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Questions from the Image:

23. A student claimed that $\sqrt{x}$ is a natural number greater than 1 and $(x+1)$ is a natural number from 1. Actually, none is correct, even though if the variables are used. How this student error?

24. Give a real number that satisfies each condition.

(a) An integer between $\frac{3}{5}$ and $\frac{5}{6}$.
(b) A whole number that is not a natural number.
(c) A natural number that is not a whole number.
(d) An irrational number between $\sqrt{2}$ and $\sqrt{3}$.

Solutions:

23. The error in the student's claim is in the interpretation of the square root function and the assumptions about the natural numbers:

• $\sqrt{x}$ is only a natural number if $x$ is a perfect square. If $x$ is not a perfect square, then $\sqrt{x}$ is irrational and not a natural number.
• The expression $(x + 1)$ being a natural number doesn't automatically make $x$ a natural number.

24. Let's find real numbers that satisfy each condition:

(a) An integer between $\frac{3}{5}$ and $\frac{5}{6}$:
The fractions $\frac{3}{5} \approx 0.6$ and $\frac{5}{6} \approx 0.833$.
Since integers are whole numbers, and there are no whole numbers between 0.6 and 0.833, this question might have an error unless it was meant to ask for a number, in which case there are none.

(b) A whole number that is not a natural number:
The only whole number that is not a natural number is 0.

(c) A natural number that is not a whole number:
This condition doesn't make sense because all natural numbers are whole numbers. There is no natural number that isn't also a whole number.

(d) An irrational number between $\sqrt{2}$ and $\sqrt{3}$:
$\sqrt{2} \approx 1.414$ and $\sqrt{3} \approx 1.732$. Any irrational number between these values, such as 1.5, 1.6, etc., would work. A specific example could be $\sqrt{2.5}$.

Do you want more details or have any other questions?

Here are some relative questions you might find interesting:

1. What defines a number as irrational, and how do you identify them?
2. How do you prove that $\sqrt{x}$ is irrational for non-perfect squares?
3. What are the differences between whole numbers, natural numbers, and integers?
4. Can you explain why zero is considered a whole number but not a natural number?
5. What are the properties of irrational numbers that distinguish them from rational numbers?

Tip: When dealing with roots and powers, always check if the base is a perfect square or cube, as this helps in identifying whether the result is rational or irrational.