Quiz: Real Numbers and Their Properties — 24 questions

1. Which statement correctly describes the relationship between rational and irrational numbers?

A. Irrational numbers are excluded from the real-number axis.

B. Both types can be represented by points on the real-number axis.

C. Rational numbers have no fractional form, unlike irrational numbers.

D. Only rational numbers belong to the set of real numbers.

Answer: (B)

Explanation

Real numbers include both rational and irrational numbers, and every real number corresponds to a point on the real-number axis. The distinction concerns whether a number can be expressed as a quotient of integers, not whether it belongs to the real numbers.

2. Which condition guarantees that a number is rational?

A. It has a decimal expansion that never terminates or repeats.

B. It cannot be represented by a point on the real-number axis.

C. It can be written as $\frac{a}{b}$ with integers $a,b$ and $b\ne0$.

D. It must be written as a quotient of two positive integers.

Answer: (C)

Explanation

A rational number has the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b\ne0$. The integers may be negative, so positivity is not required.

3. Why is $\sqrt{2}$ classified as irrational?

A. It has a decimal expansion that terminates after finitely many places.

B. It can be expressed as a quotient only when the denominator is zero.

C. It cannot be represented by a point on the real-number axis.

D. It cannot be written as $\frac{a}{b}$ with integers and $b\ne0$.

Answer: (D)

Explanation

An irrational number cannot be represented as a quotient of integers with a nonzero denominator, and its decimal expansion is therefore neither terminating nor repeating. A nonterminating decimal alone is not enough if it repeats.

4. Which expression correctly rewrites division by a nonzero number using multiplication?

A. $\frac{a}{b}=\frac{1}{a}\cdot b$ with $b\ne0$

B. $\frac{a}{b}=a\cdot\frac{1}{b}$ with $b\ne0$

C. $\frac{a}{b}=a+\frac{1}{b}$ with $b\ne0$

D. $\frac{a}{b}=a\cdot b$ with $b\ne0$

Answer: (B)

Explanation

Division by a nonzero number is defined as multiplication by its reciprocal, so $\frac{a}{b}=a\cdot\frac{1}{b}$. Multiplying by $b$ or adding the reciprocal does not preserve the meaning of division.

5. What distinguishes an algebraic identity from an ordinary equation?

A. An identity is true for every allowed value of its variables.

B. An identity requires the variables to be positive integers.

C. An identity is true for a particular value of its variables.

D. An identity contains no variables but an equation does.

Answer: (A)

Explanation

An identity remains true for every allowed assignment of its variables. An ordinary equation may hold only for particular values, so it does not have the same universal status.

6. Which factorization correctly represents the difference of cubes?

A. $a^3-b^3=(a-b)(a^2+ab+b^2)$

B. $a^3-b^3=(a-b)(a^2-ab-b^2)$

C. $a^3-b^3=(a+b)(a^2-ab+b^2)$

D. $a^3-b^3=(a+b)(a^2+ab+b^2)$

Answer: (A)

Explanation

The difference of cubes factors into a linear difference and a three-term quadratic factor: $a^3-b^3=(a-b)(a^2+ab+b^2)$. The sum factorization with alternating signs corresponds to a different expression, namely a sum of cubes.

7. Which exponent rule applies when multiplying powers with the same base, provided the powers are defined?

A. Add the integer exponents.

B. Multiply the integer exponents.

C. Divide the integer exponents.

D. Subtract the integer exponents.

Answer: (A)

Explanation

For a product of powers with the same base, the integer exponents are added when the powers are defined. Multiplying exponents is associated with a power raised to another power, not with multiplication of like-base powers.

8. What is the defining starting point of a direct proof?

A. It starts with the conclusion and reverses each statement without justification.

B. It assumes that the conclusion is false and seeks an inconsistency.

C. It begins with the hypothesis and applies valid steps toward the conclusion.

D. It presents one example that satisfies the desired conclusion.

Answer: (C)

Explanation

A direct proof starts from the given hypothesis and proceeds through successive valid transformations until the required conclusion is reached. Assuming the conclusion is false describes proof by contradiction instead.

9. How can a universal claim such as “every object has property P” be disproved?

A. By finding one object that does have property P.

B. By showing that the property is plausible for typical objects.

C. By finding one object that does not have property P.

D. By showing that the claim holds for several selected objects.

Answer: (C)

Explanation

A single counterexample is enough to disprove a universal statement because it exhibits a case in which the claim fails. A confirming example or several confirming examples cannot establish that every object has the property.

10. Which sequence describes a proof by contradiction?

A. Assume the conclusion is false, derive a contradiction, and reject that assumption.

B. Rewrite the conclusion repeatedly until it appears identical to the hypothesis.

C. Start with an example, generalize it, and declare the conclusion proved.

D. Assume the hypothesis is false, derive a contradiction, and accept the conclusion.

Answer: (A)

Explanation

A proof by contradiction assumes the negation of the desired conclusion, derives an inconsistency with a known fact or hypothesis, and concludes that the assumption was false. It differs from direct proof because it does not begin by transforming the hypothesis toward the conclusion.

11. What does the inequality $a>b$ indicate on the real-number axis?

A. The numbers $a$ and $b$ occupy the same point

B. The difference $a-b$ is less than zero

C. The number $a$ lies to the right of $b$

D. The number $a$ lies to the left of $b$

Answer: (C)

Explanation

The inequality $a>b$ means that $a-b>0$, so $a$ is positioned to the right of $b$ on the real-number axis. The opposite positioning corresponds to $a<b$, not to $a>b$.

