Finding rational numbers between two irrational numbers involves taking their average or considering fractions formed with denominators that divide the difference between the irrational numbers into smaller equal parts.
To find rational numbers between two irrational numbers a and b, follow these steps:
Calculate the Average:
- Find the average of the two given irrational numbers a and b by adding them and dividing by 2: (a+b)/2.
Divide the Difference into Smaller Parts:
- Form fractions with denominators that divide the difference b − a into smaller equal parts.
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Example: If a and b are irrational numbers, you can still find rational numbers between them by dividing the difference b − a into smaller parts.
Consider (a+b)/2 and other fractions with denominators that divide b − a into smaller equal parts.
Other Methods
To find rational numbers between two irrational numbers:
Express the irrational numbers in decimal form.
Identify a terminating decimal between the two numbers.
For example, between √2 ≈ 1.414 and √3 ≈ 1.732, the terminating decimal 1.5 lies, which can be expressed as the rational number 3/2.