Question

What is a rational number with a denominator of 7 that is between 7–√ and 8–√ . Write your answer as an improper fraction
Answer
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.
Answered by GPT 3.5
To find a rational number between 7–√ and 8–√ with a denominator of 7, we can start by converting both expressions to have the same denominator.

7–√ can be written as:
(7–√)*(7+√)/(7+√) = (49-7√+7√-√^2)/(7+√)= (49-√^2)/(7+√) = (49-(-1))/(7+√) = 50/(7+√).

8–√ can be written as:
(8–√)*(7-√)/(7-√) = (56-8√ -7√+√^2)/(7-√)= (56+√^2-15√)/(7-√) = (56+(-1)-15√)/(7-√) = 55/(7-√).

So, we have the expression 50/(7+√) as the lower bound and 55/(7-√) as the upper bound. To find a rational number between these two expressions, we can take their average:

[(50/(7+√)) + (55/(7-√))]/2 = (50(7-√) + 55(7+√))/(2((7+√)(7-√))) = (350 - 50√ + 385 + 55√)/(2(49-√^2)) = (735 + 5√)/(98) = 15/2 + √5.

Therefore, the rational number with a denominator of 7 that is between 7–√ and 8–√ is 15/2 + √5, which can be written as an improper fraction as 37/2.
Answered by GPT 3.5
The relationship between rational numbers and whole numbers is that every whole number is a rational number, but not all rational numbers are whole numbers.

Every whole number can be represented as a ratio of two integers with a denominator of 1. For example, 5 can be written as 5/1, which is a rational number. So, all whole numbers are rational numbers.

However, not all rational numbers are whole numbers. Rational numbers include fractions, decimals, and integers (including whole numbers). For example, 3/4 is a rational number, but it is not a whole number. Similarly, the decimal 0.25 is a rational number, but it is not a whole number.

In summary, all whole numbers are rational numbers, but not all rational numbers are whole numbers.