Real and Complex Numbers

1) Which of the following statement(s) is(are) false?

• The sum of two natural numbers is always a natural number.
• The difference between two integers is always an integer.
• The product of two rational numbers is always a real number.
• The product of two irrational numbers is always an irrational number.

Detailed Answer:

• The sum of two natural numbers is always a natural number: True. Example: $ 7 + 5 = 12 $ (still a natural number).
• The difference between two integers is always an integer: True. Example: $ -3 - 7 = -10 $ (still an integer).
• The product of two rational numbers is always a real number: True. All rational numbers are real numbers, so their product is always a real number.
• The product of two irrational numbers is always an irrational number: False! Example: $ \sqrt{2} \times \sqrt{2} = 2 $ (which is rational). Sometimes the product can be rational.

Correct Answer: The fourth statement is false: The product of two irrational numbers is always an irrational number.

2) How many irrational numbers are there in the given list?

Given list: $\sqrt{3},\ 2.5, \sqrt{49},\ \frac{17}{2},\ 22, \pi, -35, \sqrt{6}, 1729, -20000$

Detailed Answer:

Irrational numbers are real numbers that cannot be expressed as a simple fraction.

• $\sqrt{3}$: Irrational (not a perfect square, non-repeating, non-terminating decimal)
• $2.5$: Rational ($5/2$)
• $\sqrt{49}$: $= 7$, rational
• $\frac{17}{2}$: Rational
• $22$: Rational
• $\pi$: Irrational (never-ending, non-repeating decimal)
• $-35$: Rational (integer)
• $\sqrt{6}$: Irrational (not a perfect square)
• $1729$: Rational (integer)
• $-20000$: Rational (integer)

So, the irrational numbers are: $\sqrt{3}, \pi, \sqrt{6}$

Count: 3

Correct Answer: 3 irrational numbers in the list.

3) How many integers are there in the given list?

Given list: $\sqrt{3},\ 2.5, \sqrt{49},\ \frac{17}{2},\ 22, \pi, -35, \sqrt{6}, 1729, -20000$

Detailed Answer:

Integers are whole numbers, positive, negative, or zero (without fractional/decimal part):

• $\sqrt{3}$: Not an integer.
• $2.5$: Not an integer.
• $\sqrt{49} = 7$: Integer.
• $\frac{17}{2}=8.5$: Not an integer.
• $22$: Integer.
• $\pi$: Not an integer.
• $-35$: Integer.
• $\sqrt{6}$: Not an integer.
• $1729$: Integer.
• $-20000$: Integer.

So, the integers are: $\sqrt{49} (=7),\ 22,\ -35,\ 1729,\ -20000$

Count: 5

Correct Answer: 5 integers in the list.

4) Which of the following statement(s) is(are) true?

• $ \sqrt{2} $ is a complex number.
• Real numbers extend rational numbers.
• None of these.

Detailed Answer:

• $ \sqrt{2} $ is a real and irrational number, which is also technically a special case of a complex number (since all real numbers are complex of the form $a + 0i$), but usually when we say “complex number,” we refer to numbers with a nonzero imaginary part.
• Real numbers extend (include) all rational numbers, so this is true (every rational is real, but not every real is rational).
• “None of these” is incorrect because one correct statement is present.

Correct Answer: Real numbers extend rational numbers.

5) Which of the following rational numbers are greater than $ \sqrt{2} $ and less than $ \sqrt{3} $?

Options:

• $ \frac{9}{5} $
• $ \frac{3}{2} $
• $ \frac{5}{3} $
• $ \frac{17}{10} $

Detailed Answer:

First, find decimal values for comparison:

• $ \sqrt{2} \approx 1.414 $
• $ \sqrt{3} \approx 1.732 $

Convert each option to decimal:

• $ \frac{9}{5} = 1.8 $
• $ \frac{3}{2} = 1.5 $
• $ \frac{5}{3} \approx 1.6667 $
• $ \frac{17}{10} = 1.7 $

Now, select values $ x $ such that $ \sqrt{2} < x < \sqrt{3} $ (i.e., $ 1.414 < x < 1.732 $):

• $ \frac{9}{5} = 1.8 $: Not in the range.
• $ \frac{3}{2} = 1.5 $: In the range.
• $ \frac{5}{3} \approx 1.6667 $: In the range.
• $ \frac{17}{10} = 1.7 $: In the range.

Correct answers: $ \frac{3}{2}, \frac{5}{3}, \frac{17}{10} $ are greater than $ \sqrt{2} $ and less than $ \sqrt{3} $.