers
Consider numbers where the denominator remains the same (10 in this example), and choose numerators n such that 10 < n < 20. This gives fractions such as:
- \frac{11}{10}
- \frac{12}{10}
- \frac{13}{10}
- \frac{14}{10}
- \frac{15}{10}
Of course, all of these are rational numbers greater than 1 and less than 2.
3. Simplifying Some of These Fractions
Some of the fractions can be simplified further:
- \frac{12}{10} simplifies to \frac{6}{5}.
- \frac{15}{10} simplifies to \frac{3}{2}.
This simplification does not change their positions between 1 and 2.
Examples of Rational Numbers and Their Characteristics
To better understand the concept, let’s explore these rational numbers in more detail:
Fraction Representation
| Rational Number | Fraction Form | Simplified Form | Decimal Form |
|---|---|---|---|
| \frac{11}{10} | 1.1 | \frac{11}{10} | 1.1 |
| \frac{12}{10} | 1.2 | \frac{6}{5} | 1.2 |
| \frac{13}{10} | 1.3 | \frac{13}{10} | 1.3 |
| \frac{14}{10} | 1.4 | \frac{7}{5} | 1.4 |
| \frac{15}{10} | 1.5 | \frac{3}{2} | 1.5 |
Visualizing Rational Numbers
To further understand, consider a number line where 1 and 2 are marked. Finding and marking numbers like 1.1, 1.2, 1.3, 1.4, and 1.5 helps in visualizing these fractions between 1 and 2.
Conclusion
We can conclude that many more rational numbers exist between 1 and 2. The above examples are merely a few that demonstrate the concept. Rational numbers within any range are infinitely dense, meaning there’s always another rational number between any two given rational numbers. As you get more familiar with fractions and their decimal representations, finding intermediate rational numbers becomes more intuitive.
I hope this explanation helps in understanding how to find rational numbers between any two integers or fractions.
Feel free to practice by identifying additional numbers between other pairs of numbers to reinforce your understanding of rational numbers! If you have any further questions or need more examples, just let me know.