Тест: Understanding Rational Numbers — 10 въпроса

1. What is a rational number primarily characterized by?

A. A number that can be expressed as a fraction with integer numerator and denominator, where the denominator is not zero

B. A number that cannot be expressed as a fraction

C. Any real number that can be approximated by fractions

D. A number that is only an integer or a decimal

Answer: (A)

Обяснение

A rational number is defined as any number that can be written as a fraction p/q, where p and q are integers and q is not zero. This includes integers (where q=1) and fractions. The key characteristic is the fractional notation with integer numerator and denominator, with the denominator not equal to zero.

2. What is the defining characteristic of a rational number?

A. A number that can be written as a fraction with integers in numerator and denominator, where the denominator is not zero

B. A number that can only be written as a decimal

C. A number that cannot be expressed as a fraction

D. A number that is irrational

Answer: (A)

Обяснение

A rational number can be expressed as a fraction isplaystyle rac{p}{q} with integers p and q, q eq 0. This includes integers as those fractions with denominator 1. The other options either describe irrational numbers or are incorrect statements.

3. How can a fraction be simplified to its lowest terms?

A. By dividing both numerator and denominator by their greatest common divisor (GCD)

B. By multiplying numerator and denominator by the same number

C. By subtracting the smaller from the larger

D. By adding the numerator and denominator together

Answer: (A)

Обяснение

Fractions are simplified to their lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD). This reduces the fraction to an equivalent one with no common factors other than 1.

4. Which of the following is true about the set of rational numbers isplaystyle \\mathbb{Q} \\)?

A. It includes only positive numbers

B. It is uncountably infinite

C. It contains all real numbers except irrational ones

D. It is a countable subset of real numbers

Answer: (D)

Обяснение

The set isplaystyle \\mathbb{Q} \\) of rational numbers is countable and dense within the real numbers, but it does not include irrational numbers. Hence, it is a countable subset, making the false options incorrect.

5. Which statement accurately describes the property of equivalent fractions?

A. Two fractions are equivalent if cross-multiplied, their products are equal

B. Two fractions are equivalent if they are both greater than 1

C. Two fractions are equivalent if they have the same denominator

D. Two fractions are equivalent if they have the same numerator

Answer: (A)

Обяснение

Two fractions are equivalent if the cross-multiplied products of their numerator and denominator are equal, i.e., p×s = r×q. This means they represent the same value, even if written differently.

6. How do you simplify a fraction?

A. Divide numerator and denominator by their GCD (Greatest Common Divisor)

B. Multiply numerator and denominator by two

C. Add the numerator to the denominator

D. Subtract the numerator from the denominator

Answer: (A)

Обяснение

To simplify a fraction, divide both numerator and denominator by their GCD so that the fraction is in lowest terms. The other options do not result in the simplest form of the fraction.

7. What is an example of an equivalent fraction to isplaystyle rac{2}{3} \\)?

A. isplaystyle rac{4}{6}

B. isplaystyle rac{3}{2}

C. isplaystyle rac{2}{5}

D. isplaystyle rac{5}{3}

Answer: (A)

Обяснение

isplaystyle rac{4}{6} \\ is equivalent to isplaystyle rac{2}{3} \\ because cross-multiplied: 2 imes 6 = 12 and 3 imes 4 = 12, satisfying the condition for equivalence.

8. How is addition of two rational numbers isplaystyle rac{p}{q} \\ and isplaystyle rac{r}{s} \\) performed?

A. Find a common denominator, then add the numerators

B. Multiply the numerators and denominators directly

C. Add the numerators only

D. Divide the numerator of the first by the second

Answer: (A)

Обяснение

Addition involves finding a common denominator (isplaystyle qs \\), then adding the adjusted numerators: isplaystyle rac{ps + rq}{qs} \\.

9. Why are rational numbers considered dense?

A. Because between any two rational numbers, there exists another rational number

B. Because they include both rational and irrational numbers

C. Because they are countable

D. Because they cannot be written as fractions

Answer: (A)

Обяснение

Rational numbers are dense, meaning between any two rationals, there is another rational. This property is fundamental to their structure within the real number system.

10. How can an integer be expressed as a rational number?

A. By writing it as a fraction with denominator 1

B. By writing it as a decimal with infinite repeating 9s

C. By placing it over 0

D. By doubling it

Answer: (A)

Обяснение

Any integer n can be written as isplaystyle rac{n}{1} \\ to express it as a rational number. The other options do not correctly represent integers as rational numbers.