12. What happens to an inequality when both sides are multiplied by a negative number?

A. Its direction depends on the original variables

B. Its direction is preserved

C. Its direction is reversed

D. Its two sides become equal

Answer: (C)

Explanation

Multiplying both sides by a negative number reverses the inequality direction. Multiplication by a positive number, in contrast, preserves the direction.

13. If $a$ and $b$ are positive and $a>b$, which statement must hold for every positive integer $n$?

A. $a^n>b^n$

B. $a^n>b$

C. $a^n=b^n$

D. $a^n<b^n$

Answer: (A)

Explanation

For positive numbers, raising both sides to the same positive integer power preserves their order, so $a^n>b^n$. Equality would contradict the original strict inequality.

14. For which real number does $a^2=0$ hold?

A. Every real value of $a$

B. $a=0$

C. Any negative value of $a$

D. Any positive value of $a$

Answer: (B)

Explanation

The square of every real number is nonnegative, and it equals zero precisely when the number itself is zero. Positive and negative nonzero numbers have strictly positive squares.

15. Which condition describes membership in the closed interval $[a,b]$?

A. $a<x<b$

B. $a<x\le b$

C. $a\le x\le b$

D. $a\le x<b$

Answer: (C)

Explanation

A closed interval contains every real number between its endpoints and includes both endpoints, represented by $a\le x\le b$. Strict inequalities would exclude one or both endpoints.

16. Which statement correctly describes the open interval $(a,b)$?

A. It contains numbers satisfying $a<x\le b$

B. It contains numbers satisfying $a\le x<b$

C. It contains numbers satisfying $a<x<b$

D. It contains numbers satisfying $a\le x\le b$

Answer: (C)

Explanation

The open interval $(a,b)$ contains real numbers strictly between its endpoints, so $a<x<b$. Including either endpoint would produce a closed or half-open interval instead.

17. What is the value of $|-7|$?

A. $0$

B. It cannot be determined

C. $7$

D. $-7$

Answer: (C)

Explanation

Absolute value measures distance from zero, so $|-7|=7$. Although the original number is negative, distance is represented by a nonnegative value.

18. Which statement about the triangle inequality is correct?

A. $|a+b|=|a-b|$ for every pair

B. $|a+b|\le |a|+|b|$

C. $|a+b|\ge |a|+|b|$

D. $|a+b|=|a|+|b|$ for every pair

Answer: (B)

Explanation

The triangle inequality states that $|a+b|$ is at most $|a|+|b|$, providing an upper bound. Equality can occur in some cases, but it is not guaranteed for every pair of real numbers.

19. What is the distance between $-3$ and $5$ on the real-number axis?

A. $8$

B. $15$

C. $2$

D. $-8$

Answer: (A)

Explanation

The distance is $d(-3,5)=|-3-5|=|-8|=8$. Distance uses an absolute value, so reversing the order of the points gives the same nonnegative result.

20. Which interval is equivalent to $|x-4|<3$?

A. $x<1$ or $x>7$

B. $-3<x<3$

C. $-7<x<-1$

D. $1<x<7$

Answer: (D)

Explanation

The inequality $|x-4|<3$ means that $x$ is within 3 units of 4, giving $4-3<x<4+3$, or $1<x<7$. The outside regions correspond to a greater-than inequality rather than a less-than inequality.

21. What does $\sqrt{a}$ represent when $a\ge0$?

A. The nonnegative number whose square equals $a$

B. Any number whose square equals $a$

C. The number whose absolute value equals $a$

D. The positive number whose square equals $a$

Answer: (A)

Explanation

The square root is defined as the nonnegative solution of $x^2=a$. For positive $a$, the equation can also have a negative solution, but that solution is not denoted by $\sqrt{a}$.

22. Which statement correctly defines the nth root $\sqrt[n]{a}$ for $a\ge0$ and positive integer $n$?

A. It is the positive number whose square equals $a$

B. It is the nonnegative number whose nth power equals $a$

C. It is the number obtained by dividing $a$ by $n$

D. It is any real number whose nth power equals $a$

Answer: (B)

Explanation

The nth root is the nonnegative number that satisfies $x^n=a$. When the power is even, the equation may also have a negative solution, but the root symbol denotes the nonnegative one.

23. For nonnegative numbers, which expression correctly evaluates $\sqrt[n]{ab}$?

A. $\sqrt[n]{a-b}$

B. $\sqrt[n]{a+b}$

C. $\sqrt[n]{a}\sqrt[n]{b}$

D. $\sqrt[n]{a}+\sqrt[n]{b}$

Answer: (C)

Explanation

For nonnegative factors, the product rule gives $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$. The other expressions incorrectly treat a product of radicands as a sum or difference of roots.

24. How is $a^{m/n}$ defined when $a>0$, $m$ is an integer, and $n$ is a positive integer?

A. $\left(\sqrt[n]{a}\right)^m$

B. $\frac{a^m}{n}$

C. $\sqrt[n]{a^m}$

D. $\sqrt[m]{a^n}$

Answer: (C)

Explanation

A rational exponent is defined by $a^{m/n}=\sqrt[n]{a^m}$ under the stated conditions. The expression $\sqrt[m]{a^n}$ reverses the roles of the numerator and denominator, while dividing by $n$ is not an exponent rule.