Note: All of the above are sub-groups of Real Numbers.
C. Properties of Real Numbers
Completeness: There are no gaps on the number line. Every point is a Real number.
Density: Between any two real numbers, there is another real number.
💡 Worked Examples
1. Classify √2: It is Irrational and Real. 2. Classify -5: It is an Integer, Rational, and Real. 3. Classify 0: It is a Whole number, Integer, Rational, and Real.
✍️ Practice Exercises (20 Questions)
1. Are all rational numbers real numbers?
Yes. Rational numbers are a subset of Real numbers.
2. Are all real numbers rational?
No. Some real numbers are irrational (like √2 or π).
3. What set results from the union of Rationals and Irrationals?
Real Numbers (ℝ).
4. Is √(-1) a real number?
No. There is no real number that, when squared, equals -1. This is an "Imaginary Number."
5. Is π a real number?
Yes. It is an irrational real number.
6. True or False: 0 is a real number.
True.
7. Which symbol represents Real Numbers?
ℝ.
8. Can a number be both rational and irrational?
No. These two sets are disjoint (no overlap).
9. What property says the number line has no holes?
Completeness Property.
10. Is √25 a real number?
Yes. (√25 = 5).
11. Is every integer a real number?
Yes.
12. True or False: Real numbers are closed under addition.
True. Adding two real numbers always results in a real number.
13. Which is NOT real: 5, -2, √3, √(-4)?
√(-4). Square roots of negative numbers are not real.
14. Is the product of a rational and an irrational number always irrational?
Not always. 0 (rational) × √2 (irrational) = 0 (rational).
15. Is 0.12345... (no pattern) real?
Yes. It is an irrational real number.
16. What is the square of any non-zero real number?
Positive.
17. Can we find a real number between 1.0001 and 1.0002?
Yes (Density property).
18. True or False: 22/7 is a real number.
True (It is rational).
19. Is |-10| a real number?
Yes (10).
20. What is the most complete set of numbers on the number line?