Insert 6 rational numbers between 2.2 and 2.3. 11) Find nine rational numbers between 0 and 0.1.
The Correct Answer and Explanation is:
Answers:
1. Six rational numbers between 2.2 and 2.3:
We can express 2.2 and 2.3 as decimals:
2.2 = 22/10 = 220/100
2.3 = 23/10 = 230/100
Now, insert six rational numbers between 2.2 and 2.3:
2.21, 2.22, 2.23, 2.24, 2.25, 2.26
These numbers are rational because they can be written as fractions. For example:
2.21 = 221/100, 2.22 = 111/50, and so on.
2. Nine rational numbers between 0 and 0.1:
We can express 0.1 as a fraction:
0.1 = 1/10 = 10/100
Now choose nine rational numbers between 0 and 0.1:
0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09
These are rational numbers because they have terminating decimals and can be expressed as fractions:
0.01 = 1/100, 0.02 = 1/50, and so on.
Explanation
Rational numbers are numbers that can be written as a fraction of two integers where the denominator is not zero. They include integers, finite decimals, and repeating decimals. The task of finding rational numbers between two given values requires selecting numbers that lie within the interval and are capable of being expressed as fractions.
To find six rational numbers between 2.2 and 2.3, we can convert these to a common base. Writing both as fractions with the same denominator helps to visualize the range more clearly. When expressed as 220/100 and 230/100, it becomes easier to identify rational numbers like 221/100 (2.21), 222/100 (2.22), and so on.
Similarly, to find nine rational numbers between 0 and 0.1, we look at values such as 0.01 up to 0.09. These values increase by a common difference of 0.01 and are all terminating decimals, which confirms they are rational.
These examples show that there are infinitely many rational numbers between any two given rational numbers. This is because the set of rational numbers is dense, meaning between any two rational numbers, you can always find another rational number. Therefore, there is no limit to how many rational numbers exist within any interval. Rational numbers are fundamental in math because they form the basis of fractions and decimal operations and are used in measurements, finance, and daily life situations where precision is important.
