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Eight Standard >> Rational numbers | rational numbers between two rational numbers

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Rational numbers 

 

Finite and infinitely many rational numbers between two given rational numbers?

This video explain the technique, how to compare two rational numbers, how to write finite and infinitely many rational numbers between two given rational numbers.

Let's explore a more comprehensive explanation of how to find the number of integers between two consecutive integers, m and n, where m is greater than n.

When we have two consecutive integers, they are numbers that come right after each other in the number line, with a difference of 1 between them. For example, consecutive integers could be 5 and 6, 10 and 11, or -3 and -2.

Now, let's consider the situation where m is greater than n. In this case, the integers between m and n will be all the integers that come after n and before m on the number line.

To find the number of integers between m and n (excluding m and n themselves), we need to do the following:

  1. Calculate the difference between the two consecutive integers: m - n Since the numbers are consecutive, the difference will always be 1: m - n = 1.

  2. Subtract 1 from the difference to exclude the endpoints (m and n). This step is essential because we only want to count the integers between m and n, not m and n themselves.

So, the formula to find the number of integers between m and n is: Number of integers between m and n = (m - n) - 1

For example, let's find the number of integers between 7 and 3:

Number of integers between 7 and 3 = (7 - 3) - 1 = 4 - 1 = 3

The integers between 7 and 3 (excluding 7 and 3 themselves) are 4, 5, and 6.

If you want to include the endpoints (m and n) in the count, then you would use a slightly different formula:

Number of integers between m and n (inclusive) = (m - n) + 1

However, remember that when we say "between m and n," it usually implies excluding m and n themselves, so the first formula mentioned earlier is generally the one used to find the number of integers between two consecutive integers.

 

 Write finite and infinitely many rational numbers between two given rational numbers.

Writing Finite Rational Numbers:

To find a finite number of rational numbers between two given rational numbers, you can use the concept of averaging. For any two rational numbers (a/b) and (c/d), you can find a rational number between them by adding the two fractions and dividing by 2.

Suppose we want to find a finite number of rational numbers between (1/2) and (3/4):

Step 1: Add the fractions: (1/2) + (3/4) = (2/4) + (3/4) = 5/4

Step 2: Divide by 2 to find the average: (1/2) and (5/4) and (3/4)

So, we found one rational number between (1/2) and (3/4), which is (5/4).

You can keep repeating the process to find more finite rational numbers between the given rational numbers. For example:

(1/2) and (5/4) and (7/4) and (3/4)

Writing Infinitely Many Rational Numbers:

To find infinitely many rational numbers between two given rational numbers, you can use the concept of proportions. For any two rational numbers (a/b) and (c/d), you can generate an infinite sequence of rational numbers between them by using the formula:

(a/b) + [(n * (c/d - a/b)) / (N + 1)]

where 'n' ranges from 1 to infinity, and 'N' is any large positive number (e.g., 100, 1000).

For example, let's find infinitely many rational numbers between (1/3) and (2/3):

Step 1: Calculate the difference between the two fractions: (2/3) - (1/3) = 1/3

Step 2: Choose a large value for 'N' (e.g., 100): N = 100

Step 3: Generate rational numbers using the formula for 'n' ranging from 1 to 'N': n = 1: (1/3) + [(1 * (1/3)) / (100 + 1)] = (1/3) + (1/303) n = 2: (1/3) + [(2 * (1/3)) / (100 + 1)] = (1/3) + (2/303) n = 3: (1/3) + [(3 * (1/3)) / (100 + 1)] = (1/3) + (3/303) ... and so on, up to n = 100.

By choosing larger and larger values for 'N', you can generate more and more rational numbers between (1/3) and (2/3), effectively creating an infinite sequence of rational numbers between the given fractions.

Keep in mind that rational numbers are closely packed on the number line, which implies that between any two rational numbers, there exists an endless abundance of other rational numbers. This property allows us to find an infinite number of rational numbers between any two given ones.

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