Тест: Fundamentals of Number Sets and Operations — 10 въпроса

1. Which set of numbers includes all rational and irrational numbers?

A. Real numbers

B. Natural numbers

C. Decimal numbers

D. Integers

Answer: (A)

Обяснение

Real numbers (ℝ) include all rational and irrational numbers, making them the most comprehensive set among the options. Natural numbers are only non-negative integers, integers include negatives, and decimal numbers are a subset of rational numbers with finite decimal expansion.

2. What is the hierarchy of number sets from smallest to largest?

A. ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ

B. ℝ ⊂ ℚ ⊂ ⅅ ⊂ ℤ ⊂ ℕ

C. ℚ ⊂ ℝ ⊂ ℕ ⊂ ℤ ⊂ ⅅ

D. ℤ ⊂ ℕ ⊂ ℝ ⊂ ℚ ⊂ ⅅ

Answer: (A)

Обяснение

The number sets are nested as ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ, meaning natural numbers are included in integers, finite decimals, rationals, and reals respectively. Understanding this hierarchy helps in grasping the scope of each set.

3. What does the notation ]a; b[ represent in interval notation?

A. Interval including a but excluding b

B. Interval including b but excluding a

C. Open interval excluding endpoints a and b

D. Closed interval including endpoints a and b

Answer: (C)

Обяснение

The notation ]a; b[ represents an open interval, which includes all numbers between a and b but excludes the endpoints a and b themselves.

4. Which set of numbers includes irrationals like π and √2?

A. ℕ

B. ℝ

C. ℚ

D. ⅅ

Answer: (B)

Обяснение

The real numbers ℝ include both rationals and irrationals, such as π and √2. These are numbers that cannot be expressed as fractions.

5. If |x - c| = d, what are the solutions for x?

A. x = c + d only

B. x = c - d only

C. x = c / d

D. x = c ± d

Answer: (D)

Обяснение

The equation |x - c| = d has two solutions: x = c + d and x = c - d, representing the points at a distance d from c on the number line.

6. How is the absolute value of a negative number like -5 calculated?

A. | -5 | = -5

B. | -5 | = 5

C. | -5 | = 0

D. | -5 | = -1 / 5

Answer: (B)

Обяснение

The absolute value of a number is its magnitude, so |-5| = 5, meaning the value disregards the sign.

7. What does the inequality |x - c| ≤ d imply about x?

A. x is exactly c ± d

B. x is in the interval [c - d; c + d]

C. x is less than c or greater than d

D. x is outside the interval [c - d; c + d]

Answer: (B)

Обяснение

The inequality |x - c| ≤ d means x lies within d units of c, i.e., in the interval [c - d; c + d]. It describes all points within distance d of c.

8. Which notation expresses an infinite interval extending to positive infinity?

A. [a; +∞[

B. ]a; +∞[

C. [a; b]

D. ]a; b[

Answer: (B)

Обяснение

The notation ]a; +∞[ indicates an open interval from a to infinity, meaning all numbers greater than a.

9. What is the purpose of using scientific notation for large or small numbers?

A. To make numbers easier to read and compare

B. To convert numbers into fractions

C. To change the number's value

D. To eliminate the need for scientific understanding

Answer: (A)

Обяснение

Scientific notation condenses large or tiny numbers into a manageable form, e.g., 3.0 × 10^8, making them easier to read, compare, and use in calculations.

10. Which set contains all positive and negative whole numbers, including zero?

A. ℕ

B. ℤ

C. ⅅ

D. ℚ

Answer: (B)

Обяснение

The set ℤ comprises all integers, which include positive, negative, and zero, covering all whole numbers.