Quiz: Rational and Irrational Numbers

Multiple Choice Questions

**1. On the real number line, if point a is to the left of point b, we write:**
A) a > b
B) a < b
C) a = b
D) a ≠ b

2. What is the midpoint of the interval between 3 and 4 on the number line?
A) 3.0
B) 3.25
C) 3.5
D) 4.0

3. A rational number will have a terminating decimal representation only if its denominator is a product of:
A) threes and sevens
B) twos and fives
C) any prime numbers
D) only odd numbers

4. In Example 2, a tilted square is drawn inside a 2×2 square with area 4 square units. What is the area of the tilted square?
A) 4 square units
B) 3 square units
C) 2 square units
D) 1 square unit

5. The symbol √A represents a number a such that:
A) a + a = A
B) a × 2 = A
C) a² = A
D) a/2 = A

6. According to the resource, if N is a whole number and N is NOT a perfect square, then √N is:
A) always a terminating decimal
B) a rational number
C) an irrational number
D) undefined

7. In Example 3 (the 7×7 square), each of the four outside right triangles has legs of length 3 and 4. What is the area of the tilted square inside?
A) 49 square units
B) 24 square units
C) 25 square units
D) 12 square units

8. Newton's method for approximating square roots uses the recursion a_new = ½(a_old + N/a_old). What is this recursion used to find?
A) The cube root of N
B) The decimal expansion of the square root of N
C) The integral part of N
D) The area of a tilted square

Short-Answer Questions

9. Explain why √2 cannot be equal to a fraction (a rational number). What do we call a number with this property?

10. Using the number line procedure described in the resource, explain in your own words how the digits d₁, d₂, d₃... of a decimal expansion N.d₁d₂d₃... are found for a point a on the line.

11. The resource states that every decimal expansion represents a point on the line. Under what condition is that decimal expansion irrational?

---

Answer Key

1. B
2. C
3. B
4. C
5. C
6. C
7. C
8. B
9. √2 cannot be written as a fraction because if it could be expressed as a ratio of integers, this would lead to a contradiction (as shown through the tilted square construction of a square with area 2); such a number is called an irrational number.
10. The integer part N is found first, then the interval between N and N+1 is divided into ten equal parts; d₁ is the number of tenths parts that fit between N and a. This process is repeated by dividing each new smaller interval into ten equal parts to find each successive digit (hundredths, thousandths, etc.), getting closer and closer to the value of a.
11. The decimal expansion is irrational unless it is terminating or repeating — if it is not terminating or repeating, it represents an irrational number.